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22607 Hamburg, Germany +and +Faculty of Physics, University of Warsaw, +Pasteura 5, 02-093 Warszawa, Poland +Jürgen Reuter† +Deutsches Elektronen-Synchrotron DESY, +Notkestr. +85, 22607 Hamburg, Germany +Aleksander Filip Żarnecki‡ +Faculty of Physics, University of Warsaw, +Pasteura 5, 02-093 Warszawa, Poland +(Dated: January 9, 2023) +Neutrinos are the most elusive particles known. Heavier sterile neutrinos mixing with the standard +neutrinos might solve the mystery of the baryon asymmetry of the universe. In this letter, we show +that among all future energy frontier accelerators, muon colliders will provide the farthest search +reach for such neutrinos for mass ranges above the Z pole into the multi-TeV regime. We compare +the performance of muon with electron colliders of the same machine energy and briefly discuss the +complementarity in flavor space between the two types of accelerators. +PACS numbers: 13.35.Hb, 13.66.Lm. 14.60.Pq, 14.60.St +Introduction +Massive neutrinos are considered the first established building blocks of physics beyond the Standard +Model (SM) of particle physics. Their tiny masses are believed to originate from seesaw-like mixing with heavier +sterile neutrinos whose masses could be all the way from the electroweak (EW) to the unification scale. While long- +distance neutrino oscillation experiments like DUNE or Hyper-Kamiokande will shed more light on the mass hierarchy +and the mixing parameters, heavier neutrinos can be directly searched for at hadron colliders such as the LHC and +future lepton colliders [1–17]. For collider searches, three different regimes can be considered: light neutrinos which +are long-lived and result in displaced vertices or decay outside the detectors, intermediate-mass neutrinos that decay +promptly and are dominantly produced in Z (and W or Higgs) decays, and heavy neutrinos with masses Mν ≳ MH. +In this paper, building upon an analysis framework similar to earlier studies for searches at linear e+e− machines, we +focus on the third case and show that the most sensitive searches for direct heavy neutrino production are possible at +high-energy muon colliders. Lepton colliders are, in general, sensitive to much smaller mixing parameters and hence +to much higher scales of UV completions. In this paper, we will consider a muon collider setup with energies of 3 and +10 TeV, and integrated luminosities of 1 and 10 ab−1, respectively [18–20]. +Model setup and simulation framework +In this letter, we consider the Phenomenological Type I Seesaw Mech- +anism [21, 22], implemented within the HeavyN model with Dirac neutrinos [5, 23], i.e. +we assume it just as a +representative model candidate without any prejudice (our findings are quite generic, though specific model setups +like artificial flavor mixings could of course lead to singular cases; Refs. [24, 25] provide an example where such heavy +neutrinos appear even at a multi-TeV scale UV completion). This effective extension of the SM introduces three +flavors of right-handed neutrinos (denoted as Nk) that are singlets under the SM gauge groups. The Lagrangian of +the model reads: +L = LSM + LN + LW Nℓ + LZNν + LHNν +(1) +where LN is a sum of kinetic and mass terms for heavy neutrinos (in 4-spinor notation, which combines terms with +spinors of dotted and undotted indices): +LN = ¯Nki/∂Nk − mNk ¯NkNk +for k = 1, 2, 3, +(2) +LW Nℓ yields neutrino interactions with the W boson: +LW Nℓ = − g +√ +2W + +µ +3 +� +k=1 +τ +� +l=e +¯NkV ∗ +lkγµPLℓ− + h.c., +(3) +arXiv:2301.02602v1 [hep-ph] 6 Jan 2023 + +2 +0 +2000 +4000 +6000 +8000 +10000 + [GeV] +qql +m +1 +− +10 +1 +10 +2 +10 +3 +10 +4 +10 +5 +10 +6 +10 + Events +ee bg +sig_3000 +FIG. 1: qqℓ mass distribution for a reference scenario assuming the existence of one Dirac neutrino with a mass of +3 TeV, at a 10 TeV muon collider. The black solid line stand for the µ+µ− background and the thick green one for +the signal scenario. +LZNν interactions with the Z boson: +LZNν = − +g +2 cos θW +Zµ +3 +� +k=1 +τ +� +l=e +¯NkV ∗ +lkγµPLνl + h.c., +(4) +and LHNν interactions with the Higgs boson: +LHNν = − gmN +2MW +h +3 +� +k=1 +τ +� +l=e +¯NkV ∗ +lkPLνl + h.c. +(5) +The UFO library of the model contains 12 free parameters in addition to the SM parameters, which are three masses +of the heavy neutrinos: mNk and nine real (no CP violation expected) mixing parameters Vlk, where l = e, µ, τ and +k = N1, N2, N3. For the purpose of this analysis, we considered a scenario with only one heavy Dirac neutrino N1 ≡ N +with a mass below O(10 TeV) and equal couplings to all SM leptons (|VeN1|2 = |VµN1|2 = |VτN1|2 ≡ V 2 +lN). For sample +generation, the mixing parameter V 2 +lN has been set to 0.0003. Other values for the mixing parameters in the analysis +below were accessed via rescaling with the corresponding cross section. Although there are many different possible +signatures of such particles at future colliders, for center-of-mass energies above the Z pole, the t-channel W exchange +resulting in the production of a light-heavy neutrino pair (µ+µ− → N ν) is dominant [11] and the production cross +section is of the order of 1-10 fb for masses of the neutrinos up to the collision energy. For our choice of the parameter +space, the heavy neutrino has a microscopic lifetime (cτ ≪ 1 nm) and no displaced vertices are expected. Among +the possible decay channels of such particles, the signature of two jets and a lepton (N → qqℓ) is the most promising +because it allows for direct reconstruction of the mass of the heavy state. +In the first step, we generated event samples with Whizard 3.0.2 [26–28] at leading order (LO) in the SM coupling +constants (though recently higher-order corrections have become available in an automated manner [29] and simulated +detector response with Delphes 3.5.0 [30] using built-in Muon Collider detector cards. At the generator level, a +set of cuts was applied to remove possible singularities. +They included 10-GeV cuts on the energy of produced +jets and leptons, the invariant mass of quark and lepton pairs, and the four-momentum transfer from the incoming +muons. Furthermore, it was required that at least one lepton could be detected in the central detector (we assumed +5◦ < θ < 175◦, where θ is the lepton polar angle). For the detector simulation, the VLC clustering algorithm in the + +3 +1 +− +0.5 +− +0 +0.5 +1 +BDT response +1 +10 +2 +10 +3 +10 +4 +10 +5 +10 +Events +ee +sig +FIG. 2: Distribution of the BDT response for the reference scenario (Dirac neutrino, mN = 3 TeV) with electrons in +the final state at a 10 TeV muon collider. The red line denotes the background, and the green line the signal. +exclusive two-jet mode (R = 1.5, β = 1, γ = 1 – see [31]) was applied. Since the considered Delphes model cannot +generate fake lepton tracks, only 4- and 6-fermion background processes with at least one lepton in the final state +(qqℓν, qqℓℓ, ℓℓℓℓ, qqqqℓν, qqqqℓℓ, qqℓνℓν, qqℓννν) were generated. The most important channels in terms of cross +section (O(1 ab) at both energy stages) were qqℓν and ℓℓℓℓ; the latter could be, however, easily reduced by lepton +identification. Background channels induced by photons from collinear initial-state splittings were neglected, as their +impact on the final results was found to be marginal. +Analysis procedure +In the next step, a set of selection cuts was applied to reject events incompatible with the +expected topology of two jets and one lepton. To exclude events with significant contributions of forward deposits +assigned to the beam jets, an upper limit of 20 GeV was applied on the transverse momentum of objects not included +in the reconstructed final state. In Figure 1, we show a distribution of the invariant mass of two jets and a lepton for +a reference scenario (a 3 TeV neutrino at a 10 TeV muon collider). A peak corresponding to the mass of the heavy +neutrino is clearly visible. The left tail is due to events with leptonic τ decays, for which the escaping neutrinos +reduce the reconstructed invariant mass. On the right-hand side, the tail is an effect of finite detector resolution. +Subsequently, we applied the Boosted Decision Tree (BDT) method implemented in the TMVA package [32] to +discriminate between signal and background events. A set of eight variables describing event kinematics was chosen +to optimize the classification: +• mqqℓ – invariant mass of the dijet-lepton system, +• α – angle between the dijet system and the lepton, +• αqq – angle between the two jets, +• Eℓ – lepton energy, +• Eqqℓ – energy of the dijet-lepton system, +• pT +ℓ – lepton transverse momentum, +• pT +qq – dijet transverse momentum, +• pT +qqℓ – transverse momentum of the dijet-lepton system. +Due to the considerable difference between the numbers of expected background events, the algorithm was implemented +separately for events with reconstructed electrons and muons in the final state. The BDT response for the reference +scenario is shown in Figure 2. The two distributions are partially separated and thus, they were used to extract +expected limits on the coupling parameter V 2 +lN within the CLs method, implemented in the RooStats package [33]. + +4 +3 +10 +4 +10 + [GeV] +N +m +7 +− +10 +6 +− +10 +5 +− +10 +4 +− +10 +3 +− +10 +2 +− +10 +1 +− +10 +2 +lN +lim. V +CMS +HL-LHC +HE-LHC +FCC-hh +ILC 1 TeV +CLIC 3 TeV +Muon Collider 10 TeV +Muon Collider 3 TeV +FIG. 3: Limits on the coupling V 2 +ℓN for different Muon Collider setups (solid lines: 3 TeV – turquoise, 10 TeV – +orange). Dashed lines indicate limits from current and future hadron [1, 5] machines, dashed-dotted for e+e− +colliders [16]. See text for details. +This allowed for combining the electron and muon channels. The impact of systematic uncertainties has been neglected +at this stage, as they are not expected to significantly affect the final conclusions. +Results +In Figure 3, limits on the coupling V 2 +lN for the two Muon Collider setups are presented and compared +with the current limits coming from the CMS experiment (Majorana neutrinos, Fig. 2 in [1]), as well as with the +results obtained for future hadron colliders (Dirac neutrinos, Fig. 25b in [5]) and e+e− colliders (Dirac neutrinos, +Fig. 12 in [16]). It should be noted that in the hadron collider analyses, heavy neutrino decays into taus were not +considered, and thus their sensitivity is enhanced relative to the results presented for the lepton colliders, where the +tau-channel decays are included. As shown in Figure 3, limits expected from the e+e− colliders, ILC running at 1 TeV +and CLIC running at 3 TeV, are more stringent for masses of the heavy neutrinos up to about 700 GeV. The fact that +the results for CLIC and a Muon Collider operating at the same energy of 3 TeV do not coincide may be surprising. +However, several effects must be taken into account for a proper comparison: the most important factors are different +integrated luminosities and beam polarizations. In addition, the beam spectra and the beam-induced background +channels cannot be neglected for e+e− colliders, while their impact is significantly reduced for µ+µ− machines due +to the larger mass of the muon. +It was verified that, for the same generation setup (no beam polarization, no +beam spectrum, no beam-induced background channels, but different initial-state particles and detector designs), the +expected CLIC limits are consistent with the Muon Collider ones, giving the analysis precision. The discrepancy +visible in Figure 3 could then be explained as follows: at lower neutrino masses, the expected limits from CLIC are +more stringent due to the higher integrated luminosity and electron beam polarization, and at higher masses, they +are worse because of the impact of the luminosity spectra and beam-induced backgrounds. +In the analysis, we assumed that all the mixing parameters VlN have the same value. It is important to note that +this approach is not unique. Using data from both electron-positron and muon colliders, one could potentially loosen +this assumption and constrain the parameters VeN and VµN separately, by either excluding taus from the physical +model or implementing a proper tau tagging procedure. Such a method would give limits not only on the couplings +themselves but also on their product in the framework where couplings are treated independently, possibly hinting at +a flavor-universality violation. The details are, however, beyond the scope of this letter. +Conclusions +Extensions of the Standard Model introducing heavy neutrinos offer interesting solutions to several +of its open questions, e.g. the baryon asymmetry of the universe, dark matter and flavor. If such particles are at mass +scales well above a GeV, they can be efficiently searched for at future lepton colliders. Due to the highest achievable +energies and the clean experimental environments, muon colliders would provide the furthest discovery reach for + +5 +this kind of particles and models, vastly surpassing high-energy hadron colliders. By employing the synergy of both +different types of lepton machines, electron-positron and muon colliders, different paths in the flavor parameter space +of the models could be pursued. +Acknowledgments +The work was partially supported by the National Science Centre (Poland) under the OPUS research project no. +2021/43/B/ST2/01778. KM and JRR acknowledge the support by the Deutsche Forschungsgemeinschaft (DFG, Ger- +man Research Association) under Germany’s Excellence Strategy-EXC 2121 "Quantum Universe"-3908333. This work +has also been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 491245950. +∗ krzysztof.mekala@desy.de +† juergen.reuter@desy.de +‡ filip.zarnecki@fuw.edu.pl +[1] A. Sirunyan et al. (CMS), Search for heavy neutral leptons in events with three charged leptons in proton-proton collisions +at √s = 13 TeV, Phys. Rev. Lett. 120, 221801 (2018), arXiv:1802.02965 [hep-ex]. +[2] G. Aad et al. (ATLAS), Search for heavy neutral leptons in decays of W bosons produced in 13 TeV pp collisions using +prompt and displaced signatures with the ATLAS detector, JHEP 10, 265, arXiv:1905.09787 [hep-ex]. +[3] R. Aaij et al. (LHCb), Search for heavy neutral leptons in W + → µ+µ±jet decays, Eur. Phys. J. C 81, 248 (2021), +arXiv:2011.05263 [hep-ex]. +[4] A. Tumasyan et al. 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Reuter, Parallel Adaptive Monte Carlo Integration with the Event Generator WHIZARD, Eur. +Phys. J. C 79, 344 (2019), arXiv:1811.09711 [hep-ph]. +[29] P. M. Bredt, W. Kilian, J. Reuter, and P. Stienemeier, NLO electroweak corrections to multi-boson processes at a muon +collider, JHEP 12, 138, arXiv:2208.09438 [hep-ph]. +[30] J. de Favereau et al. (DELPHES 3), DELPHES 3, A modular framework for fast simulation of a generic collider experiment, +JHEP 02, 057, arXiv:1307.6346 [hep-ex]. +[31] M. Boronat, J. Fuster, I. Garcia, P. Roloff, R. Simoniello, and M. Vos, Jet reconstruction at high-energy electron–positron +colliders, Eur. Phys. J. C 78, 144 (2018), arXiv:1607.05039 [hep-ex]. +[32] A. Hocker et al., TMVA - Toolkit for Multivariate Data Analysis, arXiv:physics/0703039 (2007), arXiv:physics/0703039. +[33] L. Moneta et al., The RooStats Project, PoS ACAT2010, 057 (2010), arXiv:1009.1003 [physics.data-an]. + diff --git a/1NE0T4oBgHgl3EQfuQFR/content/tmp_files/load_file.txt b/1NE0T4oBgHgl3EQfuQFR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6862627d1d58b45ca5e11345fd5422fbbe1a574e --- /dev/null +++ b/1NE0T4oBgHgl3EQfuQFR/content/tmp_files/load_file.txt @@ -0,0 +1,344 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf,len=343 +page_content='Optimal search reach for heavy neutral leptons at a muon collider Krzysztof Mękała∗ Deutsches Elektronen-Synchrotron DESY, Notkestr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' 85, 22607 Hamburg, Germany and Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warszawa, Poland Jürgen Reuter† Deutsches Elektronen-Synchrotron DESY, Notkestr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' 85, 22607 Hamburg, Germany Aleksander Filip Żarnecki‡ Faculty of Physics, University of Warsaw, Pasteura 5, 02-093 Warszawa, Poland (Dated: January 9, 2023) Neutrinos are the most elusive particles known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' Heavier sterile neutrinos mixing with the standard neutrinos might solve the mystery of the baryon asymmetry of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' In this letter, we show that among all future energy frontier accelerators, muon colliders will provide the farthest search reach for such neutrinos for mass ranges above the Z pole into the multi-TeV regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' We compare the performance of muon with electron colliders of the same machine energy and briefly discuss the complementarity in flavor space between the two types of accelerators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' PACS numbers: 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content='Hb, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content='66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content='Lm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content='Pq, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content='60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content='St Introduction Massive neutrinos are considered the first established building blocks of physics beyond the Standard Model (SM) of particle physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' Their tiny masses are believed to originate from seesaw-like mixing with heavier sterile neutrinos whose masses could be all the way from the electroweak (EW) to the unification scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' While long- distance neutrino oscillation experiments like DUNE or Hyper-Kamiokande will shed more light on the mass hierarchy and the mixing parameters, heavier neutrinos can be directly searched for at hadron colliders such as the LHC and future lepton colliders [1–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' For collider searches, three different regimes can be considered: light neutrinos which are long-lived and result in displaced vertices or decay outside the detectors, intermediate-mass neutrinos that decay promptly and are dominantly produced in Z (and W or Higgs) decays, and heavy neutrinos with masses Mν ≳ MH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' In this paper, building upon an analysis framework similar to earlier studies for searches at linear e+e− machines, we focus on the third case and show that the most sensitive searches for direct heavy neutrino production are possible at high-energy muon colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' Lepton colliders are, in general, sensitive to much smaller mixing parameters and hence to much higher scales of UV completions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' In this paper, we will consider a muon collider setup with energies of 3 and 10 TeV, and integrated luminosities of 1 and 10 ab−1, respectively [18–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' Model setup and simulation framework In this letter, we consider the Phenomenological Type I Seesaw Mech- anism [21, 22], implemented within the HeavyN model with Dirac neutrinos [5, 23], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' we assume it just as a representative model candidate without any prejudice (our findings are quite generic, though specific model setups like artificial flavor mixings could of course lead to singular cases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' [24, 25] provide an example where such heavy neutrinos appear even at a multi-TeV scale UV completion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' This effective extension of the SM introduces three flavors of right-handed neutrinos (denoted as Nk) that are singlets under the SM gauge groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' The Lagrangian of the model reads: L = LSM + LN + LW Nℓ + LZNν + LHNν (1) where LN is a sum of kinetic and mass terms for heavy neutrinos (in 4-spinor notation, which combines terms with spinors of dotted and undotted indices): LN = ¯Nki/∂Nk − mNk ¯NkNk for k = 1, 2, 3, (2) LW Nℓ yields neutrino interactions with the W boson: LW Nℓ = − g √ 2W + µ 3 � k=1 τ � l=e ¯NkV ∗ lkγµPLℓ− + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=', (3) arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content='02602v1 [hep-ph] 6 Jan 2023 2 0 2000 4000 6000 8000 10000 [GeV] qql m 1 − 10 1 10 2 10 3 10 4 10 5 10 6 10 Events ee bg sig_3000 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' 1: qqℓ mass distribution for a reference scenario assuming the existence of one Dirac neutrino with a mass of 3 TeV, at a 10 TeV muon collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' The black solid line stand for the µ+µ− background and the thick green one for the signal scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' LZNν interactions with the Z boson: LZNν = − g 2 cos θW Zµ 3 � k=1 τ � l=e ¯NkV ∗ lkγµPLνl + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=', (4) and LHNν interactions with the Higgs boson: LHNν = − gmN 2MW h 3 � k=1 τ � l=e ¯NkV ∗ lkPLνl + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' (5) The UFO library of the model contains 12 free parameters in addition to the SM parameters, which are three masses of the heavy neutrinos: mNk and nine real (no CP violation expected) mixing parameters Vlk, where l = e, µ, τ and k = N1, N2, N3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' For the purpose of this analysis, we considered a scenario with only one heavy Dirac neutrino N1 ≡ N with a mass below O(10 TeV) and equal couplings to all SM leptons (|VeN1|2 = |VµN1|2 = |VτN1|2 ≡ V 2 lN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' For sample generation, the mixing parameter V 2 lN has been set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content='0003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' Other values for the mixing parameters in the analysis below were accessed via rescaling with the corresponding cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' Although there are many different possible signatures of such particles at future colliders, for center-of-mass energies above the Z pole, the t-channel W exchange resulting in the production of a light-heavy neutrino pair (µ+µ− → N ν) is dominant [11] and the production cross section is of the order of 1-10 fb for masses of the neutrinos up to the collision energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' For our choice of the parameter space, the heavy neutrino has a microscopic lifetime (cτ ≪ 1 nm) and no displaced vertices are expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' Among the possible decay channels of such particles, the signature of two jets and a lepton (N → qqℓ) is the most promising because it allows for direct reconstruction of the mass of the heavy state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' In the first step, we generated event samples with Whizard 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content='2 [26–28] at leading order (LO) in the SM coupling constants (though recently higher-order corrections have become available in an automated manner [29] and simulated detector response with Delphes 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content='0 [30] using built-in Muon Collider detector cards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' At the generator level, a set of cuts was applied to remove possible singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' They included 10-GeV cuts on the energy of produced jets and leptons, the invariant mass of quark and lepton pairs, and the four-momentum transfer from the incoming muons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' Furthermore, it was required that at least one lepton could be detected in the central detector (we assumed 5◦ < θ < 175◦, where θ is the lepton polar angle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' For the detector simulation, the VLC clustering algorithm in the 3 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content='5 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content='5 1 BDT response 1 10 2 10 3 10 4 10 5 10 Events ee sig FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' 2: Distribution of the BDT response for the reference scenario (Dirac neutrino, mN = 3 TeV) with electrons in the final state at a 10 TeV muon collider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' The red line denotes the background, and the green line the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' exclusive two-jet mode (R = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content='5, β = 1, γ = 1 – see [31]) was applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' Since the considered Delphes model cannot generate fake lepton tracks, only 4- and 6-fermion background processes with at least one lepton in the final state (qqℓν, qqℓℓ, ℓℓℓℓ, qqqqℓν, qqqqℓℓ, qqℓνℓν, qqℓννν) were generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' The most important channels in terms of cross section (O(1 ab) at both energy stages) were qqℓν and ℓℓℓℓ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' the latter could be, however, easily reduced by lepton identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' Background channels induced by photons from collinear initial-state splittings were neglected, as their impact on the final results was found to be marginal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' Analysis procedure In the next step, a set of selection cuts was applied to reject events incompatible with the expected topology of two jets and one lepton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' To exclude events with significant contributions of forward deposits assigned to the beam jets, an upper limit of 20 GeV was applied on the transverse momentum of objects not included in the reconstructed final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' In Figure 1, we show a distribution of the invariant mass of two jets and a lepton for a reference scenario (a 3 TeV neutrino at a 10 TeV muon collider).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' A peak corresponding to the mass of the heavy neutrino is clearly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' The left tail is due to events with leptonic τ decays, for which the escaping neutrinos reduce the reconstructed invariant mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' On the right-hand side, the tail is an effect of finite detector resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' Subsequently, we applied the Boosted Decision Tree (BDT) method implemented in the TMVA package [32] to discriminate between signal and background events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' A set of eight variables describing event kinematics was chosen to optimize the classification: mqqℓ – invariant mass of the dijet-lepton system, α – angle between the dijet system and the lepton, αqq – angle between the two jets, Eℓ – lepton energy, Eqqℓ – energy of the dijet-lepton system, pT ℓ – lepton transverse momentum, pT qq – dijet transverse momentum, pT qqℓ – transverse momentum of the dijet-lepton system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' Due to the considerable difference between the numbers of expected background events, the algorithm was implemented separately for events with reconstructed electrons and muons in the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' The BDT response for the reference scenario is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' The two distributions are partially separated and thus, they were used to extract expected limits on the coupling parameter V 2 lN within the CLs method, implemented in the RooStats package [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' 4 3 10 4 10 [GeV] N m 7 − 10 6 − 10 5 − 10 4 − 10 3 − 10 2 − 10 1 − 10 2 lN lim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' V CMS HL-LHC HE-LHC FCC-hh ILC 1 TeV CLIC 3 TeV Muon Collider 10 TeV Muon Collider 3 TeV FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' 3: Limits on the coupling V 2 ℓN for different Muon Collider setups (solid lines: 3 TeV – turquoise, 10 TeV – orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' Dashed lines indicate limits from current and future hadron [1, 5] machines, dashed-dotted for e+e− colliders [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' See text for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' This allowed for combining the electron and muon channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' The impact of systematic uncertainties has been neglected at this stage, as they are not expected to significantly affect the final conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' Results In Figure 3, limits on the coupling V 2 lN for the two Muon Collider setups are presented and compared with the current limits coming from the CMS experiment (Majorana neutrinos, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' 2 in [1]), as well as with the results obtained for future hadron colliders (Dirac neutrinos, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' 25b in [5]) and e+e− colliders (Dirac neutrinos, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' 12 in [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' It should be noted that in the hadron collider analyses, heavy neutrino decays into taus were not considered, and thus their sensitivity is enhanced relative to the results presented for the lepton colliders, where the tau-channel decays are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' As shown in Figure 3, limits expected from the e+e− colliders, ILC running at 1 TeV and CLIC running at 3 TeV, are more stringent for masses of the heavy neutrinos up to about 700 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' The fact that the results for CLIC and a Muon Collider operating at the same energy of 3 TeV do not coincide may be surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' However, several effects must be taken into account for a proper comparison: the most important factors are different integrated luminosities and beam polarizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' In addition, the beam spectra and the beam-induced background channels cannot be neglected for e+e− colliders, while their impact is significantly reduced for µ+µ− machines due to the larger mass of the muon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' It was verified that, for the same generation setup (no beam polarization, no beam spectrum, no beam-induced background channels, but different initial-state particles and detector designs), the expected CLIC limits are consistent with the Muon Collider ones, giving the analysis precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' The discrepancy visible in Figure 3 could then be explained as follows: at lower neutrino masses, the expected limits from CLIC are more stringent due to the higher integrated luminosity and electron beam polarization, and at higher masses, they are worse because of the impact of the luminosity spectra and beam-induced backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' In the analysis, we assumed that all the mixing parameters VlN have the same value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' It is important to note that this approach is not unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' Using data from both electron-positron and muon colliders, one could potentially loosen this assumption and constrain the parameters VeN and VµN separately, by either excluding taus from the physical model or implementing a proper tau tagging procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' Such a method would give limits not only on the couplings themselves but also on their product in the framework where couplings are treated independently, possibly hinting at a flavor-universality violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' The details are, however, beyond the scope of this letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' Conclusions Extensions of the Standard Model introducing heavy neutrinos offer interesting solutions to several of its open questions, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' the baryon asymmetry of the universe, dark matter and flavor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' If such particles are at mass scales well above a GeV, they can be efficiently searched for at future lepton colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' Due to the highest achievable energies and the clean experimental environments, muon colliders would provide the furthest discovery reach for 5 this kind of particles and models, vastly surpassing high-energy hadron colliders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' By employing the synergy of both different types of lepton machines, electron-positron and muon colliders, different paths in the flavor parameter space of the models could be pursued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' Acknowledgments The work was partially supported by the National Science Centre (Poland) under the OPUS research project no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' 2021/43/B/ST2/01778.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' KM and JRR acknowledge the support by the Deutsche Forschungsgemeinschaft (DFG, Ger- man Research Association) under Germany’s Excellence Strategy-EXC 2121 "Quantum Universe"-3908333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' This work has also been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 491245950.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content=' ∗ krzysztof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content='mekala@desy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content='de † juergen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content='reuter@desy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content='de ‡ filip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1NE0T4oBgHgl3EQfuQFR/content/2301.02602v1.pdf'} +page_content='zarnecki@fuw.' metadata={'source': 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A. Den Hartog2, J. E. Lawler2, C. Sneden3, I. U. Roederer4,5 & J. J. Cowan6 +2Department of Physics, University of Wisconsin-Madison, 1150 University Ave, Madison, WI +53706; eadenhar@wisc.edu; jelawler@wisc.edu +3Department of Astronomy and McDonald Observatory, University of Texas, Austin, TX 78712; +chris@verdi.as.utexas.edu +4Department of Astronomy, University of Michigan, 1085 S. University Ave., Ann Arbor, MI +48109, iur@umich.edu +5Joint Institute for Nuclear Astrophysics – Center for the Evolution of the Elements (JINA-CEE) +6Homer L. Dodge Department of Physics and Astronomy, University of Oklahoma, Norman, OK +73019; jjcowan1@ou.edu + + + +ORCIDS: +E. A. Den Hartog: +0000-0001-8582-0910 + +J. E. Lawler: +0000-0001-5579-9233 + C. Sneden: + +0000-0002-3456-5929 + + +I. U. Roederer +0000-0001-5107-8930 +J. J. Cowan + +0000-0002-6779-3813 + +                                                             +1 Based on observations made with the NASA/ESA Hubble Space Telescope (HST), obtained at the Space Telescope +Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc. under NASA +contract NAS 5‐26555. Other data have been obtained from the European Southern Observatory (ESO) Science +Archive Facility; and the Keck Observatory Archive, which is operated by the W. M. Keck Observatory and the NASA +Exoplanet Science Institute, under contract with NASA. These data are associated with HST programs GO-7402, +GO-14161, and GO-14232; ESO programs 66.D-0636(A), 073.D-0024(A), and 095.D-0504(A); and Keck program +H41aH.   + +Abstract +We report new measurements of branching fractions for 20 UV and blue lines in the spectrum of +neutral silicon (Si I) originating in the 3s23p4s 3Po1,2, 1Po1 and 3s3p3 1Do1,2 upper levels. +Transitions studied include both strong, nearly pure LS multiplets as well as very weak spin- +forbidden transitions connected to these upper levels. We also report a new branching fraction +measurement of the 4P1/2 – 2Po1/2,3/2 intercombination lines in the spectrum of singly-ionized +silicon (Si II). The weak spin-forbidden lines of Si I and Si II provide a stringent test on recent +theoretical calculations, to which we make comparison. The branching fractions from this study +are combined with previously reported radiative lifetimes to yield transition probabilities and +log(gf)s for these lines. We apply these new measurements to abundance determinations in five +metal-poor stars. + + + +1. +Introduction +Silicon is one of the most abundant elements in the solar system and plays an important +role in many astrophysical environments. With its high abundance and relatively low ionization +potential it is a significant source of electrons in the interior of cool stars and contributes +significantly to the interior opacity in solar-type stars (Amarsi & Asplund 2017). Because +silicon is abundant and nonvolatile, it is often used as a reference element to reconcile the +absolute scales of meteoritic (e.g. Lodders, Palme & Gail 2009) and solar photospheric +abundances (e.g. Asplund et al. 2009). Emission line ratios of Si II, and in particular the ratio of +weak resonance lines (3s23p 2Po – 3s3p2 2D) and weak intercombination lines (3s23p 2Po – 3s3p2 +4P), are potentially useful as a plasma diagnostic because of their sensitivity to temperature and +density (e.g. Bautista et al. 2009). Silicon-burning, in which 28Si is converted to 56Ni in a series +of successive alpha captures, is the final phase of fusion reactions in the interior of massive stars. +Fusion reactions involving elements heavier than 56Ni are endothermic and thus not +spontaneous. After a brief period (approximately one earth day) of Silicon-burning, the core of a +massive star collapses and may explode to release more energy as a Type II supernova. +Motivation for the current study lies in the desire to better understand stellar +nucleosynthesis. Records of the “means of production” by which the elements came into being +in the earliest epoch of our Galaxy are written into the abundance patterns of the oldest, metal- +poor stars in the halo of the Milky Way. Here can be found evidence of the early births, short +lives and violent deaths of the first massive stars. Before these abundance patterns can be +decoded to gain deeper understanding of the history of nucleosynthesis, we must first be able to +determine the abundances of the elements with accuracy and precision. This requires both +accurate atomic data and realistic stellar models. As an α–capture element7, trends of abundance +ratios such as [Si/Fe]8 with metallicity yield insight into stellar nucleosynthesis and the chemical +evolution of the Galaxy. In an earlier study of the heaviest α–element, Ca, we made detailed +comparison between new and published experimental transition probabilities for Ca I and +modern theory (Den Hartog et al. 2021). In the present study we make similar comparison with +improved transition probabilities for lines of Si I and Si II. +In §2 below, we present a discussion of our measurement method including a description +of a new radiometric calibration technique for our high-resolution spectrometer. We present our +transition probabilities for 20 lines of Si I and two Si II intercombination lines in §3 along with +comparison to the best experimental and theoretical results from the literature. In §4 we apply +the new data to derive Si abundances in five warm, very metal-poor main-sequence stars. +                                                             +7 Formally an α-element is one whose dominant isotope is composed of multiple 4He nuclei. The major natural isotopes +of Si (Z = 14) are 28Si (92.191% in the solar system), 29Si (4.645%), 30Si (3.037) (Meija et al. 2016). For astrophysical +purposes, Si is pure 28Si. Since the minor isotopes of Si collectively contribute only 7.6% to the Si elemental +abundance, they will not contribute significantly in solar and stellar optical spectra. +8 We use standard abundance notations. For elements X and Y, the relative abundances are written [X/Y] = +log10(NX/NY)star − log10(NX/NY)Ÿ. For element X, the “absolute” abundance is written log10 ε(X) = log10(NX/NH) + +12. Metallicity is defined as the stellar [Fe/H] value. We adopt the Solar reference abundances from Asplund +(2009). + + +2. Emission Branching Fractions + The technique of combining radiative lifetimes from laser-induced fluorescence +measurements with emission branching fractions (BFs) measured using high-resolution +spectrometers is now the standard method for measuring transition probabilities, or Einstein A- +values, with efficiency and accuracy (e.g. Lawler et al. 2009). The BF for a transition between +an upper level u and a lower level l is given by the ratio of its A-value to the sum of the A-values +for all transitions associated with u, which is the inverse of the radiative lifetime, u. Thus the +radiative lifetime, u, provides the absolute scale when converting a BF to an A-value. For the +purposes of measuring BFs, it can also be expressed as the ratio of relative emission intensities I +(in any units proportional to photons/time) for these transitions: +𝐵𝐹�� � 𝐴�� +∑ 𝐴�� +� +� 𝐴��𝜏� � 𝐼�� +∑ 𝐼�� +� +. �1� +BFs, by definition, sum to unity. In order to assure the correct normalization, it is therefore +important when measuring BFs to account for all possible decay paths from an upper level. If +some weak transitions cannot be measured, these “residual” BFs need to be estimated from +theory and accounted for in the total decay rate. If the sum is over significantly less than the full +complement of lines, then one has a branching ratio (BR). +In order to avoid line blends, a high-resolution spectrometer is usually required to +measure the emission branching fractions unless the spectrum is very sparse. Often a Fourier +transform spectrometer (FTS) is used as these instruments have many advantages, including +high-resolution, broad spectral coverage and excellent absolute wavenumber accuracy. FTS +instruments have one significant disadvantage in that the quantum noise in the spectrum gets +spread evenly throughout the spectrum. This “multiplex” noise results in weak lines being +swamped in the noise from the strong lines in the spectrum. To overcome the multiplex noise +the lamp current is often increased to the point that strong lines in the spectrum are affected by +optical depth, which in turn results in inaccurate BFs. Corrections for optical depth can be made, +but if the corrections are large they lead to increased uncertainties. +In the current study, BFs in Si I and II have been determined from spectra recorded with +the University of Wisconsin (UW) high-resolution echelle spectrograph. This instrument is +described in detail in Wood & Lawler (2012). As a dispersive instrument, it does not have +multiplex noise and is much better-suited than an FTS for measurement of weak lines while +keeping source currents low and avoiding significant self-absorption on the strong transitions. It +is a 3-m cross-dispersed echelle spectrograph with broad spectral coverage, resolving power R ¥ +250,000 and a 4 Mpixel CCD detector. The spectra are two-dimensional CCD images containing +multiple grating orders, with the high-resolution of each grating order running in one direction +and the orders arranged side-by-side in the other dimension. The cross-disperser utilizes a prism + +to separate the orders, so the orders are further apart at lower wavelength and get increasingly +closer together at higher wavelengths. In the far-ultraviolet (far-UV) one CCD frame covers +approximately 150 nm in the low resolution direction and three overlapping frames are required +to capture an entire grating order in the high-resolution direction. The usual mode of operation +would be to acquire five overlapping frames for each UV spectrum, to provide some redundancy +and check for source drifts. However, the wavelengths of transitions from the upper levels in the +current study are such that all transitions from each level can be studied with a single grating +setting. This serendipitous coincidence of line placement means that there is no need to combine +frames with different grating settings, eliminating the contribution to the uncertainty that such +combining generates. +The optical sources used for generating the Si I, II spectra are commercially manufactured +Si-Ne and Si-Ar hollow cathode lamps (HCLs). Each CCD frame recorded is accompanied by a +continuum lamp spectrum recorded after the frame, from which a relative radiometric calibration +for that frame is determined. In the current study a deuterium (D2) lamp is used as the +calibration light source. The only change made between these two recordings is the angle of a +steering mirror on a kinematic mount. Beyond this mirror light from each lamp encounters the +same optical path. Table 1 lists all spectra recorded for the current study of Si II and Si I BFs. +The spectra are analyzed by taking a numerical integral of each line across the width of the +grating order in which it is found and dividing that by an integral of the D2 lamp intensity at the +same CCD position. The relative irradiance of the D2 lamp can be used to put all lines on the +same relative scale. These radiometrically calibrated intensities are then converted to BFs using +Equation 1. +Multiple spectra are taken of our primary source, the Si-Ne HCL, over a range of currents +between 3 mA and 32 mA. A range of lamp currents is used to check for evidence of self- +absorption on the strongest lines of Si I. Self-absorption becomes apparent by studying the BR +of a weaker line from the same upper level compared to a strong line that connects to the lowest +term. If self-absorption is present on the strong transition this BR will increase with increasing +lamp current. We see some evidence of minor self-absorption on three strong Si I lines that +connect to the ground term. These have small corrections applied based on the extrapolation of +the BR to zero current. The largest of these extrapolations is only 2% lower than the BR +measured on the lowest current spectrum. +2.1 Detector-based Radiometric Calibration +A continuum lamp is required for the calibration of the echelle spectrometer in order to +capture the rapidly changing instrument sensitivity along the grating orders due to the sinc2 blaze +envelope of the grating. However, the calibration in the low resolution direction, which changes +slowly as a function of wavelength, can be achieved by some other means and then transferred +onto the D2 source. For this project we have chosen to use a National Institute of Standards and +Technology (NIST) calibrated photodiode detector as our standard. Switching to a detector- + +based standard from a source-based standard has the advantage that the detector will remain +stable for many years, whereas lamp sources age both with shelf life and with usage. UV +damage to the window causes changes to the radiant output, particularly in the far-UV. The +irradiance of the lamp has to be periodically checked against another little-used lamp and then +corrections applied, or the lamp must be sent out to be recalibrated at considerable expense. +Another motivation for switching to the detector-based calibration is that D2 lamps are only +calibrated between 200 nm and 400 nm and the current project required a calibration out to 410 +nm. Even the calibrated irradiance above 370 nm requires careful correction in order to use the +lamp at high resolution. This is because above 370 nm there are increasing numbers of lines in +the D2 lamp spectrum in addition to the continuum radiation. The original irradiance calibration +of our lamp was made with a 4 nm bandpass,9 effectively smoothing over the increasing forest of +lines. At high resolution these lines are resolved and care must be taken to use only continuum +radiation when calibrating the metal line intensities. For past studies we have estimated +corrections such that the corrected irradiance gave the irradiance of the continuum only rather +than an average of continuum plus lines, but such corrections introduce additional uncertainty in +the calibration. +The detector used in this calibration is a Hamamatsu S2281 silicon photodiode calibrated +at NIST over the wavelength range 200 – 1100 nm. The accuracy of this calibration is 1.2 - 0.34 +% over the 200 – 410 nm range of the present study. A line source is also required and we have +chosen a Hg pen lamp because it has a spectrum sparse enough that only one to a few lines are +transmitted through each of the narrowband optical filters employed, as described below. It is +also necessary that the source has short term stability over the period of several hours which is +the case for the Hg pen lamp. It does not need to have long term stability. Also required for this +calibration are several narrowband optical filters which allow a subset of Hg lines through each +filter. We have used filters centered at wavelengths of 250 nm, 296 nm, 313 nm, 365 nm, 405 +nm and 436 nm. In addition we have used a sharp-cut colored glass filter (Corning 0-56) to +block the strong 254 nm light from leaking through the 296 and 313 nm filters. The narrowband +filters are ½ inch diameter, and are mounted in a ten position filter wheel for ease and +reproducibility of switching from one to the next. One position in the filter wheel is left open +with no filter installed to allow unfiltered light from the D2 lamp through. +Figure 1 shows a schematic of the measurement layout. Two lamps are employed, the +Hg pen lamp and the D2 lamp, each mounted at one of the positions viewed by the steering +mirror on a kinematic mount. Light from either lamp is imaged on the entrance pinhole of the 3- +m echelle spectrometer with a focusing mirror. The Hg pen lamp is rotated in its holder such that +the pair of capillaries are viewed side-on rather than front on, to limit structure in the image. +Light from the source passes through an iris, which limits the cross section of the beam, and then +                                                             +9 private communication from Optronics Laboratories  + +through the filter wheel before reaching the pinhole. When the filter wheel is set to either the +296 nm or 313 nm filter, a two inch square colored glass filter (Corning 0-56 sharp-cut filter) is +mounted just in front of the iris (not shown in Figure 1). The calibrated photodiode is moved +into the path between the filter wheel and the entrance pinhole to measure the power of light +transmitted by each filter. This is done at both the beginning of measurement and then again at +the end, to make sure the lamp has remained stable. The photodiode is removed for echelle +measurements. A full UV spectrum (three frames) is recorded for light passing through each +filter. An unfiltered D2 spectrum is recorded on each frame. Calibrated line intensities are +determined for all lines getting through each filter by dividing integrated line intensities by the +D2 continuum intensity, using the same analysis software and method as for the Si I,II BFs, as +described above. We use the unfiltered D2 spectrum to determine the filtered line intensities so +that the D2 intensity removes the sinc2 dependence of the grating order envelope from the +intensities but does not remove the effect of the filter bandpass. The calibration of the +photodiode is transferred onto the D2 lamp relative irradiance by insisting that the sum of line +intensities through each filter be proportional to the photodiode measurement for each filter (in +Amps) divided by the responsivity of the photodiode (in Amp/Watt) and divided by the +wavenumber of the transition(s) to convert Watts into something proportional to photons/s. The +level of reproducibility for this calibration can be seen in Figure 2 which shows two such +measurements of the relative D2 irradiance made approximately one month apart. Since the new +Figure 1. Schematic of the set up for the Hg pen lamp + NIST calibrated photodiode calibration technique. + +steering mirror on +kinematicmount +focusing +mirrorpinhole +for3-m +echelle +irisPhotodiode: +moved outof +opticalpath +D2lamp +filterwheel +Hgpen lamp +duringechelle +measurementscalibration only extends down to 250 nm, we use a calibration from our windowless Ar mini-arc +lamp (λ < 232nm) and our little-used D2 lamp transferred to our everyday D2 lamp to bridge the +gap between these two calibrations. +It should be mentioned that the Hg pen lamp is not a pure line source but also has a weak +continuum component. The paper by Reader, Sansonetti & Bridges (1996) drew our attention to +this problem. The weak continuum peaks around 405 nm, but there is also significant continuum +associated with the self-absorption on the strong 254 nm line. This continuum contributes to the +power measured with the photodiode, but is not accounted for in the filtered line intensity +measurements. The problem can be mitigated to some extent by choosing a narrower bandpass +for the filter. In the current study we have employed mostly 10 nm bandpass filters, but used a 5 +nm bandpass filter at 405 nm where the continuum was strongest. The narrower bandpass +reduces the contribution of the continuum relative to the lines. The residual continuum was +accounted for by making a measurement of the ratio of line intensity to line+continuum intensity +for each filtered spectrum that had some continuum contribution (these were the 250 nm, 365 +nm, 405 nm and 436 nm filters). This ratio was then applied as a correction to the photodiode +readings in the measurements described above. +We estimate the uncertainty of the calibration to be ~3 – 5% at each point of the curve +shown in Figure 2. However, because the D2 irradiance changes smoothly and gradually with +wavelength, the uncertainty of the relative calibration between two points on the curve will be +less than this estimate and depends on the spacing of the lines being calibrated. A BR for two +closely spaced lines, such as the Si II doublet discussed below, will have little contribution to the +Figure 2. Relative D2 lamp irradiance between 250 nm and 436 +nm as measured on two separate dates using the Hg pen lamp + +NIST calibrated photodiode calibration method as described in +the text. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +250 +300 +350 +400 +450 +Relative D2 Irraddiance (W) +Wavelength (nm) +19‐Mar‐22 +16‐Apr‐22 + +uncertainty from the calibration whereas lines that are widely separated in wavelength will have +a higher contribution to the BR uncertainty. We include a systematic uncertainty of 0.001% per +cm-1 of wavenumber difference between the line and the dominant line from the upper level as a +conservative estimate of uncertainty in the radiometric calibration. This is then added in +quadrature to the statistical uncertainty. We estimate the statistical uncertainty as the larger of +twice the standard deviation of the weighted mean branching ratio and the inverse of the +weighted average signal to noise ratio. The uncertainties of the BRs are then combined using an +appropriate error propagation formula to determine the final BF uncertainties. + +3. Results and Discussion +3.1 +Si I results +The experimental work on Si I transition probabilities to date has been limited. Garz et +al. (1973) determined relative f-values for 51 lines between 250 and 800 nm from emission +measurements on a wall stabilized arc. They tied these to an absolute scale using radiative +lifetimes of Marek (1972). These were later renormalized with new radiative lifetime +measurements by Becker et al (1980). Smith et al. (1987; hereafter Sm87) reported experimental +BFs or BRs and log(gf)s (the log of the level degeneracy multiplied by the oscillator strength) for +108 lines of Si I between 163 and 410 nm. They used a combination of techniques including +emission and absorption (Hook) measurements that they tied together using the bowtie method to +produce a set of self-consistent relative f-values. They chose the beam-foil lifetime +measurements of Bashkin et al. (1980) to establish their absolute scale. O’Brian & Lawler +(1991, hereafter OL91) measured radiative lifetimes to 5% accuracy for 47 odd-parity levels of +Si I and then combined their lifetimes with the BFs of Sm87 for 36 lines originating in 13 of the +lower-lying levels that Sm87 studied. Levels above the 3s23p3d 1Po1 level at 53387 cm-1 were +deemed by OL91 to have strong infrared branches, and the BFs of Sm87, having only estimated +the strength of these transitions, were thought to be less reliable. +There have been a number of theoretical investigations of Si I. Recent studies include the +work of Froese Fischer (2005) who used the Breit-Pauli approximation for all levels in Si I up to +3s23p3d 3Do. Savukov (2016; hereafter Sav16) used the configuration-interaction plus many- +body-perturbation-theory (CI+MBPT) method to determine transition probabilities, log(gf)s and +lifetimes for levels of Si I up to the 3s23p5s 1Po1 level. Wu et al. (2016) used the multi- +configuration Dirac-Hartree-Fock (MCDHF) and active space approach to determine levels, +hyperfine structure and transition probabilities in Si I up through the 3s23p4d 3Do levels. Finally, +the thesis work of Pehlivan Rhodin (2018; hereafter PR18) used MCDHF method using the +GRASP2K package to determine transition probabilities in Si I up through the 3s23p7s and in Si +II up through the 3s27f configuration. + +Our measured BFs of Si I are presented in Table 2 organized by upper level.10 Also in +this table we compare to a subset of the experimental BFs of Sm87. Note that for several of the +weak, spin-forbidden transitions Sm87 only report an upper bound (although what is meant by +<0.000 for the 3Do1,2 – 1D2 BFs is unclear). In this study, we report the first measurements of +these very weak BFs. For lines in common between the two studies, we see an average +fractional difference (in the sense (Sm87 – UW)/UW) of +6.0% with a standard deviation of +10.3%. For lines with BFs > 0.01 the average fractional difference is +1.7% with standard +deviation of 5.6%. +As a point of reference, we also compare to BFs calculated from LS coupling (also +known as Russell-Saunders coupling) theory for the triplet multiplets in Table 2. The upper 3p4s +3Po1 and 3Po2 levels at 39760 and 39955 cm-1 are nearly pure, with NIST ASD giving the leading +percentages as 98 and 99%, respectively. The J=1 level has ~1% mixing with the 3p4s 1Po1 level +resulting in weak decays to 1D2 and 1S0 lower levels. The upper 3s3p3 3Do1 and 3Do2 levels at +45276 and 45294 cm-1 are listed in the NIST ASD as 56% from that configuration and 39% 3pnd +3Do, but probably have some mixing with nearby 1Po1 and 1Do2 levels, respectively, since both +have weak decay to the 3s23p2 1Do2 level at 6299 cm-1. The LS BFs are calculated from relative +line strengths tabulated in Appendix I of Cowan (1981). Frequency-cubed scaling is included, +and the LS BFs are renormalized to the total multiplet strength as measured in the current study. +Our measured BFs are converted to A-values and log(gf)s following the relations in +Thorne et al. (1988), +𝐴�� � 𝐵𝐹�� +𝜏� + ; log�𝑔𝑓� � log �1.499𝑔�𝐴�� +𝜎� +� , �2� + +where Aul is the transition probability in s-1, u is the radiative lifetime of the upper level in s, gu +is the degeneracy of the upper level, and  is the transition wavenumber in cm-1. We use the +radiative lifetimes measured previously in our group by OL91 to establish the absolute scale for +our BFs. The uncertainty of the A-value is the uncertainty of the BF and that of the lifetime +added in quadrature. We present A-values and log(gf)s in Table 3. Also in Table 3 we compare +to two of the recent theoretical calculations, those by Sav16 and PR18.11 + +Sav16 determined transition probabilities, log(gf)s and lifetimes only for the low-lying +levels of Si I up to the 3s23p5s 1Po1 levels at ~54870 cm-1. As such, that study is limited in scope, +                                                             +10 Throughout this paper and accompanying tables, Ritz wavelengths and energy levels are taken from the National +Institute of Standards and Technology Atomic Spectra Database (NIST ASD; Kramida, Ralchenko & Reader 2021).  +11 We do not make comparison to the best experimental measurements in Table 3. NIST ASD references the results +of OL91 (for all but the weakest lines) which combine new lifetime measurements with BFs from Sm87. Our results +are not independent from OL91 as we use their lifetimes. We would like to alert the reader that there appears to be an +error in the A-values and log(gf)s in the NIST ASD for two of the transitions included in this study: 2443.365 Å and +2452.118 Å. NIST ASD log(gf)s are +0.32 and -0.53 dex different, respectively, from those found in OL91. This +discrepancy is also found in the critical compilation on Silicon by Kelleher & Podobedova (2008).  + +but achieves relatively high precision on the transitions that it covers by fine-tuning the cavity +size, which in turn reduced the basis needed for the lowest states. Sav16 makes detailed +comparison to earlier theory of Froese-Fischer (2005) and the experimental A-values and +radiative lifetimes reported in OL91. We find that we are in good agreement with Sav16 for the +20 transitions studied here even for the weakest transitions down to log(gf) < -4. The average +fractional difference between our A-values (in the sense (Sav16 – UW)/UW) is +1.7% with a +standard deviation of 9.7%. + +We also compare in Table 3 to the MCDHF calculations of PR18 who determined +transition probabilities for Si I belonging to the even 3s23p2, 3s23pnp (n ≤ 7), and 3s23pnf (n ≤ 6) +configurations and to the odd 3s3p3, 3s23pns (n ≤ 8), and 3s23pnd (n ≤ 6) configurations. Here +we find that the agreement with our measured transition probabilities is also very good, with +average fractional difference (in the sense (PR18 – UW)/UW) of -8.5% with a standard deviation +of 13.5%. This improves to an average of -3.9% and standard deviation of 10.4% for lines with +log(gf)>-3. Unlike Sav16, the PR18 study is a comprehensive calculation involving over 100 +levels up to 61936 cm-1 and more than 1300 transitions ranging in wavelength from 6333 nm in +the infrared to 161 nm in the vacuum-UV. As such, it will prove a very valuable resource for +astronomers. + +The comparisons made in Table 2 and Table 3 are visualized in Figure 3, where we +present logarithmic differences (in the sense log(other) – log(UW) versus log(UW)) of the +experimental BFs of Sm87 in panel (a) and the log(gf)s of PR18 and Sav16 in panels (b) and (c), +respectively. In panels (a) and (b) the error bars represent the combined uncertainties added in +quadrature. (The uncertainties reported in PR18 are the relative difference between the length +and velocity gauges.) Sav16 did not give uncertainties for their A-values so no error bars are +plotted in panel (c). In panel (a) the point with an arrow beside it is the upper bound quoted in +Sm87 for the transition at 4102 Å. The weakest, spin-forbidden transitions in these comparisons +are very difficult to measure and to calculate. The level of agreement with recent theory, both +with the limited-in-scope but high precision calculations of Sav16, and with the comprehensive +calculations of PR18, is very satisfactory. + + +3.2 +Si II results +We have remeasured the BF of the very weak spin-forbidden 4P1/2 - 2Po1/2,3/2 doublet of Si +II at 2334.407 Å and 2350.172 Å, respectively, using the first eight spectra listed in Table 1. +Optical depth is not a concern in this measurement because of the weakness of the transitions. +This BF had previously been measured in our group and reported in Calamai, Smith & Bergeson +(1993, hereafter CSB93). That paper had reported the measurement of the radiative lifetimes of +the 4P1/2,3/2,5/2 levels as well as the BFs of the 4P1/2 level. We use the radiative lifetime of CSB93 +to convert our BFs to A-values. These are reported in Table 4 along with comparison to the +CSB93 measurement. CSB93 report that these lines had signal-to-noise ratios of 10-15 in their + +‐1.0 +‐0.8 +‐0.6 +‐0.4 +‐0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +‐4.5 +‐4.0 +‐3.5 +‐3.0 +‐2.5 +‐2.0 +‐1.5 +‐1.0 +‐0.5 +0.0 +log(BF)Sm87 ‐ log(BF)UW +log(gf)UW +‐1.0 +‐0.8 +‐0.6 +‐0.4 +‐0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +‐4.5 +‐4.0 +‐3.5 +‐3.0 +‐2.5 +‐2.0 +‐1.5 +‐1.0 +‐0.5 +0.0 +log(gf)PR18 ‐ log(gf)UW +log(gf)UW +‐1.0 +‐0.8 +‐0.6 +‐0.4 +‐0.2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +‐4.5 +‐4.0 +‐3.5 +‐3.0 +‐2.5 +‐2.0 +‐1.5 +‐1.0 +‐0.5 +0.0 +log(gf)Sav16 ‐ log(gf)UW +log(gf)UW +c +a +b +Figure 3. Comparison of log(BF)s or log(gf)s of Si I in the present +work to those of a) the experimental results of Sm87, b) theoretical +MCDHF calculations of PR18, and c) theoretical CI+MBPT +calculations of Sav16 versus log(gf) from this study. In each figure +the horizontal line at 0.0 represents perfect agreement. Error bars +represent combined uncertainties where available. See text for +further discussion.  + +spectra whereas we have signal-to-noise ratios ranging from 45 to 200. The radiometric +calibration does not significantly contribute to the uncertainty of our BF because of the small +wavelength span between the doublet, resulting in an uncertainty that is primarily statistical. The +superior signal-to-noise in our spectra explains why our uncertainties are lower than those of +CSB93. We also compare to recent theoretical results of PR18 and Wu et al. 2020 in Table 4. +CSB93 appear to have used the theoretical BF of Nussbaumer (1977) to convert their +lifetime for the 4P3/2 level to A-values for the 4P3/2 - 2Po1/2,3/2 doublet at 2328.517 Å and 2344.202 +Å. This is not stated clearly in their paper, and in fact they state “Thirty-four measurements of +the 4P3/2 branching fraction were made. The total uncertainty (systematic and statistical) was +about 10% at the 90% level of confidence.” This appears to be a typo, and refers to the +measurement and uncertainty of the 4P1/2 BF. It is stated clearly elsewhere in the paper that a BF +was measured for only one level, the 4P1/2 level, and the 10% uncertainty mentioned in the quote +is not consistent with the 50% uncertainty on the weak branch of the 4P3/2 level. We attempted a +BF measurement of the 4P3/2 - 2Po1/2,3/2 doublet at 2328.517 Å and 2344.202 Å, but were +unsuccessful. The weaker 2328 Å line of this pair is estimated by the theory of Nussbaumer +(1977) and that of Dufton et al. (1991) to be a ~1% branch. Although we saw a weak feature at +this wavelength in our higher current Si-Ne spectra, we decided that this feature was a blend with +a very weak neon line. There is no observed transition listed at this wavelength in the NIST +ASD neon spectrum, but there is a possible Ne II electric dipole transition nearby that obeys +parity and J selection rules. Our analysis software looks for these possibilities based on known +energy levels of both the metal and buffer gas first and second spectra. We investigated this +further by looking at this wavelength in high current Hf-Ne and Hf-Ar spectra taken for a +different study. In these spectra we also saw a very weak feature in the Hf-Ne spectra but not in +the Hf-Ar spectra, suggesting a neon blend. Unfortunately, switching to a Si-Ar lamp does not +help in this case because the other line in the doublet pair, 2344.202 Å, has a known argon blend. +We attempted to procure a third commercial HCL with krypton buffer gas which has no potential +blends on either line, but were unsuccessful. The most we can say regarding the weak line at +2328.517 Å is that it is less than a 4.5% branch with an upper bound of log(gf) < -6.7. +The 4P - 2Po intercombination lines have been part of numerous theoretical investigations +of Si II. These lines are allowed E1 transitions due to the mixing of the 3s3p2 4P levels with +doublets from the same configuration. The accuracy of calculated radiative rates depend on the +accuracy to which the mixing coefficients and the multiplet energy splittings are calculated. +Nussbaumer (1977) used the SUPERSTRUCTURE code to calculate radiative parameters from +sophisticated configuration interaction wavefunctions. Dufton et al. (1991) significantly +improved on those results by including a more extensive set of configurations. These lines were +included in the calculations of Froese Fischer (2006) and Tayal (2007) using the MCHF method. +Bautista (2009) calculated radiative rates between many configurations using several different +approximations and generated a list of recommended log(gf)s for transitions among the 15 lowest +levels in Si II. Aggarwal & Keenan (2014) used the General-purpose Relativistic Atomic + +Structure Package (GRASP()) and the Flexible Atomic Code (FAC) to calculate a large number +of radiative parameters and collision strengths in Si II, but estimate ~20% uncertainty on the +strong transitions with weak transitions such as these intercombination lines being much more +uncertain. PR18 calculate A-values for these intercombination lines using the MCDHF method +and GRASP2K package with uncertainties based on the relative difference between the length +and velocity gauges of ~19% and 12% for the 2334.407 Å and 2350.172 Å lines, respectively. +Finally Wu et al. (2020) also used the MCDHF method and the GRASP2K package in their +study of Si II. In Figure 4 we make comparison to the experimental results for the BR (4P1/2 - +2Po3/2)/(4P1/2 - 2Po1/2) of CSB93 and to the above-mentioned theoretical studies, with the exception +of the Aggarwal & Keenan (2014) study. The BR from that study lies significantly off-scale on +Figure 4 at 1.37. In this figure the horizontal line is simply a guide to the eye, and lies at the +experimental value determined in this study. It can be seen from this figure that the general level +of agreement between experiment and theory has improved dramatically over recent decades, +undoubtedly owing, at least in part, to rapid increase in computing power. We see particularly +excellent agreement between our study and the recent theoretical results of Wu et al. (2020) and +PR18 as well as that of Froese Fischer et al. (2006). +4. Silicon Abundances in Very Metal-Poor Stars +All but two of the transitions studied here lie in the ultraviolet (UV) spectral domain +below the atmospheric absorption cutoff, i.e. 𝜆 < 3000 Å. This limits high-resolution stellar +spectroscopy to the Space Telescope Imaging Spectrograph (STIS; Kimble et al. 1998; +Figure 4.  Experimental and theoretical values determined for the BR of the +(4P1/2 - 2Po3/2)/(4P1/2 - 2Po1/2) doublet of Si II. The two experimental +measurements are leftmost followed by the theoretical values in reverse +chronological order left to right. The horizontal line lies at the BR as +measured in this work as a guide for the eye. +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +BR (4P1/2 ‐ 2P3/2)/(4P1/2 ‐ 2P1/2) +This Expt. + CSB93  +Wu et al. 2020  +PR18  +Tayal 2007  +Froese Fischer 2006  +Dufton et al. 1991  +Nussbaumer 1977  +Bautista 2009 +‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ Theory ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ +‐‐‐ Expt. ‐‐‐ + +Woodgate et al. 1998), on board the Hubble Space Telescope (HST). Additionally, the UV +spectrum is crowded with strong absorption lines of light and Fe-group elements, making reliable +abundance analyses difficult to execute. The UV spectral region of cool stars features complex +blends of transitions with various pedigrees, ranging from prominent well-known lines that have +well-documented laboratory histories to many moderate and weak lines with poor or completely +unknown atomic parameters. +The UV lines of neutral Si studied here are almost all very strong, having low excitation +energies (𝜒 < 6299 cm-1 or < 0.8 eV) and relatively large transition probabilities (17 out of 20 +lines in Table 3 have log(gf) > –3). The problem here is not in identifying Si I lines; it is in +finding stars with lines that are weak enough for abundance analysis. With this unusual +constraint we concentrated on metal-poor (Fe/H] < –2) halo stars that have been observed by +HST/STIS. The list is small: 7 stars are considered in the metallicity study of Roederer et al. +(2018); the bright main sequence star HD 84937 ([Fe/H] � –2.2) has been featured in previous +papers in this series (Den Hartog et al. 2021, and references therein); the famous warm low +metallicity stars HD 19445 and HD 140283 (Chamberlain & Aller 1951) have been featured in +several UV line identification contributions (e.g., Peterson et al. 2020 and references therein); the +mildly metal-poor warm giant HD 222925 ([Fe/H] = -1.5) has been recently studied by Roederer +et al. (2022) to produce a nearly complete abundance set for 63 elements. A few other such stars +can be found but do not change the basic results which we will discuss here. +We employed HST/STIS spectra of seven of the stars included in the papers cited above +in order to explore if Si abundances derived from UV spectra could be more trustworthy than the +few optical-wavelength lines treated in the literature. We supplemented our HST/STIS spectra +with blue spectra collected using the High Resolution Echelle Spectrometer (Vogt et al. 1994) at +the Keck I telescope, and the Ultraviolet and Visual Echelle Spectrograph (Dekker et al. 2000) at +the Very Large Telescope. We accessed these data through the Keck Observatory Archives and +European Southern Observatory Archives, respectively, and Table 1 of Roederer et al. (2018) +presents a description of these data. +We derived Si abundances using synthetic/observed spectrum matches. The synthetic +spectra were computed with the plane-parallel LTE (local thermodynamic equilibrium) line +analysis code MOOG (Sneden 1973)12. Atomic line lists for these syntheses were generated with +the linemake facility (Placco et al. 2021)13, which emphasizes laboratory transition data on Fe- +group and neutron-capture neutral and singly-ionized species from the Wisconsin atomic physics +group and on molecular species from the Old Dominion University group (e.g., Brooke et al. +2016, and references therein). We adopted the atmosphere parameters of Roederer et al. 2018, +2022) to produce model atmospheres interpolated from the ATLAS grid (Kurucz 2011, 2018)14. +                                                             +12 Available at https://www.as.utexas.edu/~chris/moog.html +13 https://github.com/vmplacco/linemake +14 http://kurucz.harvard.edu/grids.html  + +For almost all stars the lower wavelength boundary of our HST/STIS spectra was 𝜆 � 2300 Å, +thus ruling out work on the five lowest-wavelength Si I transitions. +Our initial synthetic spectrum tests yielded results that further narrowed the range of +stellar parameters that are useful for this abundance exercise. For stars that have [Fe/H] > –2.5 +and effective temperatures Teff < 6000 K, many of the promising Si I lines simply are too strong +and/or too blended with other strong neutral and ionized species features to yield reliable +abundances. In particular, we discarded the giant star HD 222925 (Teff/log(g)/[M/H]/vt = +5636K/2.54/–1.5/2.20km s-1; Roederer et al. 2022) and the subgiant HD 140283 (5600K/3.66/- +2.6/1.15km s-1; Roederer et al. 2018). We report here on five very metal-poor main sequence +turnoff stars that have Teff � 6050 K. +In Table 5 we list the model parameters, individual line abundances, and final species +abundances for both Si I and Si II transitions in the program stars. The mean abundances are +based on 10-11 Si I lines and 2 Si II lines, all in the vacuum UV spectral domain, whereas in +previous studies the Si abundances of these kinds of stars have come almost exclusively from the +optical Si I transitions at 3905.5 and 4102.9 Å. We derive <[Si/Fe]I> = 0.43 (𝜎 = 0.11). The +inclusion of the ionized species in Si abundance studies is a rarity, and for our program stars the +abundance agreement between neutral and ion is excellent. From Table 5 we find <[Si/Fe]II – +[Si/Fe]I> = +0.03 (𝜎 = 0.05). In Figure 5 we show small spectral regions around both Si II lines +and around six representative Si I lines in the program star BD+03º 740. For this star and the +other two lowest metallicity stars BD-13º 3442 and CD-33º 1173 the Si II lines are essentially on +the weak-line linear part of the curve of growth. They are easy to detect, and to employ in +abundance analyses. Many Si I lines are also reliable abundance indicators. However, the 2516, +2519, and 2881 Å transitions are clearly saturated and thus less sensitive to abundance changes. +In cooler, higher metallicity stars such as HD 19445 and HD 84937 these and other lines become +so strong that they are untrustworthy for abundance determinations. Some caution should be +used in interpreting the Si abundances of those stars. +We also derived abundances for the Si I 3905 Å line and list them in Table 5. The 4102.9 +Å Si I line was too weak and too blended with the strong H𝛿 4101.75 Å feature in our stars. +However we did not include the 3905 Å line in the mean abundance calculations because this +transition is known to yield temperature-dependent abundances in LTE calculations. Si in metal- +poor giants from the 𝜆3905 line is almost uniformly overabundant, <[Si/Fe]> ~ +0.4 ± 0.1 (e.g, +Cayrel et al. 2004), but is much less abundant in main sequence stars near the turnoff region, +<[Si/Fe]> ~ +0.1 ± 0.1 (e.g, Cohen et al. 2004). The sample of horizontal-branch stars +investigated by Preston et al. (2006) covers a large temperature range and shows this effect +clearly in their Figure 8. A summary of the observational issues in LTE abundances was +discussed by Sneden & Lawler (2008). From Table 5 we compute <[Si/Fe]> = +0.28 (𝜎 = 0.11) +from the 3905 Å line, clearly lower than the mean from the UV Si I lines discussed above. +Amarsi & Asplund (2017) computed NLTE corrections for optical-wavelength Si I transitions in + + +Figure 5: Observed and synthesized spectra for both Si II lines (the 2 bottom panels) +and for representative lines of Si I (the 6 upper panels) in the star BD+03º 740. In each +panel, the filled circles are the observations. The red line is a synthesis without any +contribution from Si. The best fit to a line is given by the black line, and the blue and +green lines show the synthetic spectra for Si abundances 0.4 dex lesser and greater than +the best match.  + +the solar photosphere, and have published on-line tables of NLTE corrections for many +(Teff/log(g)/[Fe/H]/vt) combinations.15 Their suggested correction for the 3905 Å line in stars with +parameters (6000 K/4.0/-3.0/1-2 km s-1) is D[Si/Fe] @ +0.1 dex. Applying this adjustment to the +abundances from this line for our stars would bring the 3905 Å line into better agreement with +our abundances derived from the UV Si I transitions. Abundances from the UV lines should be +preferred. +5. Discussion + In Figure 6 we illustrate the Galactic Chemical Evolution (GCE) trends of [Si/Fe] as a +function of metallicity ([Fe/H]). Silicon is synthesized in explosive oxygen burning, and thus is +formed in core-collapse supernovae early in the history of the Galaxy and then ejected into the +gas that eventually forms the halo stars. (Curtis et al. 2019) We show a compilation of +                                                             +15 Anish Amarsi - Theoretical Astrophysics, Department of Physics and Astronomy, Uppsala University - Astronomy +and Space Physics Theoretical astrophysics Department of Physics and Astronomy Uppsala University Box 516, +75120 Uppsala Sweden; Email: anish.amarsi@physics.uu.se ; www.astro.uu.se  +Figure 6. The [Si/Fe] abundance ratios as a function of metallicity ([Fe/H]) +for metal-poor stars from Roederer et al. (2014) (blue open squares) and this +paper (red filled circles). + +口 +口 +0.5 +口口口 +I/Fe] +口 +0 +中 +S口 +Roederer et al. (2014) +Thispaper +3 +2 +[Fe I /H]abundance data, [Si/Fe], from an earlier survey of low-metallicity Galactic stars (Roederer et al. +2014; shown as open squares). The values of [Si/Fe] exhibit significant scatter over the observed +metallicity range. This could be the result of comparing different types of stars (i.e., dwarfs with +giants) or due to the choice of the atomic lines used for the abundance determinations and/or the +source of the log(gf)s employed. Employing our new experimental silicon data (discussed +above, see Tables 3 and 4) leads to a more consistent pattern with less scatter. For the five stars +in this study (shown as filled red circles in Figure 6) the average value of [Si/Fe] = 0.44, +significantly higher than the solar value of 0. This value can serve as a constraint on GCE +models and, in particular, on supernovae nucleosynthesis model predictions for early Galactic +times. +It would be expected that the [Si/Fe] values illustrated in Figure 6 would begin to exhibit +a downward pattern at metallicities closer to [Fe/H] = -1 with the onset of Type Ia supernovae +(the main producer of iron) throughout the Galaxy. The abundance data from Roederer et al. +(2014) does hint at such a downward trend, but clearly more studies employing the new precise +atomic data in somewhat more metal-rich stars will be needed to confirm such a trend. + +6. Conclusions +We have made new BF measurements for 20 UV and blue lines of Si I as well as the +4P1/2 intercombination lines of Si II. Comparisons are made to earlier experiment as well as +theory. These BF have been combined with radiative lifetimes measured previously to determine +A-values and log(gf)s for these transitions. The current study represents a significant +improvement in measurement of the very weak spin-forbidden lines of both Si I and Si II. These +new data have been applied to abundance determinations in five metal-poor main sequence +turnoff stars. We find that many of the Si I UV transitions can be used as reliable abundance +indicators in very metal-poor stars and we obtain excellent agreement between abundances +determined using Si I transitions and the Si II intercombination lines. + +ACKNOWLEDGEMENTS +This work is supported by NSF grant AST-1814512 and AST-2206050 (E.D.H. and J.E.L). +I.U.R. acknowledges support from NSF grants AST 2205847 and PHY 14-30152 (Physics +Frontier Center/JINA-CEE), and NASA grants GO-14232, GO-15657 and AR-16630 from the +Space Telescope Science Institute, which is operated by the Association of Universities for +Research in Astronomy, Incorporated, under NASA contract NAS5-26555. We are grateful to +Hampus Nilsson for sharing the Si I, II data from the Pehlivan Rhodin (2018) thesis prior to its +publication, and to Karen Lind for helpful discussions. +Facilities: HST (STIS), Keck I (HIRES), VLT (UVES). +Software: LINEMAKE (Placco et al. 2021), MOOG (Sneden 1973). + + +REFERENCES +Aggarwal, K. M. & Keenan, F. P. 2014, MNRAS, 442, 388 +Amarsi, A. M. & Asplund, M. 2017, MNRAS, 464, 264 +Asplund M., Grevesse N., Sauval A. J., Scott P., 2009, ARA&A, 47, 481 +Bashkin, S., Astner, G., Mannervik, S., et al. 1980, PhyS, 21, 820 +Bautista, M. A., Quinet, P., Palmeri, P., et al. 2009, A&A, 508, 1527 +Becker, U., Zimmermann, P., & Holweger, H. 1980, GeCoA, 44, 2145 +Brooke, J. S. A., Bernath, P. F., Western, C. M. et al. 2016, JQSRT, 168, 142 +Calamai, A. G., Smith, P. & Bergeson, S. D. 1993, ApJ, 415, L59 +Cayrel, R., Depagne, E., Spite, M. et al. 2004, A&A, 416, 1117 +Chamberlain, J. W., & Aller, L. H. 1951, ApJ, 114, 52 +Cohen, J. 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Echelle spectra of commercial HCLs used in the study of Si II and Si I BFs.a +Indexb +Date +Serial +Number +Buffer +Gas +Lamp +Current +(mA) +Spectral Coverage +(Å) +Coadds +Total +Exposure +(min) +11 +2021 Jul 31 +2 +Neon +12 +2090 - 2955 +120 +360 +12 +2021 Aug 03 +2 +Neon +12 +2090 - 2955 +5 +150 +13 +2021 Aug 12 +1 +Neon +12 +2090 - 2955 +6 +180 +14 +2021 Aug 27 +1 +Neon +20 +2090 - 2955 +15 +150 +15 +2021 Aug 29 +1 +Neon +22 +2040 - 2790 +18 +180 +16 +2021 Sept 04 +1 +Neon +15 +2040 - 2790 +3 +180 +17 +2022 Apr 02 +1 +Neon +25 +2040 - 2790 +4 +120 +18 +2022 Apr 05 +1 +Neon +25 +2040 - 2790 +16 +160 +31 +2021 Dec 17 +1 + Neon +12 +2150 - 3245 +80 +120 +32 +2021 Dec 17 +3 + Neon +12 +2350 - 4300 +144 +120 +33 +2022 Jan 06 +3 + Neon +18 +2150 - 3245 +90 +60 +34 +2022 Jan 06 +1 + Neon +18 +2350 - 4300 +120 +60 +35 +2022 Jan 06 +5 + Neon +6 +2150 - 3245 +15 +75 +36 +2022 Jan 06 +7 + Neon +6 +2350 - 4300 +37 +74 +37 +2022 Jan 08 +1 + Neon +12 +2350 - 4300 +720 +120 +38 +2022 Jan 08 +3 + Neon +18 +2350 - 4300 +90 +90 +39 +2022 Jan 08 +5 + Neon +6 +2350 - 4300 +60 +60 +40 +2022 Jan 13 +1 + Neon +24 +2350 - 4300 +120 +60 +41 +2022 Jan 13 +3 +Neon +29 +2350 - 4300 +120 +60 +42 +2022 Mar 29 +1 + Neon +24 +2350 - 4300 +240 +60 +43 +2022 Mar 29 +3 + Neon +28 +2350 - 4300 +240 +60 +44 +2022 Apr 05 +3 +Neon +25 +2350 - 4300 +180 +60 +45 +2022 Apr 07 +1 +Neon +12 +2280 – 4200 +60 +60 +46 +2022 Apr 07 +3 + Neon +18 +2280 - 4200 +90 +60 +47 +2022 Apr 07 +5 + Neon +25 +2150 – 3245 +103 +60 +48 +2022 May 14 +1 + Argon +18 +2050 – 2800 +100 +100 +49 +2022 May 19 +1 + Argon +20 +2050 – 2800 +60 +60 +50 +2022 May 19 +3 + Argon +20 +2150 – 3245 +60 +60 +51 +2022 May 19 +5 + Neon +28 +2150 – 3245 +144 +60 +52 +2022 May 21 +1 + Neon +30 +2150 - 3245 +240 +60 +53 +2022 May 21 +3 + Argon +19 +2350 - 4300 +240 +60 +54 +2022 May 22 +1 + Neon +21 +2350 -4300 +120 +120 +55 +2022 May 30 +1 + Neon +3 +2350 - 4300 +24 +120 +56 +2022 May 30 +3 +Neon +3 +2280 - 4200 +24 +120 +Note: + aAll echelle spectra were taken from commercially manufactured Si-Ne or Si-Ar HCLs, and have a spectral +resolving power of ~250,000 although the effective resolving power is somewhat lower due to line broadening. +Each of the spectra were calibrated with a D2 lamp spectrum, which was recorded immediately following the +completion of each HCL spectrum. Each spectrum listed is a single CCD frame, and does not cover an entire +echelle grating order, but is sufficient coverage to determine branching fractions of all transitions from one or more +upper levels studied. +bThe first eight spectra list (indices 11 – 18) were used to study the BF of the Si II intercombination lines. The +remaining spectra (indices 31 – 56) were used in the Si I BF study. + +Table 2. +Branching Fractions of Si I +Upper levela + +Lower levela +lair +svac +This Expt. + +Other Expt.b +LSc +Configuration +and Term +Ek (cm-1) +Termd +Ei (cm-1) +(Å) +(cm-1) +BF +(±%) +BF +(±%) +BF +3s23p4s 3Po1 +39760.285 +3P0 +0.000 +2514.316 +39760.20 +0.337 +(1) +0.333 +(0.9) +0.333 + + +3P1 +77.115 +2519.202 +39683.17 +0.244 +(1) +0.247 +(1.6) +0.248 + + +3P2 +223.157 +2528.508 +39537.11 +0.409 +(1) +0.407 +(1.0) +0.409 + + +1D2 +6298.850 +2987.643 +33461.42 +0.0103 +(6) +0.012 +(8) + + + +1S0 +15394.370 +4102.936 +24365.91 +0.00056 +(17) +<0.0020 +(30) + +3s23p4s 3Po2 +39955.053 +3P1 +77.115 +2506.897 +39877.90 +0.243 +(1) +0.246 +(1.2) +0.252 + + +3P2 +223.157 +2516.112 +39731.88 +0.757 +(1) +0.754 +(0.4) +0.748 + + +1D2 +6298.850 +2970.353 +33656.18 +0.00020 +(10) +0.00027 +(13) + +3s23p4s 1Po1 +40991.884 +3P0 +0.000 +2438.768 +40991.80 +0.0030 +(7) +0.0034 +(5.9) + + + +3P1 +77.115 +2443.365 +40914.80 +0.0024 +(7) +0.0027 +(7.4) + + + +3P2 +223.157 +2452.118 +40768.70 +0.0022 +(7) +0.0025 +(8.7) + + + +1D2 +6298.850 +2881.578 +34693.02 +0.940 +(0.5) +0.934 +(0.2) + + + +1S0 +15394.370 +3905.523 +25597.51 +0.052 +(9) +0.057 +(2.2) + +3s3p3 3Do1 +45276.188 +3P0 +0.000 +2207.978 +45276.10 +0.566 +(1) +0.577 +(1.4) +0.557 + + +3P1 +77.115 +2211.745 +45199.20 +0.409 +(1) +0.398 +(2.3) +0.415 + + +3P2 +223.157 +2218.916 +45053.10 +0.025 +(4) +0.023 +(13) +0.027 + + +1D2 +6298.850 +2564.825 +38977.34 +0.00044 +(26) +<0.000 +(15) + +3s3p3 3Do2 +45293.629 +3P1 +77.115 +2210.892 +45216.60 +0.763 +(0.5) +0.760 +(0.4) +0.751 + + +3P2 +223.157 +2218.057 +45070.40 +0.236 +(1) +0.240 +(1.3) +0.248 + + +1D2 +6298.850 +2563.679 +38994.78 +0.00053 +(19) +<0.000 +(5) + +Notes: +a Upper and lower levels are taken from NIST ASD and are ordered by term. +b Sm87: Smith et al. 1987, ApJ 322, 573. The BFs of weak lines at 2438.768 Å and 2443.365 Å have only one significant digit listed in +Sm87 Table 1. We have calculated to two significant digits from their log(gf)s. +c LS BFs within the triplet multiplets calculated from the relative line strengths in Appendix I of Cowan (1981) and with frequency-cubed +scaling. They are renormalized to the total multiplet strength from the current measurements. +d The configuration of all lower levels is 3s23p2 + +Table 3. +A-values and log(gf)s for 20 transitions of Si I +lair +Eupper +Jupper +Elower +Jlower +This Expt. +Sav16a + +PR18 +(Å) +(cm-1) + +(cm-1) + +Aki (s-1) +(±%) +log(gf) +log(gf) +log(gf) +(±%) +2207.978 +45276.188 +1 +0.000 +0 +2.57E+07 +(5) +-1.248 +-1.229 +-1.318 +(6.5) +2210.892 +45293.629 +2 +77.115 +1 +3.47E+07 +(5) +-0.895 +-0.876 +-0.965 +(7.5) +2211.745 +45276.188 +1 +77.115 +1 +1.86E+07 +(5) +-1.388 +-1.372 +-1.459 +(5.8) +2218.057 +45293.629 +2 +223.157 +2 +1.07E+07 +(5) +-1.402 +-1.392 +-1.477 +(6) +2218.916 +45276.188 +1 +223.157 +2 +1.13E+06 +(6) +-2.603 +-2.586 +-2.670 +(4.4) +2438.768 +40991.884 +1 +0.000 +0 +7.06E+05 +(9) +-2.723 +-2.684 +-2.705 +(5.6) +2443.365 +40991.884 +1 +77.115 +1 +5.52E+05 +(9) +-2.828 +-2.788 +-2.805 +(0.7) +2452.118 +40991.884 +1 +223.157 +2 +5.01E+05 +(9) +-2.868 +-2.829 +-2.850 +(3.4) +2506.897 +39955.053 +2 +77.115 +1 +5.39E+07 +(5) +-0.595 +-0.566 +-0.578 +(0.9) +2514.316 +39760.285 +1 +0.000 +0 +7.48E+07 +(5) +-0.672 +-0.667 +-0.679 +(0.8) +2516.112 +39955.053 +2 +223.157 +2 +1.68E+08 +(5) +-0.098 +-0.088 +-0.101 +(0.9) +2519.202 +39760.285 +1 +77.115 +1 +5.42E+07 +(5) +-0.810 +-0.793 +-0.805 +(0.9) +2528.508 +39760.285 +1 +223.157 +2 +9.08E+07 +(5) +-0.583 +-0.567 +-0.579 +(0.8) +2563.679 +45293.629 +2 +6298.850 +2 +2.43E+04 +(20) +-3.922 +-3.953 +-4.078 +(25.2) +2564.825 +45276.188 +1 +6298.850 +2 +2.00E+04 +(26) +-4.228 +-4.298 +-4.389 +(28.7) +2881.578 +40991.884 +1 +6298.850 +2 +2.19E+08 +(5) +-0.088 +-0.044 +-0.061 +(1.4) +2970.353 +39955.053 +2 +6298.850 +2 +4.44E+04 +(11) +-3.531 +-3.577 +-3.613 +(6.4) +2987.643 +39760.285 +1 +6298.850 +2 +2.30E+06 +(8) +-2.035 +-2.082 +-2.113 +(3) +3905.523 +40991.884 +1 +15394.370 +0 +1.22E+07 +(10) +-1.077 +-0.999 +-1.018 +(3.5) +4102.936 +39760.285 +1 +15394.370 +0 +1.24E+05 +(18) +-3.026 +-3.126 +-3.154 +(3) +Notes +a log(gf)s calculated from A-values presented in Sav16 using equation 2. + +Table 4. +Branching Fractions, A-values and log(gf)s for the 4P1/2 - 2P1/2,3/2 doublet of Si II from experiment and recent theory. + +3s3p2 4P1/2 – 3s23p 2Po1/2 ; +lair = 2334.407 Å +3s3p2 4P1/2 – 3s23p 2Po3/2 ; +lair = 2350.172 Å + +BF +A (s-1) +log(gf) +BF +A (s-1) +log(gf) +This Expt:a +0.519 ≤ 1% +4990 ± 16% +-5.088 +0.481 ≤ 1% +4630 ± 16% +-5.116 +Other Expt: CSB93 +0.541 ≤ 10% +5200 ± 19% +-5.070 +0.459 ≤ 10% +4410 ± 21% +-5.136 +Theory: PR18b +0.520 +5280 ± 18.9% +-5.064 +0.480 +4882 ± 11.7% +-5.092 +Theory: Wu20b +0.514 +5230 +-5.068 +0.486 +4940 +-5.087 +Notes: +a our BFs are combined with the lifetime of CSB93 (104 ± 16 μs) to determine our A-value and log(gf) +b PR18 and Wu20 do not report BFs. We calculate BFs from their A-values to show the excellent agreement with the BFs measured in this study. + + + + + +Table 5. +Line-by-line abundances from Si I and Si II lines for the five metal-poor stars investigated. +Stellar Parameters + + +star +BD+03º 740 +BD-13º 3442 +CD-33º 1173 +HD 19445 +HD 84937 + + +Teff +6351 +6405 +6625 +6055 +6300 + + +log(g) +3.97 +4.04 +4.29 +4.49 +4.00 + + +vt +1.7 +1.6 +1.6 +1.2 +1.5 + + +[Fe I/H] +-2.89 +-2.84 +-2.98 +-2.14 +-2.24 + + +[Fe II/H] +-2.78 +-2.73 +-2.90 +-2.17 +-2.26 + + +source +Cowan20 +Cowan20 +Cowan20 +Roederer18 +Sneden16 +Line-by-Line Abundances – Si I +λ (Å) +χ (eV) +log(gf) +[Si I/Fe] +[Si I/Fe] +[Si I/Fe] +[Si I/Fe] +[Si I/Fe] +2438.768 +0.000 +-2.723 +0.34 +0.29 +0.48 +0.64 +0.57 +2443.365 +0.010 +-2.828 +0.37 +0.34 +0.48 +0.59 +0.47 +2452.118 +0.028 +-2.868 +0.41 +0.34 +0.43 +… +0.42 +2506.897 +0.010 +-0.595 +0.44 +0.46 +0.58 +0.64 +0.52 +2514.316 +0.000 +-0.672 +0.49 +0.46 +0.58 +0.59 +0.52 +2516.112 +0.028 +-0.098 +0.49 +0.49 +0.63 +0.49 +0.52 +2519.202 +0.010 +-0.810 +0.34 +… +0.58 +… +0.57 +2528.508 +0.028 +-0.583 +0.34 +0.49 +0.58 +… +0.57 +2564.825 +0.780 +-3.922 +… +… +… +0.39 +… +2881.578 +0.780 +-0.088 +0.24 +0.29 +0.33 +0.39 +0.32 +2970.353 +0.780 +-3.531 +0.41 +0.24 +… +0.29 +0.37 +2987.643 +0.780 +-2.035 +0.19 +0.24 +0.38 +0.29 +0.17 +3905.523a +1.907 +-3.026 +0.14 +0.24 +0.28 +0.44 +0.32 + + +meana +0.37 +0.36 +0.51 +0.48 +0.46 + + + +0.09 +0.10 +0.10 +0.14 +0.13 +Line-by-Line Abundances – Si II +λ (Å) +χ (eV) +log(gf) +[Si II/Fe] +[Si II/Fe] +[Si II/Fe] +[Si II/Fe] +[Si II/Fe] +2334.407 +0.036 +-5.088 +0.43 +0.48 +0.50 +0.57 +0.52 +2350.172 +0.036 +-5.116 +0.36 +0.43 +0.65 +0.32 +0.37 + + +mean +0.40 +0.46 +0.58 +0.45 +0.45 + + + +0.05 +0.04 +0.11 +0.18 +0.11 +Note +aThe mean and standard deviation of [Si I/Fe] are calculated without the λ3905 line data as this this transition is known to yield +temperature-dependent abundances in LTE calculations. See text for further discussion. + + diff --git a/3dFIT4oBgHgl3EQf5yug/content/tmp_files/load_file.txt b/3dFIT4oBgHgl3EQf5yug/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..669be1786d76da62dc620deeebe8a3c0f3ee61c8 --- /dev/null +++ b/3dFIT4oBgHgl3EQf5yug/content/tmp_files/load_file.txt @@ -0,0 +1,1626 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf,len=1625 +page_content='Atomic Transition Probabilities for Transitions of Si I and Si II and the Silicon Abundances of Several Very Metal-Poor Stars1 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Den Hartog2, J.' metadata={'source': 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University of Wisconsin-Madison, 1150 University Ave, Madison, WI 53706;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' eadenhar@wisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' jelawler@wisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='edu 3Department of Astronomy and McDonald Observatory, University of Texas, Austin, TX 78712;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' chris@verdi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='edu 4Department of Astronomy, University of Michigan, 1085 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' University Ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=', Ann Arbor, MI 48109, iur@umich.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='edu 5Joint Institute for Nuclear Astrophysics – Center for the Evolution of the Elements (JINA-CEE) 6Homer L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Dodge Department of Physics and Astronomy, University of Oklahoma, Norman, OK 73019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' jjcowan1@ou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='edu ORCIDS: E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Den Hartog: 0000-0001-8582-0910 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Lawler: 0000-0001-5579-9233 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Sneden: 0000-0002-3456-5929 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Roederer 0000-0001-5107-8930 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Cowan 0000-0002-6779-3813 1 Based on observations made with the NASA/ESA Hubble Space Telescope (HST), obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' under NASA contract NAS 5‐26555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Other data have been obtained from the European Southern Observatory (ESO) Science Archive Facility;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' and the Keck Observatory Archive, which is operated by the W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Keck Observatory and the NASA Exoplanet Science Institute, under contract with NASA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' These data are associated with HST programs GO-7402, GO-14161, and GO-14232;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' ESO programs 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='D-0636(A), 073.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='D-0024(A), and 095.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='D-0504(A);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' and Keck program H41aH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Abstract We report new measurements of branching fractions for 20 UV and blue lines in the spectrum of neutral silicon (Si I) originating in the 3s23p4s 3Po1,2, 1Po1 and 3s3p3 1Do1,2 upper levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Transitions studied include both strong, nearly pure LS multiplets as well as very weak spin- forbidden transitions connected to these upper levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' We also report a new branching fraction measurement of the 4P1/2 – 2Po1/2,3/2 intercombination lines in the spectrum of singly-ionized silicon (Si II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The weak spin-forbidden lines of Si I and Si II provide a stringent test on recent theoretical calculations, to which we make comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The branching fractions from this study are combined with previously reported radiative lifetimes to yield transition probabilities and log(gf)s for these lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' We apply these new measurements to abundance determinations in five metal-poor stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Introduction Silicon is one of the most abundant elements in the solar system and plays an important role in many astrophysical environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' With its high abundance and relatively low ionization potential it is a significant source of electrons in the interior of cool stars and contributes significantly to the interior opacity in solar-type stars (Amarsi & Asplund 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Because silicon is abundant and nonvolatile, it is often used as a reference element to reconcile the absolute scales of meteoritic (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Lodders, Palme & Gail 2009) and solar photospheric abundances (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Asplund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Emission line ratios of Si II, and in particular the ratio of weak resonance lines (3s23p 2Po – 3s3p2 2D) and weak intercombination lines (3s23p 2Po – 3s3p2 4P), are potentially useful as a plasma diagnostic because of their sensitivity to temperature and density (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Bautista et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Silicon-burning, in which 28Si is converted to 56Ni in a series of successive alpha captures, is the final phase of fusion reactions in the interior of massive stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Fusion reactions involving elements heavier than 56Ni are endothermic and thus not spontaneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' After a brief period (approximately one earth day) of Silicon-burning, the core of a massive star collapses and may explode to release more energy as a Type II supernova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Motivation for the current study lies in the desire to better understand stellar nucleosynthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Records of the “means of production” by which the elements came into being in the earliest epoch of our Galaxy are written into the abundance patterns of the oldest, metal- poor stars in the halo of the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Here can be found evidence of the early births, short lives and violent deaths of the first massive stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Before these abundance patterns can be decoded to gain deeper understanding of the history of nucleosynthesis, we must first be able to determine the abundances of the elements with accuracy and precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' This requires both accurate atomic data and realistic stellar models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' As an α–capture element7, trends of abundance ratios such as [Si/Fe]8 with metallicity yield insight into stellar nucleosynthesis and the chemical evolution of the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' In an earlier study of the heaviest α–element, Ca, we made detailed comparison between new and published experimental transition probabilities for Ca I and modern theory (Den Hartog et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' In the present study we make similar comparison with improved transition probabilities for lines of Si I and Si II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' In §2 below, we present a discussion of our measurement method including a description of a new radiometric calibration technique for our high-resolution spectrometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' We present our transition probabilities for 20 lines of Si I and two Si II intercombination lines in §3 along with comparison to the best experimental and theoretical results from the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' In §4 we apply the new data to derive Si abundances in five warm, very metal-poor main-sequence stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 7 Formally an α-element is one whose dominant isotope is composed of multiple 4He nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The major natural isotopes of Si (Z = 14) are 28Si (92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='191% in the solar system), 29Si (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='645%), 30Si (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='037) (Meija et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' For astrophysical purposes, Si is pure 28Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Since the minor isotopes of Si collectively contribute only 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='6% to the Si elemental abundance, they will not contribute significantly in solar and stellar optical spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 8 We use standard abundance notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' For elements X and Y, the relative abundances are written [X/Y] = log10(NX/NY)star − log10(NX/NY)Ÿ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' For element X, the “absolute” abundance is written log10 ε(X) = log10(NX/NH) + 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Metallicity is defined as the stellar [Fe/H] value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' We adopt the Solar reference abundances from Asplund (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Emission Branching Fractions The technique of combining radiative lifetimes from laser-induced fluorescence measurements with emission branching fractions (BFs) measured using high-resolution spectrometers is now the standard method for measuring transition probabilities, or Einstein A- values, with efficiency and accuracy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Lawler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The BF for a transition between an upper level u and a lower level l is given by the ratio of its A-value to the sum of the A-values for all transitions associated with u, which is the inverse of the radiative lifetime, \uf074u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Thus the radiative lifetime, \uf074u, provides the absolute scale when converting a BF to an A-value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' For the purposes of measuring BFs, it can also be expressed as the ratio of relative emission intensities I (in any units proportional to photons/time) for these transitions: 𝐵𝐹�� � 𝐴�� ∑ 𝐴�� � � 𝐴��𝜏� � 𝐼�� ∑ 𝐼�� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' �1� BFs, by definition, sum to unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' In order to assure the correct normalization, it is therefore important when measuring BFs to account for all possible decay paths from an upper level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' If some weak transitions cannot be measured, these “residual” BFs need to be estimated from theory and accounted for in the total decay rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' If the sum is over significantly less than the full complement of lines, then one has a branching ratio (BR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' In order to avoid line blends, a high-resolution spectrometer is usually required to measure the emission branching fractions unless the spectrum is very sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Often a Fourier transform spectrometer (FTS) is used as these instruments have many advantages, including high-resolution, broad spectral coverage and excellent absolute wavenumber accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' FTS instruments have one significant disadvantage in that the quantum noise in the spectrum gets spread evenly throughout the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' This “multiplex” noise results in weak lines being swamped in the noise from the strong lines in the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' To overcome the multiplex noise the lamp current is often increased to the point that strong lines in the spectrum are affected by optical depth, which in turn results in inaccurate BFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Corrections for optical depth can be made, but if the corrections are large they lead to increased uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' In the current study, BFs in Si I and II have been determined from spectra recorded with the University of Wisconsin (UW) high-resolution echelle spectrograph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' This instrument is described in detail in Wood & Lawler (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' As a dispersive instrument, it does not have multiplex noise and is much better-suited than an FTS for measurement of weak lines while keeping source currents low and avoiding significant self-absorption on the strong transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' It is a 3-m cross-dispersed echelle spectrograph with broad spectral coverage, resolving power R ¥ 250,000 and a 4 Mpixel CCD detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The spectra are two-dimensional CCD images containing multiple grating orders, with the high-resolution of each grating order running in one direction and the orders arranged side-by-side in the other dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The cross-disperser utilizes a prism to separate the orders, so the orders are further apart at lower wavelength and get increasingly closer together at higher wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' In the far-ultraviolet (far-UV) one CCD frame covers approximately 150 nm in the low resolution direction and three overlapping frames are required to capture an entire grating order in the high-resolution direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The usual mode of operation would be to acquire five overlapping frames for each UV spectrum, to provide some redundancy and check for source drifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' However, the wavelengths of transitions from the upper levels in the current study are such that all transitions from each level can be studied with a single grating setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' This serendipitous coincidence of line placement means that there is no need to combine frames with different grating settings, eliminating the contribution to the uncertainty that such combining generates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The optical sources used for generating the Si I, II spectra are commercially manufactured Si-Ne and Si-Ar hollow cathode lamps (HCLs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Each CCD frame recorded is accompanied by a continuum lamp spectrum recorded after the frame, from which a relative radiometric calibration for that frame is determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' In the current study a deuterium (D2) lamp is used as the calibration light source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The only change made between these two recordings is the angle of a steering mirror on a kinematic mount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Beyond this mirror light from each lamp encounters the same optical path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Table 1 lists all spectra recorded for the current study of Si II and Si I BFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The spectra are analyzed by taking a numerical integral of each line across the width of the grating order in which it is found and dividing that by an integral of the D2 lamp intensity at the same CCD position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The relative irradiance of the D2 lamp can be used to put all lines on the same relative scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' These radiometrically calibrated intensities are then converted to BFs using Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Multiple spectra are taken of our primary source, the Si-Ne HCL, over a range of currents between 3 mA and 32 mA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' A range of lamp currents is used to check for evidence of self- absorption on the strongest lines of Si I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Self-absorption becomes apparent by studying the BR of a weaker line from the same upper level compared to a strong line that connects to the lowest term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' If self-absorption is present on the strong transition this BR will increase with increasing lamp current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' We see some evidence of minor self-absorption on three strong Si I lines that connect to the ground term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' These have small corrections applied based on the extrapolation of the BR to zero current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The largest of these extrapolations is only 2% lower than the BR measured on the lowest current spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='1 Detector-based Radiometric Calibration A continuum lamp is required for the calibration of the echelle spectrometer in order to capture the rapidly changing instrument sensitivity along the grating orders due to the sinc2 blaze envelope of the grating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' However, the calibration in the low resolution direction, which changes slowly as a function of wavelength, can be achieved by some other means and then transferred onto the D2 source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' For this project we have chosen to use a National Institute of Standards and Technology (NIST) calibrated photodiode detector as our standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Switching to a detector- based standard from a source-based standard has the advantage that the detector will remain stable for many years, whereas lamp sources age both with shelf life and with usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' UV damage to the window causes changes to the radiant output, particularly in the far-UV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The irradiance of the lamp has to be periodically checked against another little-used lamp and then corrections applied, or the lamp must be sent out to be recalibrated at considerable expense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Another motivation for switching to the detector-based calibration is that D2 lamps are only calibrated between 200 nm and 400 nm and the current project required a calibration out to 410 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Even the calibrated irradiance above 370 nm requires careful correction in order to use the lamp at high resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' This is because above 370 nm there are increasing numbers of lines in the D2 lamp spectrum in addition to the continuum radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The original irradiance calibration of our lamp was made with a 4 nm bandpass,9 effectively smoothing over the increasing forest of lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' At high resolution these lines are resolved and care must be taken to use only continuum radiation when calibrating the metal line intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' For past studies we have estimated corrections such that the corrected irradiance gave the irradiance of the continuum only rather than an average of continuum plus lines, but such corrections introduce additional uncertainty in the calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The detector used in this calibration is a Hamamatsu S2281 silicon photodiode calibrated at NIST over the wavelength range 200 – 1100 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The accuracy of this calibration is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='2 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='34 % over the 200 – 410 nm range of the present study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' A line source is also required and we have chosen a Hg pen lamp because it has a spectrum sparse enough that only one to a few lines are transmitted through each of the narrowband optical filters employed, as described below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' It is also necessary that the source has short term stability over the period of several hours which is the case for the Hg pen lamp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' It does not need to have long term stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Also required for this calibration are several narrowband optical filters which allow a subset of Hg lines through each filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' We have used filters centered at wavelengths of 250 nm, 296 nm, 313 nm, 365 nm, 405 nm and 436 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' In addition we have used a sharp-cut colored glass filter (Corning 0-56) to block the strong 254 nm light from leaking through the 296 and 313 nm filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The narrowband filters are ½ inch diameter, and are mounted in a ten position filter wheel for ease and reproducibility of switching from one to the next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' One position in the filter wheel is left open with no filter installed to allow unfiltered light from the D2 lamp through.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Figure 1 shows a schematic of the measurement layout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Two lamps are employed, the Hg pen lamp and the D2 lamp, each mounted at one of the positions viewed by the steering mirror on a kinematic mount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Light from either lamp is imaged on the entrance pinhole of the 3- m echelle spectrometer with a focusing mirror.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The Hg pen lamp is rotated in its holder such that the pair of capillaries are viewed side-on rather than front on, to limit structure in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Light from the source passes through an iris, which limits the cross section of the beam, and then 9 private communication from Optronics Laboratories through the filter wheel before reaching the pinhole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' When the filter wheel is set to either the 296 nm or 313 nm filter, a two inch square colored glass filter (Corning 0-56 sharp-cut filter) is mounted just in front of the iris (not shown in Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The calibrated photodiode is moved into the path between the filter wheel and the entrance pinhole to measure the power of light transmitted by each filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' This is done at both the beginning of measurement and then again at the end, to make sure the lamp has remained stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The photodiode is removed for echelle measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' A full UV spectrum (three frames) is recorded for light passing through each filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' An unfiltered D2 spectrum is recorded on each frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Calibrated line intensities are determined for all lines getting through each filter by dividing integrated line intensities by the D2 continuum intensity, using the same analysis software and method as for the Si I,II BFs, as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' We use the unfiltered D2 spectrum to determine the filtered line intensities so that the D2 intensity removes the sinc2 dependence of the grating order envelope from the intensities but does not remove the effect of the filter bandpass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The calibration of the photodiode is transferred onto the D2 lamp relative irradiance by insisting that the sum of line intensities through each filter be proportional to the photodiode measurement for each filter (in Amps) divided by the responsivity of the photodiode (in Amp/Watt) and divided by the wavenumber of the transition(s) to convert Watts into something proportional to photons/s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The level of reproducibility for this calibration can be seen in Figure 2 which shows two such measurements of the relative D2 irradiance made approximately one month apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Since the new Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Schematic of the set up for the Hg pen lamp + NIST calibrated photodiode calibration technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' steering mirror on kinematicmount focusing mirrorpinhole for3-m echelle irisPhotodiode: moved outof opticalpath D2lamp filterwheel Hgpen lamp duringechelle measurementscalibration only extends down to 250 nm, we use a calibration from our windowless Ar mini-arc lamp (λ < 232nm) and our little-used D2 lamp transferred to our everyday D2 lamp to bridge the gap between these two calibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' It should be mentioned that the Hg pen lamp is not a pure line source but also has a weak continuum component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The paper by Reader, Sansonetti & Bridges (1996) drew our attention to this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The weak continuum peaks around 405 nm, but there is also significant continuum associated with the self-absorption on the strong 254 nm line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' This continuum contributes to the power measured with the photodiode, but is not accounted for in the filtered line intensity measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The problem can be mitigated to some extent by choosing a narrower bandpass for the filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' In the current study we have employed mostly 10 nm bandpass filters, but used a 5 nm bandpass filter at 405 nm where the continuum was strongest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The narrower bandpass reduces the contribution of the continuum relative to the lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The residual continuum was accounted for by making a measurement of the ratio of line intensity to line+continuum intensity for each filtered spectrum that had some continuum contribution (these were the 250 nm, 365 nm, 405 nm and 436 nm filters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' This ratio was then applied as a correction to the photodiode readings in the measurements described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' We estimate the uncertainty of the calibration to be ~3 – 5% at each point of the curve shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' However, because the D2 irradiance changes smoothly and gradually with wavelength, the uncertainty of the relative calibration between two points on the curve will be less than this estimate and depends on the spacing of the lines being calibrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' A BR for two closely spaced lines, such as the Si II doublet discussed below, will have little contribution to the Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Relative D2 lamp irradiance between 250 nm and 436 nm as measured on two separate dates using the Hg pen lamp + NIST calibrated photodiode calibration method as described in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='2 250 300 350 400 450 Relative D2 Irraddiance (W) Wavelength (nm) 19‐Mar‐22 16‐Apr‐22 uncertainty from the calibration whereas lines that are widely separated in wavelength will have a higher contribution to the BR uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' We include a systematic uncertainty of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='001% per cm-1 of wavenumber difference between the line and the dominant line from the upper level as a conservative estimate of uncertainty in the radiometric calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' This is then added in quadrature to the statistical uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' We estimate the statistical uncertainty as the larger of twice the standard deviation of the weighted mean branching ratio and the inverse of the weighted average signal to noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The uncertainties of the BRs are then combined using an appropriate error propagation formula to determine the final BF uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Results and Discussion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='1 Si I results The experimental work on Si I transition probabilities to date has been limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Garz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' (1973) determined relative f-values for 51 lines between 250 and 800 nm from emission measurements on a wall stabilized arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' They tied these to an absolute scale using radiative lifetimes of Marek (1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' These were later renormalized with new radiative lifetime measurements by Becker et al (1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' (1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' hereafter Sm87) reported experimental BFs or BRs and log(gf)s (the log of the level degeneracy multiplied by the oscillator strength) for 108 lines of Si I between 163 and 410 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' They used a combination of techniques including emission and absorption (Hook) measurements that they tied together using the bowtie method to produce a set of self-consistent relative f-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' They chose the beam-foil lifetime measurements of Bashkin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' (1980) to establish their absolute scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' O’Brian & Lawler (1991, hereafter OL91) measured radiative lifetimes to 5% accuracy for 47 odd-parity levels of Si I and then combined their lifetimes with the BFs of Sm87 for 36 lines originating in 13 of the lower-lying levels that Sm87 studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Levels above the 3s23p3d 1Po1 level at 53387 cm-1 were deemed by OL91 to have strong infrared branches, and the BFs of Sm87, having only estimated the strength of these transitions, were thought to be less reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' There have been a number of theoretical investigations of Si I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Recent studies include the work of Froese Fischer (2005) who used the Breit-Pauli approximation for all levels in Si I up to 3s23p3d 3Do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Savukov (2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' hereafter Sav16) used the configuration-interaction plus many- body-perturbation-theory (CI+MBPT) method to determine transition probabilities, log(gf)s and lifetimes for levels of Si I up to the 3s23p5s 1Po1 level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' (2016) used the multi- configuration Dirac-Hartree-Fock (MCDHF) and active space approach to determine levels, hyperfine structure and transition probabilities in Si I up through the 3s23p4d 3Do levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Finally, the thesis work of Pehlivan Rhodin (2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' hereafter PR18) used MCDHF method using the GRASP2K package to determine transition probabilities in Si I up through the 3s23p7s and in Si II up through the 3s27f configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Our measured BFs of Si I are presented in Table 2 organized by upper level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='10 Also in this table we compare to a subset of the experimental BFs of Sm87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Note that for several of the weak, spin-forbidden transitions Sm87 only report an upper bound (although what is meant by <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='000 for the 3Do1,2 – 1D2 BFs is unclear).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' In this study, we report the first measurements of these very weak BFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' For lines in common between the two studies, we see an average fractional difference (in the sense (Sm87 – UW)/UW) of +6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='0% with a standard deviation of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' For lines with BFs > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='01 the average fractional difference is +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='7% with standard deviation of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' As a point of reference, we also compare to BFs calculated from LS coupling (also known as Russell-Saunders coupling) theory for the triplet multiplets in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The upper 3p4s 3Po1 and 3Po2 levels at 39760 and 39955 cm-1 are nearly pure, with NIST ASD giving the leading percentages as 98 and 99%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The J=1 level has ~1% mixing with the 3p4s 1Po1 level resulting in weak decays to 1D2 and 1S0 lower levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The upper 3s3p3 3Do1 and 3Do2 levels at 45276 and 45294 cm-1 are listed in the NIST ASD as 56% from that configuration and 39% 3pnd 3Do, but probably have some mixing with nearby 1Po1 and 1Do2 levels, respectively, since both have weak decay to the 3s23p2 1Do2 level at 6299 cm-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The LS BFs are calculated from relative line strengths tabulated in Appendix I of Cowan (1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Frequency-cubed scaling is included, and the LS BFs are renormalized to the total multiplet strength as measured in the current study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Our measured BFs are converted to A-values and log(gf)s following the relations in Thorne et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' (1988), 𝐴�� � 𝐵𝐹�� 𝜏� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' log�𝑔𝑓� � log �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='499𝑔�𝐴�� 𝜎� � , �2� where Aul is the transition probability in s-1, \uf074u is the radiative lifetime of the upper level in s, gu is the degeneracy of the upper level, and \uf073 is the transition wavenumber in cm-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' We use the radiative lifetimes measured previously in our group by OL91 to establish the absolute scale for our BFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The uncertainty of the A-value is the uncertainty of the BF and that of the lifetime added in quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' We present A-values and log(gf)s in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Also in Table 3 we compare to two of the recent theoretical calculations, those by Sav16 and PR18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='11 Sav16 determined transition probabilities, log(gf)s and lifetimes only for the low-lying levels of Si I up to the 3s23p5s 1Po1 levels at ~54870 cm-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' As such, that study is limited in scope, 10 Throughout this paper and accompanying tables, Ritz wavelengths and energy levels are taken from the National Institute of Standards and Technology Atomic Spectra Database (NIST ASD;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Kramida, Ralchenko & Reader 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 11 We do not make comparison to the best experimental measurements in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' NIST ASD references the results of OL91 (for all but the weakest lines) which combine new lifetime measurements with BFs from Sm87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Our results are not independent from OL91 as we use their lifetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' We would like to alert the reader that there appears to be an error in the A-values and log(gf)s in the NIST ASD for two of the transitions included in this study: 2443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='365 Å and 2452.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='118 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' NIST ASD log(gf)s are +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='32 and -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='53 dex different, respectively, from those found in OL91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' This discrepancy is also found in the critical compilation on Silicon by Kelleher & Podobedova (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' but achieves relatively high precision on the transitions that it covers by fine-tuning the cavity size, which in turn reduced the basis needed for the lowest states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Sav16 makes detailed comparison to earlier theory of Froese-Fischer (2005) and the experimental A-values and radiative lifetimes reported in OL91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' We find that we are in good agreement with Sav16 for the 20 transitions studied here even for the weakest transitions down to log(gf) < -4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The average fractional difference between our A-values (in the sense (Sav16 – UW)/UW) is +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='7% with a standard deviation of 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' We also compare in Table 3 to the MCDHF calculations of PR18 who determined transition probabilities for Si I belonging to the even 3s23p2, 3s23pnp (n ≤ 7), and 3s23pnf (n ≤ 6) configurations and to the odd 3s3p3, 3s23pns (n ≤ 8), and 3s23pnd (n ≤ 6) configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Here we find that the agreement with our measured transition probabilities is also very good, with average fractional difference (in the sense (PR18 – UW)/UW) of -8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='5% with a standard deviation of 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' This improves to an average of -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='9% and standard deviation of 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='4% for lines with log(gf)>-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Unlike Sav16, the PR18 study is a comprehensive calculation involving over 100 levels up to 61936 cm-1 and more than 1300 transitions ranging in wavelength from 6333 nm in the infrared to 161 nm in the vacuum-UV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' As such, it will prove a very valuable resource for astronomers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The comparisons made in Table 2 and Table 3 are visualized in Figure 3, where we present logarithmic differences (in the sense log(other) – log(UW) versus log(UW)) of the experimental BFs of Sm87 in panel (a) and the log(gf)s of PR18 and Sav16 in panels (b) and (c), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' In panels (a) and (b) the error bars represent the combined uncertainties added in quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' (The uncertainties reported in PR18 are the relative difference between the length and velocity gauges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=') Sav16 did not give uncertainties for their A-values so no error bars are plotted in panel (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' In panel (a) the point with an arrow beside it is the upper bound quoted in Sm87 for the transition at 4102 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The weakest, spin-forbidden transitions in these comparisons are very difficult to measure and to calculate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The level of agreement with recent theory, both with the limited-in-scope but high precision calculations of Sav16, and with the comprehensive calculations of PR18, is very satisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='2 Si II results We have remeasured the BF of the very weak spin-forbidden 4P1/2 - 2Po1/2,3/2 doublet of Si II at 2334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='407 Å and 2350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='172 Å, respectively, using the first eight spectra listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Optical depth is not a concern in this measurement because of the weakness of the transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' This BF had previously been measured in our group and reported in Calamai, Smith & Bergeson (1993, hereafter CSB93).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' That paper had reported the measurement of the radiative lifetimes of the 4P1/2,3/2,5/2 levels as well as the BFs of the 4P1/2 level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' We use the radiative lifetime of CSB93 to convert our BFs to A-values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' These are reported in Table 4 along with comparison to the CSB93 measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' CSB93 report that these lines had signal-to-noise ratios of 10-15 in their ‐1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='0 ‐0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='8 ‐0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='6 ‐0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='4 ‐0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='0 ‐4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='5 ‐4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='0 ‐3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='5 ‐3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='0 ‐2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='5 ‐2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='0 ‐1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='5 ‐1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='0 ‐0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='0 log(BF)Sm87 ‐ log(BF)UW log(gf)UW ‐1.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='0 ‐2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='5 ‐2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='0 ‐1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='5 ‐1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='0 ‐0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='0 log(gf)Sav16 ‐ log(gf)UW log(gf)UW c a b Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Comparison of log(BF)s or log(gf)s of Si I in the present work to those of a) the experimental results of Sm87, b) theoretical MCDHF calculations of PR18, and c) theoretical CI+MBPT calculations of Sav16 versus log(gf) from this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' In each figure the horizontal line at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='0 represents perfect agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Error bars represent combined uncertainties where available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' See text for further discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' spectra whereas we have signal-to-noise ratios ranging from 45 to 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The radiometric calibration does not significantly contribute to the uncertainty of our BF because of the small wavelength span between the doublet, resulting in an uncertainty that is primarily statistical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The superior signal-to-noise in our spectra explains why our uncertainties are lower than those of CSB93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' We also compare to recent theoretical results of PR18 and Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 2020 in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' CSB93 appear to have used the theoretical BF of Nussbaumer (1977) to convert their lifetime for the 4P3/2 level to A-values for the 4P3/2 - 2Po1/2,3/2 doublet at 2328.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='517 Å and 2344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='202 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' This is not stated clearly in their paper, and in fact they state “Thirty-four measurements of the 4P3/2 branching fraction were made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The total uncertainty (systematic and statistical) was about 10% at the 90% level of confidence.” This appears to be a typo, and refers to the measurement and uncertainty of the 4P1/2 BF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' It is stated clearly elsewhere in the paper that a BF was measured for only one level, the 4P1/2 level, and the 10% uncertainty mentioned in the quote is not consistent with the 50% uncertainty on the weak branch of the 4P3/2 level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' We attempted a BF measurement of the 4P3/2 - 2Po1/2,3/2 doublet at 2328.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='517 Å and 2344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='202 Å, but were unsuccessful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The weaker 2328 Å line of this pair is estimated by the theory of Nussbaumer (1977) and that of Dufton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' (1991) to be a ~1% branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Although we saw a weak feature at this wavelength in our higher current Si-Ne spectra, we decided that this feature was a blend with a very weak neon line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' There is no observed transition listed at this wavelength in the NIST ASD neon spectrum, but there is a possible Ne II electric dipole transition nearby that obeys parity and J selection rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Our analysis software looks for these possibilities based on known energy levels of both the metal and buffer gas first and second spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' We investigated this further by looking at this wavelength in high current Hf-Ne and Hf-Ar spectra taken for a different study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' In these spectra we also saw a very weak feature in the Hf-Ne spectra but not in the Hf-Ar spectra, suggesting a neon blend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Unfortunately, switching to a Si-Ar lamp does not help in this case because the other line in the doublet pair, 2344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='202 Å, has a known argon blend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' We attempted to procure a third commercial HCL with krypton buffer gas which has no potential blends on either line, but were unsuccessful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The most we can say regarding the weak line at 2328.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='517 Å is that it is less than a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='5% branch with an upper bound of log(gf) < -6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The 4P - 2Po intercombination lines have been part of numerous theoretical investigations of Si II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' These lines are allowed E1 transitions due to the mixing of the 3s3p2 4P levels with doublets from the same configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The accuracy of calculated radiative rates depend on the accuracy to which the mixing coefficients and the multiplet energy splittings are calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Nussbaumer (1977) used the SUPERSTRUCTURE code to calculate radiative parameters from sophisticated configuration interaction wavefunctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Dufton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' (1991) significantly improved on those results by including a more extensive set of configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' These lines were included in the calculations of Froese Fischer (2006) and Tayal (2007) using the MCHF method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Bautista (2009) calculated radiative rates between many configurations using several different approximations and generated a list of recommended log(gf)s for transitions among the 15 lowest levels in Si II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Aggarwal & Keenan (2014) used the General-purpose Relativistic Atomic Structure Package (GRASP()) and the Flexible Atomic Code (FAC) to calculate a large number of radiative parameters and collision strengths in Si II, but estimate ~20% uncertainty on the strong transitions with weak transitions such as these intercombination lines being much more uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' PR18 calculate A-values for these intercombination lines using the MCDHF method and GRASP2K package with uncertainties based on the relative difference between the length and velocity gauges of ~19% and 12% for the 2334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='407 Å and 2350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='172 Å lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Finally Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' (2020) also used the MCDHF method and the GRASP2K package in their study of Si II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' In Figure 4 we make comparison to the experimental results for the BR (4P1/2 - 2Po3/2)/(4P1/2 - 2Po1/2) of CSB93 and to the above-mentioned theoretical studies, with the exception of the Aggarwal & Keenan (2014) study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The BR from that study lies significantly off-scale on Figure 4 at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' In this figure the horizontal line is simply a guide to the eye, and lies at the experimental value determined in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' It can be seen from this figure that the general level of agreement between experiment and theory has improved dramatically over recent decades, undoubtedly owing, at least in part, to rapid increase in computing power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' We see particularly excellent agreement between our study and the recent theoretical results of Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' (2020) and PR18 as well as that of Froese Fischer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Silicon Abundances in Very Metal-Poor Stars All but two of the transitions studied here lie in the ultraviolet (UV) spectral domain below the atmospheric absorption cutoff, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 𝜆 < 3000 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' This limits high-resolution stellar spectroscopy to the Space Telescope Imaging Spectrograph (STIS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Kimble et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Experimental and theoretical values determined for the BR of the (4P1/2 - 2Po3/2)/(4P1/2 - 2Po1/2) doublet of Si II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The two experimental measurements are leftmost followed by the theoretical values in reverse chronological order left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The horizontal line lies at the BR as measured in this work as a guide for the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='0 BR (4P1/2 ‐ 2P3/2)/(4P1/2 ‐ 2P1/2) This Expt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' CSB93 Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 2020 PR18 Tayal 2007 Froese Fischer 2006 Dufton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 1991 Nussbaumer 1977 Bautista 2009 ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ Theory ‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐ ‐‐‐ Expt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' ‐‐‐ Woodgate et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 1998), on board the Hubble Space Telescope (HST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Additionally, the UV spectrum is crowded with strong absorption lines of light and Fe-group elements, making reliable abundance analyses difficult to execute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The UV spectral region of cool stars features complex blends of transitions with various pedigrees, ranging from prominent well-known lines that have well-documented laboratory histories to many moderate and weak lines with poor or completely unknown atomic parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The UV lines of neutral Si studied here are almost all very strong, having low excitation energies (𝜒 < 6299 cm-1 or < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='8 eV) and relatively large transition probabilities (17 out of 20 lines in Table 3 have log(gf) > –3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The problem here is not in identifying Si I lines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' it is in finding stars with lines that are weak enough for abundance analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' With this unusual constraint we concentrated on metal-poor (Fe/H] < –2) halo stars that have been observed by HST/STIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The list is small: 7 stars are considered in the metallicity study of Roederer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' (2018);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' the bright main sequence star HD 84937 ([Fe/H] � –2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='2) has been featured in previous papers in this series (Den Hartog et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 2021, and references therein);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' the famous warm low metallicity stars HD 19445 and HD 140283 (Chamberlain & Aller 1951) have been featured in several UV line identification contributions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=', Peterson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 2020 and references therein);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' the mildly metal-poor warm giant HD 222925 ([Fe/H] = -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='5) has been recently studied by Roederer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' (2022) to produce a nearly complete abundance set for 63 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' A few other such stars can be found but do not change the basic results which we will discuss here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' We employed HST/STIS spectra of seven of the stars included in the papers cited above in order to explore if Si abundances derived from UV spectra could be more trustworthy than the few optical-wavelength lines treated in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' We supplemented our HST/STIS spectra with blue spectra collected using the High Resolution Echelle Spectrometer (Vogt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 1994) at the Keck I telescope, and the Ultraviolet and Visual Echelle Spectrograph (Dekker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 2000) at the Very Large Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' We accessed these data through the Keck Observatory Archives and European Southern Observatory Archives, respectively, and Table 1 of Roederer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' (2018) presents a description of these data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' We derived Si abundances using synthetic/observed spectrum matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The synthetic spectra were computed with the plane-parallel LTE (local thermodynamic equilibrium) line analysis code MOOG (Sneden 1973)12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Atomic line lists for these syntheses were generated with the linemake facility (Placco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 2021)13, which emphasizes laboratory transition data on Fe- group and neutron-capture neutral and singly-ionized species from the Wisconsin atomic physics group and on molecular species from the Old Dominion University group (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=', Brooke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 2016, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' We adopted the atmosphere parameters of Roederer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 2018, 2022) to produce model atmospheres interpolated from the ATLAS grid (Kurucz 2011, 2018)14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 12 Available at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='utexas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='edu/~chris/moog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='html 13 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='com/vmplacco/linemake 14 http://kurucz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='edu/grids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='html For almost all stars the lower wavelength boundary of our HST/STIS spectra was 𝜆 � 2300 Å, thus ruling out work on the five lowest-wavelength Si I transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Our initial synthetic spectrum tests yielded results that further narrowed the range of stellar parameters that are useful for this abundance exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' For stars that have [Fe/H] > –2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='5 and effective temperatures Teff < 6000 K, many of the promising Si I lines simply are too strong and/or too blended with other strong neutral and ionized species features to yield reliable abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' In particular, we discarded the giant star HD 222925 (Teff/log(g)/[M/H]/vt = 5636K/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='54/–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='5/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='20km s-1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Roederer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 2022) and the subgiant HD 140283 (5600K/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='66/- 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='6/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='15km s-1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Roederer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' We report here on five very metal-poor main sequence turnoff stars that have Teff � 6050 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' In Table 5 we list the model parameters, individual line abundances, and final species abundances for both Si I and Si II transitions in the program stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The mean abundances are based on 10-11 Si I lines and 2 Si II lines, all in the vacuum UV spectral domain, whereas in previous studies the Si abundances of these kinds of stars have come almost exclusively from the optical Si I transitions at 3905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='5 and 4102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='9 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' We derive <[Si/Fe]I> = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='43 (𝜎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The inclusion of the ionized species in Si abundance studies is a rarity, and for our program stars the abundance agreement between neutral and ion is excellent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' From Table 5 we find <[Si/Fe]II – [Si/Fe]I> = +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='03 (𝜎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='05).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' In Figure 5 we show small spectral regions around both Si II lines and around six representative Si I lines in the program star BD+03º 740.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' For this star and the other two lowest metallicity stars BD-13º 3442 and CD-33º 1173 the Si II lines are essentially on the weak-line linear part of the curve of growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' They are easy to detect, and to employ in abundance analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Many Si I lines are also reliable abundance indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' However, the 2516, 2519, and 2881 Å transitions are clearly saturated and thus less sensitive to abundance changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' In cooler, higher metallicity stars such as HD 19445 and HD 84937 these and other lines become so strong that they are untrustworthy for abundance determinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Some caution should be used in interpreting the Si abundances of those stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' We also derived abundances for the Si I 3905 Å line and list them in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The 4102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='9 Å Si I line was too weak and too blended with the strong H𝛿 4101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='75 Å feature in our stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' However we did not include the 3905 Å line in the mean abundance calculations because this transition is known to yield temperature-dependent abundances in LTE calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Si in metal- poor giants from the 𝜆3905 line is almost uniformly overabundant, <[Si/Fe]> ~ +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='1 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='g, Cayrel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 2004), but is much less abundant in main sequence stars near the turnoff region, <[Si/Fe]> ~ +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='1 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='g, Cohen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The sample of horizontal-branch stars investigated by Preston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' (2006) covers a large temperature range and shows this effect clearly in their Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' A summary of the observational issues in LTE abundances was discussed by Sneden & Lawler (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' From Table 5 we compute <[Si/Fe]> = +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='28 (𝜎 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='11) from the 3905 Å line, clearly lower than the mean from the UV Si I lines discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Amarsi & Asplund (2017) computed NLTE corrections for optical-wavelength Si I transitions in Figure 5: Observed and synthesized spectra for both Si II lines (the 2 bottom panels) and for representative lines of Si I (the 6 upper panels) in the star BD+03º 740.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' In each panel, the filled circles are the observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The red line is a synthesis without any contribution from Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The best fit to a line is given by the black line, and the blue and green lines show the synthetic spectra for Si abundances 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='4 dex lesser and greater than the best match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' the solar photosphere, and have published on-line tables of NLTE corrections for many (Teff/log(g)/[Fe/H]/vt) combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='15 Their suggested correction for the 3905 Å line in stars with parameters (6000 K/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='0/-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='0/1-2 km s-1) is D[Si/Fe] @ +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='1 dex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Applying this adjustment to the abundances from this line for our stars would bring the 3905 Å line into better agreement with our abundances derived from the UV Si I transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Abundances from the UV lines should be preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Discussion In Figure 6 we illustrate the Galactic Chemical Evolution (GCE) trends of [Si/Fe] as a function of metallicity ([Fe/H]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Silicon is synthesized in explosive oxygen burning, and thus is formed in core-collapse supernovae early in the history of the Galaxy and then ejected into the gas that eventually forms the halo stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' (Curtis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 2019) We show a compilation of 15 Anish Amarsi - Theoretical Astrophysics, Department of Physics and Astronomy, Uppsala University - Astronomy and Space Physics Theoretical astrophysics Department of Physics and Astronomy Uppsala University Box 516, 75120 Uppsala Sweden;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Email: anish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='amarsi@physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='uu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='se ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='astro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='uu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='se Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The [Si/Fe] abundance ratios as a function of metallicity ([Fe/H]) for metal-poor stars from Roederer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' (2014) (blue open squares) and this paper (red filled circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 口 口 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='5 口口口 I/Fe] 口 0 中 S口 Roederer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' (2014) Thispaper 3 2 [Fe I /H]abundance data, [Si/Fe], from an earlier survey of low-metallicity Galactic stars (Roederer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' shown as open squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The values of [Si/Fe] exhibit significant scatter over the observed metallicity range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' This could be the result of comparing different types of stars (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=', dwarfs with giants) or due to the choice of the atomic lines used for the abundance determinations and/or the source of the log(gf)s employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Employing our new experimental silicon data (discussed above, see Tables 3 and 4) leads to a more consistent pattern with less scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' For the five stars in this study (shown as filled red circles in Figure 6) the average value of [Si/Fe] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='44, significantly higher than the solar value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' This value can serve as a constraint on GCE models and, in particular, on supernovae nucleosynthesis model predictions for early Galactic times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' It would be expected that the [Si/Fe] values illustrated in Figure 6 would begin to exhibit a downward pattern at metallicities closer to [Fe/H] = -1 with the onset of Type Ia supernovae (the main producer of iron) throughout the Galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The abundance data from Roederer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' (2014) does hint at such a downward trend, but clearly more studies employing the new precise atomic data in somewhat more metal-rich stars will be needed to confirm such a trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Conclusions We have made new BF measurements for 20 UV and blue lines of Si I as well as the 4P1/2 intercombination lines of Si II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Comparisons are made to earlier experiment as well as theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' These BF have been combined with radiative lifetimes measured previously to determine A-values and log(gf)s for these transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The current study represents a significant improvement in measurement of the very weak spin-forbidden lines of both Si I and Si II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' These new data have been applied to abundance determinations in five metal-poor main sequence turnoff stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' We find that many of the Si I UV transitions can be used as reliable abundance indicators in very metal-poor stars and we obtain excellent agreement between abundances determined using Si I transitions and the Si II intercombination lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' ACKNOWLEDGEMENTS This work is supported by NSF grant AST-1814512 and AST-2206050 (E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' acknowledges support from NSF grants AST 2205847 and PHY 14-30152 (Physics Frontier Center/JINA-CEE), and NASA grants GO-14232, GO-15657 and AR-16630 from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Incorporated, under NASA contract NAS5-26555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' We are grateful to Hampus Nilsson for sharing the Si I, II data from the Pehlivan Rhodin (2018) thesis prior to its publication, and to Karen Lind for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=', & Lawler, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 2012, ApOpt, 51, 8407 Woodgate, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=', Kimble, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=', Bowers, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 1998, PASP, 110, 1183 Wu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=', Bian, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=', Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 2016, CaJPh, 94, 359 Wu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=', He, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=', Hu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' & Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 2020, InJPh, 95, 807 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Echelle spectra of commercial HCLs used in the study of Si II and Si I BFs.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='2022 May 19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='Argon ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='2150 – 3245 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='56 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='2022 May 30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='Neon ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='2280 - 4200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='Note: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='aAll echelle spectra were taken from commercially manufactured Si-Ne or Si-Ar HCLs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' and have a spectral resolving power of ~250,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='000 although the effective resolving power is somewhat lower due to line broadening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Each of the spectra were calibrated with a D2 lamp spectrum, which was recorded immediately following the completion of each HCL spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Each spectrum listed is a single CCD frame, and does not cover an entire echelle grating order, but is sufficient coverage to determine branching fractions of all transitions from one or more upper levels studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' bThe first eight spectra list (indices 11 – 18) were used to study the BF of the Si II intercombination lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' The remaining spectra (indices 31 – 56) were used in the Si I BF study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Branching Fractions of Si I Upper levela Lower levela lair svac This Expt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Other Expt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='b LSc Configuration and Term Ek (cm-1) Termd Ei (cm-1) (Å) (cm-1) BF (±%) BF (±%) BF 3s23p4s 3Po1 39760.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='285 3P0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='000 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2528.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='508 39537.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='409 (1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='407 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='409 1D2 6298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='850 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='4) 2970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='353 39955.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='053 2 6298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='850 2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='44E+04 (11) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='531 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='577 3.' 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+page_content='24E+05 (18) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='026 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='126 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='154 (3) Notes a log(gf)s calculated from A-values presented in Sav16 using equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Branching Fractions, A-values and log(gf)s for the 4P1/2 - 2P1/2,3/2 doublet of Si II from experiment and recent theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' 3s3p2 4P1/2 – 3s23p 2Po1/2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' lair = 2334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='407 Å 3s3p2 4P1/2 – 3s23p 2Po3/2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' lair = 2350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='172 Å BF A (s-1) log(gf) BF A (s-1) log(gf) This Expt:a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='519 ≤ 1% 4990 ± 16% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='088 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='481 ≤ 1% 4630 ± 16% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='116 Other Expt: CSB93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='541 ≤ 10% 5200 ± 19% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='070 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='459 ≤ 10% 4410 ± 21% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='136 Theory: PR18b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='520 5280 ± 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='9% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='064 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='480 4882 ± 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='7% 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='092 Theory: Wu20b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='514 5230 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='486 4940 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='087 Notes: a our BFs are combined with the lifetime of CSB93 (104 ± 16 μs) to determine our A-value and log(gf) b PR18 and Wu20 do not report BFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' We calculate BFs from their A-values to show the excellent agreement with the BFs measured in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Line-by-line abundances from Si I and Si II lines for the five metal-poor stars investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' Stellar Parameters star BD+03º 740 BD-13º 3442 CD-33º 1173 HD 19445 HD 84937 Teff 6351 6405 6625 6055 6300 log(g) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='97 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='04 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='29 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='49 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='00 vt 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='5 [Fe I/H] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='89 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='84 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='98 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='24 [Fe II/H] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='78 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='73 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='90 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='26 source Cowan20 Cowan20 Cowan20 Roederer18 Sneden16 Line-by-Line Abundances – Si I λ (Å) χ (eV) log(gf) [Si I/Fe] [Si I/Fe] [Si I/Fe] [Si I/Fe] [Si I/Fe] 2438.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='768 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='723 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content='11 Note aThe mean and standard deviation of [Si I/Fe] are calculated without the λ3905 line data as this this transition is known to yield temperature-dependent abundances in LTE calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} +page_content=' See text for further discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFIT4oBgHgl3EQf5yug/content/2301.11391v1.pdf'} diff --git a/3dFKT4oBgHgl3EQf8i5H/content/tmp_files/2301.11949v1.pdf.txt b/3dFKT4oBgHgl3EQf8i5H/content/tmp_files/2301.11949v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..67af073ebfcba24e3cb18a291bc0abe709bd60ad --- /dev/null +++ b/3dFKT4oBgHgl3EQf8i5H/content/tmp_files/2301.11949v1.pdf.txt @@ -0,0 +1,2162 @@ + +1 +Magnetic Amplification at Yb3+ "Designer Defects" +in the van der Waals Ferromagnet, CrI3 + +Kimo Pressler, Thom J. Snoeren, Kelly M. Walsh, Daniel R. Gamelin* +Department of Chemistry, University of Washington, Seattle, WA 98195, United States +Email: gamelin@uw.edu + + +Abstract. The two-dimensional (2D) van der Waals ferromagnet CrI3 has been doped with the +magnetic optical impurity Yb3+ to yield materials that display sharp multi-line Yb3+ +photoluminescence (PL) controlled by the magnetism of CrI3. Magneto-PL shows that Yb3+ +magnetization is pinned to the magnetization of CrI3. An effective internal field of ~10 T at Yb3+ +is estimated, attributed to strong in-plane Yb3+-Cr3+ superexchange coupling. The anomalously +low energy of Yb3+ PL in CrI3 reflects relatively high Yb3+-I- covalency, contributing to Yb3+- +Cr3+ superexchange coupling. The Yb3+ PL energy and linewidth both reveal the effects of +spontaneous zero-field CrI3 magnetic ordering within 2D layers below TC, despite the absence of +net magnetization in multilayer samples. These results illustrate the use of optical impurities as +"designer defects" to introduce unique functionality to 2D magnets. +Keywords: 2D Ferromagnet, Lanthanide Doping, Molecular Field, Chromium Triiodide, +Photoluminescence + + +Defects have the power to transform the physical properties of crystals, imparting new and +potentially useful functionalities from conductivity to quantum photon emission.1-6 In magnetic +materials, defects can strongly affect spin-wave propagation, magnetic domain-wall propagation, +skyrmion dynamics, and magnetic vortex pinning.7-9 Recently, the layered van der Waals +ferromagnet CrI3 has emerged as a promising platform for exploring strongly correlated spin +physics, magnetic proximity effects, and next-generation spin-based device architectures in the + + +2 +two-dimensional (2D) limit,10-14 but the potential to expand CrI3 functionality through +introduction of defects remains untapped. Here, we report that doping CrI3 with Yb3+ as a +"designer point defect" transforms its normally broad and featureless d-d photoluminescence +(PL) into narrow-line sensitized f-f emission, without compromising its attractive magnetic +properties. We further show that Yb3+ in CrI3 experiences a large internal effective field that +makes it extremely sensitive to small external magnetic fields. Using this property, we +demonstrate magnetically saturated circular polarization of Yb3+ emission at anomalously small +applied fields. Strikingly, the internal effective field also transmits magnetic information to Yb3+ +even in the absence of any applied field, making Yb3+ a unique embedded luminescent probe of +spontaneous zero-field magnetic ordering within the 2D monolayers of bulk CrI3. These +discoveries establish optical impurity doping as an effective strategy for expanding the +functionality of 2D magnets, with potential ramifications for both basic science and future spin- +photonic technologies. +CrI3 has become a model system for exploring magnetic exchange in 2D van der Waals +structures,10-14 stimulated by recent discoveries of Ising-like hard ferromagnetism in exfoliated +monolayer CrI3 and layer- and stacking-dependent magnetism in multi-layer CrI3.15,16 Layering +CrI3 with non-magnetic 2D materials introduces magnetic functionality to the non-magnetic +material via inter-layer exchange coupling, allowing magnetic manipulation of properties such as +WSe2 valley polarization and valley Zeeman splittings.17 Extension from few to many (bulk) +layers preserves the strong Ising-like intralayer ferromagnetic ordering, but facile motion of +domain walls unblocks demagnetization.18 Despite its rich magnetic properties, CrI3 itself has not +garnered much attention as an optical material. Bulk CrI3 has been investigated for its very large +Kerr and Faraday rotation strengths in relation to optical isolators and associated + + +3 +technologies.19,20 PL of bulk CrI3 has apparently not been reported, and few-layer CrI3 shows17 +only the very broad d-d PL characteristic of weak-field pseudo-octahedral Cr3+.21 Circular +polarization of this d-d PL was used to probe the magnetism of few-layer CrI3,17 but the +emission's breadth limits its further utility for fundamental studies or in spin-photonics, +stimulating efforts to narrow the band via cavity coupling.22 Doping CrI3 with optically active +impurities has also not been reported, either in bulk or exfoliated samples. +To investigate intralayer "proximity" effects resulting from magnetic exchange coupling, we +have prepared CrI3 doped with luminescent and spin-bearing Yb3+ ions. Large-diameter single- +crystal flakes of CrI3 were prepared by chemical vapor transport. Yb3+ was introduced by adding +Yb(0) to the precursor mix. The Yb3+ concentration in the resulting Yb3+:CrI3 crystals is +controllable, and samples with up to ~5% Yb3+ (cation mole fraction, [Yb3+]/([Cr3+]+[Yb3+])) are +described here. Further experimental details are provided in the Supporting Information (SI). +Figure 1a shows a photograph of representative Yb3+:CrI3 flakes in their growth tube. The flakes +are between 5 and 10 mm across, with typical thicknesses of 5-20 µm (see SI). Figure 1b plots +XRD data collected on undoped and 4.9% Yb3+-doped CrI3 single-crystal flakes using a powder +diffractometer. Only (00l) peaks are observed, corresponding to the interlayer lattice spacing and +reflecting the flake's alignment. Figure 1c highlights the shift to smaller angle of the 001 peak +upon doping. From fitting the XRD peak positions of undoped and 4.9% Yb3+-doped CrI3 +samples, the interlayer lattice parameter was found to increase 0.24% from 6.996 ± 0.002 to +7.013 ± 0.002 Å, attributed to the larger ionic radius of Yb3+ than Cr3+ (87 vs 62 pm, +respectively) (see SI). These data suggest that the local strain of doping is relieved by distorting +the lattice along its softest dimension, as expected. Substitutional incorporation of Yb3+ at the +Cr3+ site is verified by single-crystal XRD measurements (see SI), which also show the increased + + +4 +interlayer spacing. The single-crystal data show no detectable electron density between layers, +ruling out Yb3+ intercalation. + + +Figure 1. (a) Photograph of 4.9% Yb3+:CrI3 crystals prepared by chemical vapor +transport. The scale bar shows 5 mm. All experiments were performed on +individual single-crystal flakes from such a reaction tube. (b) XRD data collected +on undoped and Yb3+-doped CrI3 single crystals using a powder diffractometer. +Only (00l) peaks are observed, indicating an oriented sample. Reference peaks for +c-oriented CrI3 diffraction are included (black, ICSD Coll. Code 251654). (c) +Magnified view of the 001 reflection for the same samples, displaying an increase +in the interlayer lattice spacing upon Yb3+ doping. The x axis in (c) was +determined as described in the SI. + + +Figure 2a plots the PL spectra of CrI3 and Yb3+:CrI3 single flakes measured at several +temperatures between 4 and 200 K. The CrI3 spectrum broadens and decreases in intensity with +increasing temperature, eventually reaching only 7.5% of its 4 K intensity at 200 K (see SI). +Although the broadening to higher energies is expected from thermal hot bands, the broadening +to lower energies is abnormal and suggests an additional feature. Upon introduction of Yb3+, the +broad featureless d-d emission of Cr3+ disappears and is replaced by a series of sharp f-f +transitions of Yb3+ around 1.15 eV. Assignment of the PL fine structure is discussed later. In +some samples, Yb3+ doping also reveals another broad emission band centered at ~0.95 eV, +which is responsible for the red tail of the CrI3 PL here and in some literature spectra. This + +a +b +C +Yb3+: Crl3 +Intensity (rel.) +Intensity (rel.) +Crl3 +001 +002 +ref. +001 +003 +004 +005 +006 +I x10 +[x10 +20 +40 +60 +80 +13.0 +14.0 +2θ (deg.) +20 (deg.) +5 +feature has been traced to Ni2+ impurities (<0.4%) found in some Cr(0) precursors, and it can be +mostly eliminated by using 5N Cr(0) precursors (Fig. 2a, bottom). The Yb3+ PL is not influenced +by this Ni2+ impurity (see SI). + + +Figure 2. (a) Variable-temperature PL spectra of CrI3 (top) and 4.9% Yb3+:CrI3 +(bottom), measured from 4 to 200 K under 1.88 eV CW excitation at 4 mW/cm2. +(b) Single-configurational-coordinate diagram (A1g coordinate) describing +vibronic broadening of the absorption and luminescence bands associated with +transitions between the 4A2g and 4T2g ligand-field states of pseudo-octahedral +Cr3+. In Yb3+-doped CrI3, energy transfer from the Cr3+ 4T2g excited state to Yb3+ +yields sensitized 2F5/2 ! 2F7/2 f-f luminescence. + + +Figure 2b illustrates the photophysics of Yb3+:CrI3 schematically. The lowest-energy excited +state of CrI3 is the Cr3+ 4T2g ligand-field state, involving excitation of a t2g electron into a σ- +antibonding eg orbital (in idealized Oh symmetry). The resulting change in equilibrium geometry +is described by the single-configurational-coordinate (SCC) diagram of Fig. 2b, which illustrates +the totally symmetric distortion coordinate. This 4T2g excited state also distorts along a +symmetry-breaking Jahn-Teller coordinate (not illustrated).21 These distortions lead to extensive + +b +a +1.0 +Crls Undoped +, Intensity (norm.) +4 - 200 K +0.8 +Cr3+ +g +0.6 +ET +0.4- + 0.2 +Cr3+ +Cr3+ +1.0 +Yb3+:Crl3 +Abs +PL + Intensity (norm.) +0.8 +4 - 200 K +Yb3+ +0.6. +A +12g +PL +0.4- +0.2 +2 +7/2 +0.0 +1.2 +1.1 +1.0 +0.9 +Energy (eV) +6 +vibronic progressions in the absorption and PL spectra associated with this transition, and cause +a large PL Stokes shift. Doping CrI3 with Yb3+ introduces a set of 2F5/2 states just below the Cr3+ +4T2g excited state, favorably positioned for efficient Cr3+ ! Yb3+ energy transfer. At 4.9% Yb3+ +doping, the Cr3+ 4T2g PL is entirely quenched and strong Yb3+ 2F5/2 emission is observed in its +place (Fig. 2a). Because both Cr3+ and Yb3+ states are localized at single ions, energy migration +within the CrI3 lattice is required for this complete quenching. In undoped CrI3, energy migration +among equivalent Cr3+ sites may occur but is not readily apparent. In Yb3+:CrI3, this energy +migration is interrupted when energy is captured by Yb3+ dopants. In 4.9% Yb3+:CrI3, the +average Cr3+ ion has only ~14% probability of having a neighboring Yb3+, and ~50% probability +of having at least one Yb3+ within its first two cation shells. Energy must therefore migrate over +at least a few lattice sites within the 4T2g lifetime to fully quench the Cr3+ emission as observed in +Fig. 2a. +Figure 3a shows the anticipated electronic structure of Yb3+ in CrI3. In the free ion, spin-orbit +coupling splits the 2F term into 2F5/2 (excited) and 2F7/2 (ground) states by an amount ΔE = 7/2ζ, +where ζ = 361.8 meV is the free-ion spin-orbit coupling constant.23 In crystals, each of these +states is further split by the crystal field. Figure 3b shows circularly polarized PL spectra of 4.9% +Yb3+:CrI3 measured in a 0.5 T field applied parallel to the crystal's c axis (vide infra). Three +zero-phonon electronic origins are observed and assigned to the Γ8 ! Γ6, Γ8, and Γ7 transitions +anticipated from Fig. 3a using idealized Oh notation. The actual cation site symmetry in CrI3 is +lower (Fig. 3a, right),24 but the expected low-symmetry splitting of the Γ8 origin is not clearly +identifiable. The Γ6 peak is broad with observable structure on its high-energy shoulder, thus +making the precise energy of this origin unclear within ~20 cm-1 (~2.5 meV). Analysis of these +PL energies within the Angular Overlap Model (AOM)25 reproduces the 2F7/2 splittings well, + + +7 +predicting a 2F5/2 splitting of ~34 meV and splittings of the two Γ8 levels by <0.5 meV each (see +SI). Additional satellite features are observed ~127 cm-1 (15.7 meV) below the Γ8 and Γ7 +electronic origins and assigned as phonon sidebands. Raman spectra show a totally symmetric +lattice breathing mode of CrI3 at this energy (ν = 127 cm-1).26 +A striking aspect of this Yb3+:CrI3 PL is its very low energy relative to other Yb3+ PL. This +energy is primarily determined by spin-orbit coupling (Fig. 3a). Yb3+ spin-orbit coupling can be +reduced from that in the free ion by covalent expansion of the f-electron wavefunctions +(nephelauxetic effect),27,28 but f-orbital covalency in trivalent lanthanides is typically very small +and this effect is usually considered negligible at ambient pressure. A survey of Yb3+-doped +crystals shows that the energy gap between Yb3+ 2F5/2 and 2F7/2 barycenters remains very near the +free-ion value of ΔE ~ 1.266 eV across doped oxide, fluoride, chloride, bromide, sulfide, and +phosphide lattices (see SI).29-33 We note that we have been unable to find any reports of PL from +other Yb3+-doped iodide crystals, perhaps because Yb3+ is easily reduced to Yb2+ under common +iodide crystal-growth conditions. Yb3+:CrI3 deviates from this typical behavior substantially: ΔE +is only ~1.163 eV, or ~9% smaller than in the free ion, representing the smallest spin-orbit +coupling yet reported for Yb3+. Covalency in Yb3+:CrI3 is certainly enhanced by the large ionic +radius and polarizability of the iodides, but this consideration alone likely cannot explain the +anomaly. The atomic spin-orbit coupling of I is also much greater than those of other common +ligands for Yb3+, and should contribute to the spectroscopic spin-orbit splitting via covalency. +Furthermore, the large ionic radius of Yb3+ compared to Cr3+ means that Yb3+ experiences an +internal pressure imposed by the surrounding lattice, which may also increase covalency. +Importantly, Yb3+-I- covalency is essential for strong Yb3+-Cr3+ superexchange coupling. + + + +8 + + + + + +Figure 3. (a) Splitting of the Yb3+ free-ion 2F term due to spin-orbit (ζ) and +crystal-field (Oh, TC +(4) + + +12 +where E0 is the energy at 0 K, αs describes the electron-phonon interaction strength, and Δ is the +energy of the activating phonon mode, fixed at Δ = 127 cm-1 (15.7 meV, Fig. 3b). +The solid curve in the high-temperature portion of Fig. 4b (>TC) shows a fit to the high- +temperature data using eq 4, floating E0 and αs and yielding best-fit values of 1.1242 eV and -6.3 +meV, respectively. Eq 4 plateaus at E0 in the limit of 0 K (dashed line < TC in Fig. 4b), but the +experimental peak energy shows a discontinuity at TC, dropping sharply and decreasing with +decreasing temperature until reaching ~7 meV below E0 in the low-temperature limit. With its +link to TC and its characteristic curvature, this trend in Yb3+ PL energy is associated with the +spontaneous magnetization of individual CrI3 monolayers, even though there is no net +magnetization in these samples. +Spontaneous ferromagnetic ordering is classified as a second-order phase transition and, +within the theory of universal scaling laws, is characterized by the order parameter β shown in eq +5 describing the magnetization temperature dependence.44 +!(!) = !! − +!!!! +!! +! + + + + + +(5) +M0 is the saturation moment per magnetic ion and equals 3.1 µB for CrI3.18 The precise value of β +depends on the underlying spin physics, but it is commonly around 1/3.12 Previous examination +of bulk CrI3 found a critical exponent of β = 0.284, between that expected from the 3D Ising +model (β = 0.325) and that of the tri-critical mean-field model (β = 0.250).34 Accordingly, the +data in Fig. 4b below TC were simulated using eq 6 (sum of eq 4 and eq 5, with eq 4 parameters +fixed by the high-temperature data). The scaling parameter (γ) in eq 6 relates magnetization to +PL energy shift. The data are reproduced well using fixed values of β = 1/3, TC = 60 K, and Δ = +127 cm-1 (15.7 meV), with γ as the only adjustable parameter. Relating eqs 5 and 6, these results +indicate an Yb3+ PL energy shift of -2.2 meV/µB during spontaneous CrI3 intralayer + + +13 +magnetization. We stress that the zero-field PL data in Fig. 4 are not magnetic data, but highlight +the strong influence of CrI3 spontaneous magnetization on the Yb3+ PL. Because TC in these +samples is indistinguishable from that of bulk CrI3 (Figs. 3f, S15), we tentatively attribute the +small apparent broadening of the PL energy discontinuity around TC in Fig. 4b to additional PL +hot bands that are not spectrally resolved. +!(!) = !! + +!! +!! !!!!! + ! − +!!!! +!! +! + + +T < TC +(6) + + +Figure 4. (a) False-color plot of the Yb3+ PL intensities vs temperature measured +for 4.9% Yb3+:CrI3 from 4 to 150 K at zero external magnetic field. The +horizontal dashed line indicates TC = 61 K. (b) Peak position of the Γ8 ! Γ7 +transition plotted vs temperature. The solid red curve shows the behavior +predicted from the combination of resonant phonon interactions (eq 4) and +spontaneous magnetization (below TC, eq 6). The dashed red curve shows the +behavior predicted from eq 4 alone below TC. The solid curve was obtained using +eqs 4 and 6 with fixed parameters of Δ = 127 cm-1 (15.7 meV), TC = 60 K, and β = +1/3, adjusting only the amplitude scaling. (c) Plot of the Γ8 ! Γ7 PL linewidth vs +temperature, from the same VTPL measurements. + + + +a +b +150 +1124 +(meV) +6 +Peak +1.0 +125- +1122 +Position + Position +4 +Peak Position +1120. +0.8 +2 +100. +Temperature (K) +Peak +PL +(meV) +1118 +. Intensity ( +0.6 +0 +1116 +75 +C +9. +FWHM (meV) +8 +50. +0.2 +Peak FWHM +7 +6. +25. +-0.0 +5. +5 . +4 +1.181.161.141.121.101.08 +0 +20 +40 +60 +80 +100 +120 +Energy (ev) +Temperature (K) +14 +Figure 4c plots the temperature dependence of the Γ8 ! Γ7 linewidth (full-width-at-half- +maximum, FWHM). These data show similar trends as observed in the peak energies of Fig. 4b. +Below TC, the FWHM decreases from ~9 meV to ~4.5 meV in the low-temperature limit, +attributed to the reduction in spin disorder around Yb3+. These data thus also reflect spontaneous +magnetic ordering in monolayers of CrI3. Although distinct low-energy shoulders are not +resolved in these data, we hypothesize that the energy and linewidth changes below TC both +ultimately stem from loss of hot-magnon sideband intensity as CrI3 monolayers order +magnetically.45 It will be an interesting future direction to explore magnon coupling to f-f +transitions in these and related doped 2D magnetic materials. +In summary, doping Yb3+ into the 2D van der Waals ferromagnet CrI3 transforms this +material's PL from broad-band to sharp multi-line, while retaining its key magnetic functionality. +The f-f PL of Yb3+:CrI3 is anomalously low in energy, reflecting relatively covalent Yb3+-I- +bonding. Yb3+ magnetization is pinned to CrI3 by strong superexchange interactions, which +contribute an effective internal field of ~10 T that is greater than the field required for magnetic +saturation of paramagnetic Yb3+ and much greater than the field required for full CrI3 +magnetization at low temperature (~0.2 T). Flipping the magnetization of CrI3 with a small +external field thus also flips the Yb3+ magnetization and inverts its PL circular polarization. +Magnetic pinning is maintained up to the TC of CrI3, but is rapidly lost above TC. We further +showed that the Yb3+ PL energy and linewidth both sense this internal field even at zero applied +field, mapping spontaneous intralayer magnetic ordering below TC despite the absence of net +magnetization. Because each Yb3+ ion is a local lattice defect within an individual CrI3 +monolayer, we expect these induced functionalities to persist down to the monolayer, prompting +future studies on exfoliated Yb3+:CrI3 and associated stacked van der Waals heterostructures and + + +15 +layered devices. These results demonstrate the power of designer defects to add functionality to +2D magnetic materials, enrich their fundamental physics, and create new materials of potential +utility for future spin-photonics applications. + +Acknowledgments. Support of this project by the US NSF (DMR-1807394) is gratefully +acknowledged. Initial stages of this work were performed as part of Programmable Quantum +Materials, an Energy Frontier Research Center funded by the U.S. Department of Energy (DOE), +Office of Science, Basic Energy Sciences (BES), under award DESC0019443. Additional +support was received from the UW Clean Energy Institute (graduate fellowships to T.J.S. and +K.M.W.). Part of this work was conducted at the Molecular Analysis Facility, a National +Nanotechnology Coordinated Infrastructure (NNCI) site at the University of Washington that is +supported in part by the National Science Foundation (NNCI-1542101 and NNCI-2025489), the +University of Washington, the Molecular Engineering & Sciences Institute, the Clean Energy +Institute, and the National Institutes of Health. The authors thank Dr. Werner Kaminsky and +Paige M. Gannon for single-crystal XRD measurements, Dr. Xi Wang for assistance with optical +microscope measurements, Prof. Jiun-Haw Chu and Dr. Zhaoyu Liu for VSM measurements, +and Prof. Robert Glaum, Maximilian Jähnig, and Julia Spitz for provision of and assistance with +the BonnMag code. + + +Author Information + +Corresponding Author +Daniel R. Gamelin - Department of Chemistry, University of Washington, +Seattle, Washington 98195-1700, United States; orcid.org/0000-0003-2888-9916; +Email: gamelin@chem.washington.edu + + +Authors +Kimo Pressler - Department of Chemistry, University of Washington, Seattle, +Washington 98195-1700, United States; orcid.org/0000-0003-2788-1592 +Thom J. Snoeren - Department of Chemistry, University of Washington, Seattle, +Washington 98195-1700, United States; orcid.org/0000-0001-8055-3710 +Kelly M. Walsh - Department of Chemistry, University of Washington, Seattle, +Washington 98195-1700, United States; orcid.org/0000-0001-5349-8816 + + + +Supporting Information + + +16 +The Supporting Information is available free of charge at https://pubs.acs.org/doi/XXXX +Additional experimental details, including about sample preparation and characterization. +Additional variable-temperature PL data, PL polarization vs magnetic field data, +excitation-power-dependence data, results from Yb3+ crystal-field calculations, and +comparison of Yb3+ crystal-field barycenter energies in various lattices (PDF). + + +References +(1) Cox, P. A., Electronic Structure and Chemistry of Solids. Oxford University Press: Oxford, +1987. +(2) Henderson, B.; Imbusch, G. F., Optical Spectroscopy of Inorganic Solids. Oxford University +Press: Oxford, 1989. +(3) Kittel, C., Introduction to Solid State Physics. 8 ed.; Wiley: New York, 2004. +(4) Bassett, L. C.; Alkauskas, A.; Exarhos, A. L.; Fu, K.-M. 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S., Spectroscopic studies of the two-dimensional magnetic +insulators chromium trichloride and chromium tribromide—II. J. Phys. Chem. Solids 1979, +40, 149-173. + + + + + + + + + + + +19 +Table of Contents Graphic + + + + + + +q+ +q +g+ +++++个 ++++++ +Yb3+:Crl3 +Supporting Information for + + +Magnetic Amplification at Yb3+ "Designer Defects" +in the van der Waals Ferromagnet, CrI3 + +Kimo Pressler, Thom J. Snoeren, Kelly M. Walsh, Daniel R. Gamelin* +Department of Chemistry, University of Washington, Seattle, WA 98195, United States +Email: gamelin@uw.edu + +Experimental Methods +General Considerations. All sample preparation and manipulation was performed in a +glovebox under an atmosphere of purified dinitrogen. +Chemicals. Chromium metal powder (200 mesh, 99.94%, lot X15E028) was purchased from +Alfa Aesar. According to the manufacturer's certificate of analysis, the majority of the impurity +in this sample lot was Ni at 343 ppm. A chromium chip (99.995%, lot MKCH4484) was also +purchased from Sigma Aldrich as a higher-purity Cr source. The Cr chip was ground to a powder +using a mortar and pestle and used in an analogous manner as the powder precursor. I2 +(≥99.99%) was purchased from Sigma Aldrich. Ytterbium metal powder 40 mesh (99.9%) was +purchased from BeanTown chemical. All chemicals were used as received without further +purification. +Synthesis of CrI3 and Yb3+-Doped CrI3 Single Crystals. Single crystals of the doped and +undoped CrI3 were grown by chemical vapor transport in a manner similar to that described in +previous literature reports.1 For undoped CrI3, Cr(0) metal and I2 were loaded as a 1:3 +stochiometric ratio into a quartz tube and sealed under an evacuated atmosphere. For Yb3+-doped +CrI3, additional Yb(0) metal was loaded along with the other starting materials. The quartz tubes +were 15 cm long with inner and outer diameters of 14 and 16 mm, respectively. Sealed tubes +were placed in an open-ended horizontal tube furnace with the starting materials in the hot zone +set at 650 ˚C and the other end at a temperature of ca. 500 ˚C. Samples were heated for 5 days +and then allowed to slowly cool to room temperature. Once cooled, the tubes were brought into a +glove box and cracked open to yield shiny dark plate-like crystals that had formed at the cold end +of the quartz tube. Elemental analysis of the Yb3+-doped samples was performed by inductively +coupled plasma mass spectrometry (ICP-MS) using a PerkinElmer NexION 2000B. Samples +were prepared by digesting single crystals in concentrated nitric acid with sonication and then +further diluted in ultrapure H2O. Yb3+ doping levels are reported as cation mole fraction, +[Yb3+]/([Cr3+]+[Yb3+]), in percentage, with an estimated uncertainty of ±0.1%. Crystal thickness +was measured by mounting a representative flake to a glass slide using double-sided tape and +imaging the flake with an optical microscope in a glovebox at various magnifications. The edge +length was calculated in ImageJ2 using known pixel resolutions. +X-ray Diffraction (XRD) Characterization. Samples were prepared for XRD on the +powder diffractometer by placing single crystals onto silicon substrates and sealing under Kapton +films to reduce exposure to air. Data were collected using a Bruker D8 Discover powder +diffractometer with a high-efficiency IµS microfocus x-ray source for Cu Kα radiation (50 kV, 1 +mA). For single-crystal XRD, a crystal measuring 0.10 x 0.05 x 0.01 mm3 was mounted on a +loop with oil. Data were collected at 263 K on a Bruker APEX II single-crystal X-ray +diffractometer using Mo-radiation, equipped with a Miracol X-ray optical collimator. The data +were integrated and scaled using SAINT, SADABS within the APEX2 software package by + + +S-2 +Bruker.3 Solution by direct methods (SHELXT4, 5 or SIR976, 7) produced a complete heavy-atom +phasing model consistent with the proposed structure. The structure was completed by difference +Fourier synthesis with SHELXL.8, 9 Scattering factors are from Waasmair and Kirfel.10 All atoms +were refined anisotropically by full-matrix least-squares. +Including intrinsic disorder, a least squares refinement optimization of the data yields the +lattice structure that we report. From the 983 reflections collected covering the indices, -8 ≤ h ≤ +8, -14 ≤ k ≤ 14, -8 ≤ l ≤ 8, 518 reflections were found that were symmetry independent and an R1 +value of 0.0521 was obtained, indicating a good fit. R1 is calculated as: +!! = +!!"# − !!"#! +!!"# + +There is no detectable electron density between layers, indicating that Yb3+ does not intercalate +between layers in CrI3. +Variable-Temperature Photoluminescence (VTPL). Samples for VTPL measurements +were prepared by placing a single crystal between two quartz disks and loading into a closed- +cycle helium cryostat. PL spectra were collected by exciting the sample with a continuous-wave +660 nm (1.88 eV) diode at 4 mW/cm2. Emission was collected and focused into a +monochromator with a spectral bandwidth of 0.627 nm and detected by a Hamamatsu +InGaAs/InP NIR photomultiplier tube, with signal recorded using a photon counter. Temperature +was varied from 4 to 300 K, starting at low temperature. All spectra were corrected for +instrument response. +Magnetic Circularly Polarized Luminescence (MCPL). Samples for MCPL measurements +were prepared as single crystals placed between two quartz disks and loaded into a +superconducting magneto-optical cryostat (Cryo-Industries SMC-1659 OVT) oriented in the +Faraday configuration. For full-spectrum measurements at static fields, samples were excited +with a 660 nm (1.88 eV) diode at approximately 40 mW/cm2. For field-sweep measurements, +samples were excited with a linearly polarized HeNe laser (632.8 nm/1.96 eV, 27 mW/cm2 for -6 +to +6 T scans, 55 mW/cm2 for -0.4 to +0.4 T scans). No distinguishable difference was found in +the either the PL spectra or variable-field data between the two excitation sources. For field- +sweep measurements, the monochromator was centered at 1.117 eV with a 6 nm spectral +bandwidth, and the signal was continuously monitored as the field was swept at a rate of 0.10 +T/min and 0.45 T/min for the 0.4 T and 6 T scans, respectively. PL was collected along the +magnetic-field axis and passed through a liquid-crystal variable retardation plate set at λ/4, +followed by a linear polarizer to separate the left- and right-circularly polarized components. The +PL was then focused into a fiber-optic cable and fed into a monochromator with a spectral +bandwidth of 0.627 nm and detected by a Hamamatsu InGaAs/InP NIR photomultiplier, with +signals recorded using a photon counter. Polarization ratios are defined as ρ = (σ- – σ+)/(σ- + σ+) += (IL – IR)/(IL + IR) = ΔI/I, following the sign conventions outlined in Piepho and Schatz.11 +Magnetic Measurements. Magnetic data on individual single-crystal flakes (Fig. 1) were +collected using a Quantum Design PPMS DynaCool vibrating sample magnetometer (VSM). A +flake was affixed to the end of a quartz paddle with varnish (VGE 7031). The paddle was then +snapped into the VSM brass sample holder with another quartz paddle placed symmetrically +above the sample. The weak background signal from the sample holder was removed in the data +analysis. The sample was probed with the external field aligned perpendicular to the face of the +crystal, and magnetization data were collected as a function of applied field and temperature. The +masses of individual flakes are below 0.1 mg and could not be accurately measured, so the +magnetic data are reported in units of emu. + + +S-3 +Ligand-field calculations within the Angular Overlap Model (AOM). Yb3+ +ligand(crystal)-field energies and g factors were calculated using the BonnMag package.12 +Crystallographic data13 on CrI3 were used to create an [YbI6]3- unit with reduced symmetry +(point group C2). Crystallographic parameters were not adjusted for size differences between +Cr3+ and Yb3+. The electronic structure of Yb3+ was calculated using the spin-orbit coupling +parameter ζ as well as AOM parameters eσ and eπ to describe σ and π interactions with the I- +ligands, respectively. The value for eπ was taken to be isotropic. The Slater-Condon-Shortley +(SCS) parameters F2, F4, and F6 were taken to be 0, as is typically the case for Yb3+ (4f13 +configuration). The Stevens orbital reduction factor k was taken to be equal to 1.0. Increasing +(decreasing) ζ while keeping all other parameters constant results in an increase (decrease) in all +transition energies while retaining peak splitting energies. Adjusting eσ or eπ alters the relative +energies of the peaks but maintains the barycenters. + + + + + +Figure S1. Images of an individual Yb3+:CrI3 single-crystal flake under an optical microscope at +various magnification levels, viewing the flake's (a,b) edge, and (c) face. The flake thickness is +estimated to be 5.1 ± 0.3 µm. + +Side +View! +Yb3+:CrI3! +Tape! +Side +View! +Top +View! +a! +b! +c! + +100μm10 μm100 μm +S-4 + +Figure S2. Analysis of XRD reflections collected using a powder diffractometer for 4.9% Yb3+- +doped and undoped CrI3 single-crystal flakes (same data as shown in Fig. 1bc). Using the +method described by Jesche,14 the lattice parameter c for oriented single crystals with a +monoclinic space group can be extracted from XRD data from a powder diffractometer using the +following equation: +2c ∙ sinβ ∙ sin θ − S cosθ +2 += λℓ +Here, β is the obtuse angle in the monoclinic unit cell (108.507° for CrI3), λ is the x-ray +wavelength (Cu, 1.5406 Å), ℓ is the Miller index of each reflection in the XRD spectrum and +S +!"#! +! is a correction factor related to the displacement of the x-ray focal plane relative to the +sample surface. Plotting 2θ values of the peak maxima vs λℓ, the data can be fit using the +equation above. For fitting, β and θ were taken in radians. By this method, the c lattice +parameters were found to be 6.996 ± 0.002 and 7.013 ± 0.002 Å for the undoped and doped +samples, respectively. From the lattice parameter c, the position of the (00ℓ) powder +diffractometer XRD peaks for a monoclinic single crystal can be calculated using the following +equation: +2θ = 2sin!! +λ +2sinβ +ℓ +c +The zero-shift in 2θ was determined by adding an offset to the experimental 2θ values and +adjusting the offset to minimize the difference between experimental and calculated peak +positions across all peaks in the XRD spectrum. This offset accounts for the measurement +discrepancy due to the thickness of the single crystals displacing the x-ray focal plane. For CrI3, +a zero-shift of -0.015° was found, contrasted to a zero-shift of +0.164° for Yb3+-doped CrI3. The +displacement-corrected XRD spectra are shown in Fig. 1c in the main text. + + + + +10 +8 +6 +4 +2 +0 +λℓ (Å) +90 +80 +70 +60 +50 +40 +30 +20 +10 +2θ (deg.) +CrI3 +Yb +3+:CrI3 + + +S-5 + +Table S1. Single-crystal X-ray diffraction data for 2.5% Yb3+:CrI3 measured at 263 K, +compared to literature data for CrI3. + + +Yb3+:CrI3 +CrI3 (250 K, ref. 13) +Space group +C2/m +C2/m + + + +a +6.86 Å +6.87 Å +b +11.89 Å +11.89 Å +c +6.99 Å +6.98 Å +α +90.0° +90.0° +β +108.7° +108.5° +γ +90.0° +90.0° + + + +[(Yb/Cr) – Cr]avg +3.96 Å +3.96 Å +[(Yb/Cr) – I]avg +2.72 Å +2.72 Å +[(Yb/Cr) – I – (Yb/Cr)]avg +93.3° +93.6° +[I – (Yb/Cr) – I]avg +86.8° +86.9° + + + + +Figure S3. Visualization of the experimental room-temperature single-crystal XRD structure as +viewed along the a, b, and c principal axes (left to right). Yb3+ (cyan) is found to substitute for +Cr3+ (blue) in the edge-sharing octahedra formed by I- (purple) anions. No excess electron +density is observed between layers. Intralayer disorder is observed. The structure refines to the +expected high-temperature C2/m monoclinic symmetry. Some intralayer disorder was observed +(not shown). + +aa +S-6 +Figure S4. (a) Variable-temperature PL spectra of CrI3 measured from 4 to 200 K under 1.88 eV +CW excitation (from Fig. 2 of the main text). (b) Scatter plot depicting total integrated area of +the CrI3 PL from panel (a). The 200 K intensity is 7.5% that of the 4 K value. (c) Variable- +temperature PL spectra of 4.9% Yb3+:CrI3 measured from 4 to 200 K under 1.88 eV CW +excitation (from Fig. 2 of the main text). (d) Scatter plot depicting total integrated area of the +Yb3+ PL from panel (c). The 200 K intensity is 0.8% that of the 4 K value. (e) Variable- +temperature PL spectra of 5.0% Yb3+:CrI3 measured from 4 to 200 K under 1.88 eV CW +excitation (from Fig. 2 of the main text). (f) Scatter plot depicting total integrated area of the +Yb3+ PL from panel (e). The 200 K intensity is 7.5% that of the 4 K value. Note that a second, +broad "trap" PL band is observed at ~0.98 eV in samples made from Cr metal powder precursor +(99.94%, panel (c)) but not in samples made from Cr chip precursor (99.995%, panel (e)). Ni is +the primary impurity in the powder precursor (see Methods), and Ni is detected in this CrI3 +sample at 0.4% cation mole fraction. Ni2+ 3A2g ! 3T2g absorption in NiI2 and Ni2+:CdI2 is +centered around 0.93 eV,15 and the broad "trap" PL band in panel (c) is thus tentatively attributed +to Ni2+ impurities in CrI3. + +a +b +1.0 +Crl3 Undoped +Integrated Area (norm.) +1.0 +Crl3 +PL Intensity (norm.) +4 - 200 K +0.89 - 1.24 eV +0.8 +Cr3+ +0.8 +0.6 +0.6 +0.4 +0.4 +0.2 +0.2 +0.0 +0.0- +1.2 +1.1 +1.0 +0.9 +0 +50 +100 +150 +200 +Energy (eV) +Temperature (K) +c +d +1.0 +Crl3 +Integrated Area (norm.) +1.0 +(wuou) / +0.8 +Yb 3+ +4 - 200 K +1.07 - 1.2 eV +0.8 +_ Intensity +0.6. +0.6- +Trap +0.4 - +0.4 +0.2 +0.2- +0.0 +0.0- +1.2 +1.1 +1.0 +0.9 +T +0 +50 +100 +150 +200 +Energy (eV) +h +e +Temperature (K) +1.0 +1.0 +Yb° ++ :Crl3 +Integrated Area (norm.) + Intensity (norm.) +0.8 +4 - 200 K +0.8 +1.07 - 1.2 eV +3+ +h +0.6 +0.6 +0.4. +0.4 +0.2 +0.2 +0.0 +0.0. +1.0 +0 +50 +100 +1.2 +1.1 +0.9 +150 +200 +Energy (eV) +Temperature (K) +S-7 + +Figure S5. Comparison of the 5 K experimental data and calculated (AOM) f-f PL transition +energies for 4.9% Yb3+:CrI3. A best fit to the experimental PL data resulted in the following +values: ζ = 2665 cm-1 (330.4 meV), eσ = 176.5 cm-1 (21.9 meV), eπ = 122.5 cm-1 (15.2 meV). The +calculated transition energies using these parameters are shown as the vertical red lines in both +panels. (a) Comparison of calculated transition energies obtained by changing from ζ = 2665 cm- +1 (red) to ζ = 2675 cm-1 (blue), with all other parameters constant to the best-fit (red). (b) +Comparison of calculated transition energies obtained by individually changing the values of eσ +and eπ. The gray traces show the effect of changing from eσ = 176.5 cm-1 (red) to eσ = 206.5 cm-1 +with all other parameters constant to the best fit (red). The green traces show the effect of +changing from eπ = 122.5 cm-1 (red) to eπ = 152.5 cm-1 with all other parameters constant to the +best fit (red). From the best-fit parameters, g is anisotropic (g1 = 2.672, g2 = 2.686, g3 = 2.642) +and an average ground-state g value of ~2.7 is predicted. + + + + + + +a +b +1.0 +1.0 +{ = 2665 cm + = 2665 cm + Intensity (norm.) +. Intensity (norm.) +0.8 +0.8 - +e。 = 176.5 cm +e = 122.5 cm +e = 122.5 cm +-1 + = 2665 cm +0.6 +0.6 - +- +e. = 206.5 cm +e= 122.5 cm + = 2675 cm +0.4 +0.4 - +{ = 2665 cm +e。 = 176.5 cm +-1 +PL +0.2 +e = 122.5 cm +P +0.2 +=152.5cm +0.0 +0.0 +9400 +9200 +9000 +8800 +9400 +9200 +9000 +8800 +Wavenumber (cm-") +Wavenumber (cm"") +S-8 + + +Figure S6. The Yb3+ valence energy level diagram described by the best-fit parameters of Fig. +S5. The energies of the crystal field states in eV are: 0.0000, (0.0179, 0.0182), 0.0496, (1.1667, +1.1668), 1.2013 eV. + + + + + +Energy (cm"1) +10000 +9689 +(9410, 9411) +8000 +Energy (cm +6000 +4000 +2000 +400 +-0 +(144, 147) +0 +Free lon + +Spin Orbit Coupling + Crystal Field +S-9 +Table S2. Energies (cm-1) of the valence electronic states, 2F5/2 and 2F7/2 barycenter +energies,a and ΔE(Barycenter) for Yb3+ ions in several host crystals, and for the free ion. +These data were used to generate Fig. S7 (after converting to eV). Many of these entries are +compiled in ref. 16. +Host Lattice +0 +1 +2 +3 +2F7/2 +Barycenter +0' +1' +2' +2F5/2 +Barycenter + +ΔE(Bary) +ref. +Ca2Ga2SiO7 (CGS) +0.0 +300 +490 +970 +440 +10250 +10570 +11010 +10610 +10170 +17 +SrLaGa3O7 (SLG) +0.0 +220 +386 +910 +379 +10190 +10450 +11025 +10555 +10176 +17 +Ca4GdO(BO3)3 +(GdCOB) (site I, Gd) +0.0 +423 +668 +1003 +524 +10246 +10706 +11089 +10680 +10157 +18 +GdCOB (site II, Ca) +0.0 +437 +694 +1003 +534 +10261 +10737 +11150 +10716 +10183 +18 +GdCOB (site III, Ca) +0.0 +417 +688 +1003 +527 +10240 +10682 +11026 +10649 +10122 +18 +Ca4YO(BO3)3 (YCOB) +0.0 +427 +556 +1023 +502 +10242 +10537 +11109 +10629 +10128 +19 +Sc2O3 +0.0 +474 +634 +1076 +546 +10250 +10640 +11198 +10696 +10150 +20 +Ca5(PO4)3F (CFAP) +0.0 +409 +597 +1099 +526 +10178 +10496 +11069 +10581 +10055 +21 +Sr5(PO4)3F (SFAP) +0.0 +362 +618 +1190 +543 +10150 +10512 +11108 +10590 +10048 +22 +Sr5(VO4)3F (SVAP) +0.0 +321 +562 +1078 +490 +10141 +10740 +11050 +10644 +10153 +23 +Y3Al5O12 (YAG) +0.0 +584 +635 +783 +501 +10328 +10752 +10917 +10666 +10165 +24 +BaCaBO3F (BCBF) +0.0 +303 +533 +902 +435 +10204 +10570 +11000 +10591 +10157 +25 +LiNbO3 +0.0 +352 +448 +788 +397 +10204 +10471 +11090 +10588 +10191 +26 +KGd(WO4)2 (KGW) +0.0 +163 +385 +535 +271 +10188 +10471 +10682 +10447 +10176 +27 +KY(WO4)2 (KYW) +0.0 +169 +407 +568 +286 +10187 +10476 +10695 +10453 +10167 +27 +CaWO4 +0.0 +220 +366 +492 +270 +10278 +10366 +10665 +10436 +10167 +28 +YAlO3 +0.0 +209 +341 +590 +285 +10220 +10410 +10730 +10453 +10168 +28 +LiYF4 +0.0 +216 +371 +479 +267 +10288 +10409 +10547 +10415 +10148 +28 +YAl3(BO3)4 (YAB) +0.0 +94 +185 +581 +215 +10194 +10277 +10672 +10381 +10166 +29 +Cs2NaYbCl6 +0 +225 +225 +573 +256 +10243 +10243 +10708 +10398 +10142 +30, 31 +Cs3Yb2Br9 +0.0 +144 +201 +421 +192 +10277 +10301 +10578 +10385 +10194 +32 +CsCdBr3 +0.0 +114 +140 +441 +174 +10119 +10146 +10590 +10285 +10111 +32 +CuInS2 +0.0 +32 +87 +182 +75 +10033 +10060 +--- +10095a +10020 +33 +InP +0 +35.5 +35.5 +97.5 +42 +10018 +10064 +10064 +10049 +10007 +34 +Free ion +--- +--- +--- +--- +0.0 +--- +--- +--- +10213 +10213 +35 +CrI3 +0.0 +146 +146 +400 +173 +9410 + +--- +9551a +9379 +this +work +aFor the entire data set of complete entries, the ratio of 2F5/2:2F7/2 CF splitting energies, (E(2F5/2 Barycenter) - +E0')/(E(2F7/2 Barycenter)) is 0.82 ± 0.14. The 2F5/2 barycenter energies for Yb3+:CrI3 and Yb3+:CuInS2 were thus set +equal to the 2F7/2 barycenter energies for the same compounds. The resulting uncertainties in ΔE(Bary) are estimated +to be < ~1%, close to or smaller than the data points in Fig. S7. For comparison, the Yb3+:CrI3 AOM calculations +above yield: 2F7/2 barycenter = 173 cm-1 (21 meV), 2F5/2 barycenter = 9503 cm-1 (1.178 eV), ΔΕ(Bary) = 9330 cm-1 +(1.157 eV), within this uncertainty range. + + + + + + + + + +S-10 + +Figure S7. Plot of the difference between experimental Yb3+ 2F5/2 and 2F7/2 barycenter energies +(ΔE(Bary)) for the compounds listed in Table S2, and for the free ion, vs the barycenter energy +for the 2F7/2 ground multiplet. The compounds associated with select data points are labeled. The +dashed blue line shows the value of the free ion. + + +Figure S8. (a) Power dependence of !- (red) and !+ (black) PL peak intensities and circular +polarization (ρ, blue) of the Γ8 ! Γ7 transition. The data were collected at 0.5 T and 5 K and the +sample was excited with linearly polarized light at 1.96 eV. The PL intensities show a linear +increase with power, resulting in a constant polarization ratio. The error bars represent +uncertainty estimated from the linear fit of the polarization intensities. (b) The !- (red) and !+ +(black) component of the Γ8 ! Γ7 transition normalized across all powers. The traces overlay +each other well, showing no detectable power dependence. + + +b +a +0.20 +, Signal Intensity (norm.) +1.0 +1.0 +a- +Polarization Ratio ( +Intensity (norm.) +0.8 +0.8 +a+ +0.15 +a+ +0.6 +0.6 +.0.10 +0.4. +0.4 +0.05 +0.2 +0.2 +0.0- +-0.00 +0.0 - +0 +100 +200 +300 +400 +1.120 +1.116 +1.112 +Energy (eV) +Power Density (mW/cm3)KGW +1.28 +CaWO4 +Free lon +Cs3Yb2Brg +LiYF4_ KYW +1.26- +QOYAIO3 +YAB +CsCdBr3 +Cs2NaYbCl6 +Oxides +00 +△E(Bary) (eV) +1.24- +CulnS2 +InP +1.22- +1.20 - +1.18- +1.16- +0 +10 +20 +30 +40 +50 +60 +70 +2F7/2 Barycenter (meV) +S-11 + +Figure S9. Comparison of full MCPL spectra across two different samples, measured at 0.5 T, 5 +K. (a) The sample used in Fig. 3b,c,e,f of the main text. (b) The sample used in Fig. 3d of the +main text. The two samples show very similar spectra, with slight differences in polarization +magnitude. + +Figure S10. Magnetic data for a single-crystal flake of 5% Yb3+:CrI3, measured by VSM. The +sample was probed with the external field aligned perpendicular to the face of the crystal. (a) +Plots of magnetization vs external field measured at various temperatures. The data are similar to +those collected on undoped CrI3 bulk crystals (e.g., Fig S11). At 2 K, a coercive field of ~44 mT +was found. (b) Plot of magnetization vs temperature measured in the field-cooled and field- +warmed directions. The inset shows the derivative of the field-cooled data as a function of +temperature, where the Curie temperature is found to be 60.4 K. These data show that Yb3+ +doping has no significant effect on the magnetism of CrI3 in these samples. + + +a +b +1.0 +1.0. +0.5 T +0.5 T + Intensity (norm.) +5 K + Intensity (norm.) +5 K +0.8 +0.8 - +a+ +a+ +0.6 - +0.6 - +0.4 - +0.4 - +0.2 +0.2 - +P +0.0 - +0.0 - +1.18 +1.16 +1.14 +1.12 +1.10 +1.08 +1.18 +1.16 +1.14 +1.12 +1.10 +1.08 +Energy (eV) +Energy (eV)a +b +(emu) +0.008 +Magnetic Moment (emu) +0.02 T +0.0008 +Magnetic Moment +0.004 +0.0006 +dM/dT +0.000 +Field Warmed +2 K +0.0004 +Tc = 60.4 K +Field Cooled +20 K +0.004 +40 K +60 K +0.0002 +0 +50 +100 +80 K +T(K) +0.008 +100 K +-3 +-2 +-1 +0 +1 +2 +3 +20 +40 +60 +80 +100 +Field (T) +Temperature(K) +S-12 + +Figure S11. The same polarization data as featured in Fig. 3d of the main text, overlayed with +CrI3 magnetization data measured from -3 to +3 T with the field oriented parallel to the +crystallographic c axis (blue) by single-crystal vibrating sample magnetometry (VSM).36 For +comparison, the magnetization perpendicular to c (green) is also shown. The Yb3+ MCPL +polarization ρ is superimposable with the CrI3 magnetization measured in the same +configuration. + + + +Figure S12. (a) Individual circularly polarized MCPL components measured during continuous +field sweeps from -6 to +6 T and back at 5 K. (b) The same data, displayed as the polarization +ratio (ρ, normalized). Panel (b) is shown as Fig. 3d of the main text. Data measured using 14 +mW/cm2 excitation. + + + +b +a +9000 +1.0 - +5 K + Intensity (counts) +8000 +(norm.) +0.5 +g+ +7000 +0.0 +2 +6000- +6 +-0.5 +p +P +5000 +-1.0 +T +9- +-4 +-2 +0 +2 +4 +6 +-6 +-4 +-2 +0 +2 +4 +6 +Field (T) +Field (T)1 +Crls Single +1.0- +1.0 +Magnetization (norm.) +Crystal VSM +p 1.117 ev (norm.) +0.5- +0.5 +BIl c BIla,b +0.0. +0.0 +-0.5- +-0.5 +-1.0- +-1.0 +9- +-2 +0 +2 +4 +6 +Field (T) +S-13 + + +Figure S13. Comparison of field-dependent polarization ratios (ρ, normalized) measured with +(a) linearly polarized and (b) unpolarized excitation at 5 K. In panel (b), no data were collected +above 2 T. Panel (a) is shown as Fig. 3d of the main text. + + + +Figure S14. (a,b) Effect of excitation power on the polarization ratio (ρ, normalized). Magnetic +hystereses measured under (a) low- and (b) higher-power excitation (14 vs 55 mW/cm2, 5 K) +show no difference. The black (red) trace corresponds to the sweep from negative (positive) to +positive (negative) fields. (c, d) The separate circularly polarized PL components from the same +(c) low- and (d) high-power measurements. + + +a +b +1.0 +5 K +1.0 +5 K +(wou) +('wuou) +0.5 +0.5 +1.117 ev +0.0 +1.117 eV +0.0 +-0.5 +-0.5 +p +p +-1.0 +-1.0- +-6 +-4 +-2 +0 +2 +4 +6 +-2 +-1 +0 +1 +2 +Field (T) +Field (T)a +b +1.0 - +1.0 - +55 mW/cm +2 +14 mW/cm +0.5 +0.5 - +0.0 +0.0 +-0.5 - +-0.5- +Q +-1.0- +-- +-1.0 +-0.4 +-0.2 +0.0 +0.2 +0.4 +-0.4 +-0.2 +0.0 +0.2 +0.4 +Field (T) +Field (T) +c +d +PL Intensity (counts) +PL Intensity (counts) +120 x10 +30 x10 +115- +g+ +28 +9+ +55 mW/cm +14 mW/cm +110- +105- +26 +100 - +24 - +95 +-0.4 +-0.2 +0.0 +0.2 +0.4 +-0.4 +-0.2 +0.0 +0.2 +0.4 +Field (T) +Field (T) +S-14 + +Figure S15. Temperature dependence of the Γ8 ! Γ7 PL feature of 4.9% Yb3+:CrI3 measured +from 4 to 200 K under no external magnetic field (from Fig. 4 of the main text, T = 4, 15, 30, 40, +50, 55, 58, 60, 62, 65, 70, 85, 100, 125, 150 K). A linear baseline was subtracted from each +spectrum here to facilitate viewing and determination of the peak's FWHM. + + +1.0- +Baseline Subtracted +PL Intensity (norm.) +0.8 - +0.6- +0.4 - +0.2 +0.0 +1.135 +1.130 +1.125 +1.120 +1.115 +1.110 +Energy (eV) +S-15 + + +Figure S16. (a,b) False-color plots of the Yb3+ PL intensities vs temperature measured for the +two samples shown in Fig. S4c,e, respectively, from 4 to 150 K at zero external magnetic field. +The horizontal dashed line indicates TC = 61 K. The two samples show the same temperature +dependence, but the features are slightly better resolved in panel (a). Panel (a) is shown as Fig. +4a of the main text. + + + +References +(1) Seyler, K. L.; Zhong, D.; Klein, D. R.; Gao, S.; Zhang, X.; Huang, B.; Navarro-Moratalla, +E.; Yang, L.; Cobden, D. H.; McGuire, M. A.; Yao, W.; Xiao, D.; Jarillo-Herrero, P.; Xu, X. +Ligand-field helical luminescence in a 2D ferromagnetic insulator. Nat. 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Nano Lett. 2020, 20 (3), 2100-2106. + + diff --git a/3dFKT4oBgHgl3EQf8i5H/content/tmp_files/load_file.txt b/3dFKT4oBgHgl3EQf8i5H/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a80e0dad3fb4fa4d7d81474f127fcd2e091d4336 --- /dev/null +++ b/3dFKT4oBgHgl3EQf8i5H/content/tmp_files/load_file.txt @@ -0,0 +1,2282 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf,len=2281 +page_content='1 Magnetic Amplification at Yb3+ "Designer Defects" in the van der Waals Ferromagnet, CrI3 Kimo Pressler, Thom J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Snoeren, Kelly M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Walsh, Daniel R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Gamelin* Department of Chemistry, University of Washington, Seattle, WA 98195, United States Email: gamelin@uw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='edu Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' The two-dimensional (2D) van der Waals ferromagnet CrI3 has been doped with the magnetic optical impurity Yb3+ to yield materials that display sharp multi-line Yb3+ photoluminescence (PL) controlled by the magnetism of CrI3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Magneto-PL shows that Yb3+ magnetization is pinned to the magnetization of CrI3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' An effective internal field of ~10 T at Yb3+ is estimated, attributed to strong in-plane Yb3+-Cr3+ superexchange coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' The anomalously low energy of Yb3+ PL in CrI3 reflects relatively high Yb3+-I- covalency, contributing to Yb3+- Cr3+ superexchange coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' The Yb3+ PL energy and linewidth both reveal the effects of spontaneous zero-field CrI3 magnetic ordering within 2D layers below TC, despite the absence of net magnetization in multilayer samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' These results illustrate the use of optical impurities as "designer defects" to introduce unique functionality to 2D magnets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Keywords: 2D Ferromagnet, Lanthanide Doping, Molecular Field, Chromium Triiodide, Photoluminescence Defects have the power to transform the physical properties of crystals, imparting new and potentially useful functionalities from conductivity to quantum photon emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='1-6 In magnetic materials, defects can strongly affect spin-wave propagation, magnetic domain-wall propagation, skyrmion dynamics, and magnetic vortex pinning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='7-9 Recently, the layered van der Waals ferromagnet CrI3 has emerged as a promising platform for exploring strongly correlated spin physics, magnetic proximity effects, and next-generation spin-based device architectures in the 2 two-dimensional (2D) limit,10-14 but the potential to expand CrI3 functionality through introduction of defects remains untapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Here, we report that doping CrI3 with Yb3+ as a "designer point defect" transforms its normally broad and featureless d-d photoluminescence (PL) into narrow-line sensitized f-f emission, without compromising its attractive magnetic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' We further show that Yb3+ in CrI3 experiences a large internal effective field that makes it extremely sensitive to small external magnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Using this property, we demonstrate magnetically saturated circular polarization of Yb3+ emission at anomalously small applied fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Strikingly, the internal effective field also transmits magnetic information to Yb3+ even in the absence of any applied field, making Yb3+ a unique embedded luminescent probe of spontaneous zero-field magnetic ordering within the 2D monolayers of bulk CrI3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' These discoveries establish optical impurity doping as an effective strategy for expanding the functionality of 2D magnets, with potential ramifications for both basic science and future spin- photonic technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' CrI3 has become a model system for exploring magnetic exchange in 2D van der Waals structures,10-14 stimulated by recent discoveries of Ising-like hard ferromagnetism in exfoliated monolayer CrI3 and layer- and stacking-dependent magnetism in multi-layer CrI3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='15,16 Layering CrI3 with non-magnetic 2D materials introduces magnetic functionality to the non-magnetic material via inter-layer exchange coupling, allowing magnetic manipulation of properties such as WSe2 valley polarization and valley Zeeman splittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='17 Extension from few to many (bulk) layers preserves the strong Ising-like intralayer ferromagnetic ordering, but facile motion of domain walls unblocks demagnetization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='18 Despite its rich magnetic properties, CrI3 itself has not garnered much attention as an optical material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Bulk CrI3 has been investigated for its very large Kerr and Faraday rotation strengths in relation to optical isolators and associated 3 technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='19,20 PL of bulk CrI3 has apparently not been reported, and few-layer CrI3 shows17 only the very broad d-d PL characteristic of weak-field pseudo-octahedral Cr3+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content="21 Circular polarization of this d-d PL was used to probe the magnetism of few-layer CrI3,17 but the emission's breadth limits its further utility for fundamental studies or in spin-photonics, stimulating efforts to narrow the band via cavity coupling." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='22 Doping CrI3 with optically active impurities has also not been reported, either in bulk or exfoliated samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' To investigate intralayer "proximity" effects resulting from magnetic exchange coupling, we have prepared CrI3 doped with luminescent and spin-bearing Yb3+ ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Large-diameter single- crystal flakes of CrI3 were prepared by chemical vapor transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Yb3+ was introduced by adding Yb(0) to the precursor mix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' The Yb3+ concentration in the resulting Yb3+:CrI3 crystals is controllable, and samples with up to ~5% Yb3+ (cation mole fraction, [Yb3+]/([Cr3+]+[Yb3+])) are described here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Further experimental details are provided in the Supporting Information (SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Figure 1a shows a photograph of representative Yb3+:CrI3 flakes in their growth tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' The flakes are between 5 and 10 mm across, with typical thicknesses of 5-20 µm (see SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Figure 1b plots XRD data collected on undoped and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='9% Yb3+-doped CrI3 single-crystal flakes using a powder diffractometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=" Only (00l) peaks are observed, corresponding to the interlayer lattice spacing and reflecting the flake's alignment." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Figure 1c highlights the shift to smaller angle of the 001 peak upon doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' From fitting the XRD peak positions of undoped and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='9% Yb3+-doped CrI3 samples, the interlayer lattice parameter was found to increase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='24% from 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='996 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='002 to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='013 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='002 Å, attributed to the larger ionic radius of Yb3+ than Cr3+ (87 vs 62 pm, respectively) (see SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' These data suggest that the local strain of doping is relieved by distorting the lattice along its softest dimension, as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Substitutional incorporation of Yb3+ at the Cr3+ site is verified by single-crystal XRD measurements (see SI), which also show the increased 4 interlayer spacing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' The single-crystal data show no detectable electron density between layers, ruling out Yb3+ intercalation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' (a) Photograph of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='9% Yb3+:CrI3 crystals prepared by chemical vapor transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' The scale bar shows 5 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' All experiments were performed on individual single-crystal flakes from such a reaction tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' (b) XRD data collected on undoped and Yb3+-doped CrI3 single crystals using a powder diffractometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Only (00l) peaks are observed, indicating an oriented sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Reference peaks for c-oriented CrI3 diffraction are included (black, ICSD Coll.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Code 251654).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' (c) Magnified view of the 001 reflection for the same samples, displaying an increase in the interlayer lattice spacing upon Yb3+ doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' The x axis in (c) was determined as described in the SI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Figure 2a plots the PL spectra of CrI3 and Yb3+:CrI3 single flakes measured at several temperatures between 4 and 200 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' The CrI3 spectrum broadens and decreases in intensity with increasing temperature, eventually reaching only 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='5% of its 4 K intensity at 200 K (see SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Although the broadening to higher energies is expected from thermal hot bands, the broadening to lower energies is abnormal and suggests an additional feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Upon introduction of Yb3+, the broad featureless d-d emission of Cr3+ disappears and is replaced by a series of sharp f-f transitions of Yb3+ around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='15 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Assignment of the PL fine structure is discussed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' In some samples, Yb3+ doping also reveals another broad emission band centered at ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='95 eV, which is responsible for the red tail of the CrI3 PL here and in some literature spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' This a b C Yb3+: Crl3 Intensity (rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=') Intensity (rel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=') Crl3 001 002 ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' 001 003 004 005 006 I x10 [x10 20 40 60 80 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='0 2θ (deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=') 20 (deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=') 5 feature has been traced to Ni2+ impurities (<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='4%) found in some Cr(0) precursors, and it can be mostly eliminated by using 5N Cr(0) precursors (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' 2a, bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' The Yb3+ PL is not influenced by this Ni2+ impurity (see SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' (a) Variable-temperature PL spectra of CrI3 (top) and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='9% Yb3+:CrI3 (bottom), measured from 4 to 200 K under 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='88 eV CW excitation at 4 mW/cm2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' (b) Single-configurational-coordinate diagram (A1g coordinate) describing vibronic broadening of the absorption and luminescence bands associated with transitions between the 4A2g and 4T2g ligand-field states of pseudo-octahedral Cr3+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' In Yb3+-doped CrI3, energy transfer from the Cr3+ 4T2g excited state to Yb3+ yields sensitized 2F5/2 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' 2F7/2 f-f luminescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Figure 2b illustrates the photophysics of Yb3+:CrI3 schematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' The lowest-energy excited state of CrI3 is the Cr3+ 4T2g ligand-field state, involving excitation of a t2g electron into a σ- antibonding eg orbital (in idealized Oh symmetry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' The resulting change in equilibrium geometry is described by the single-configurational-coordinate (SCC) diagram of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' 2b, which illustrates the totally symmetric distortion coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' This 4T2g excited state also distorts along a symmetry-breaking Jahn-Teller coordinate (not illustrated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='21 These distortions lead to extensive b a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='0 Crls Undoped , Intensity (norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=') 4 - 200 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='8 Cr3+ g 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='6 ET 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='4- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='2 Cr3+ Cr3+ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='0 Yb3+:Crl3 Abs PL Intensity (norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=') 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='8 4 - 200 K Yb3+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' A 12g PL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='4- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='2 2 7/2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='9 Energy (eV) 6 vibronic progressions in the absorption and PL spectra associated with this transition, and cause a large PL Stokes shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Doping CrI3 with Yb3+ introduces a set of 2F5/2 states just below the Cr3+ 4T2g excited state, favorably positioned for efficient Cr3+ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Yb3+ energy transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' At 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='9% Yb3+ doping, the Cr3+ 4T2g PL is entirely quenched and strong Yb3+ 2F5/2 emission is observed in its place (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Because both Cr3+ and Yb3+ states are localized at single ions, energy migration within the CrI3 lattice is required for this complete quenching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' In undoped CrI3, energy migration among equivalent Cr3+ sites may occur but is not readily apparent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' In Yb3+:CrI3, this energy migration is interrupted when energy is captured by Yb3+ dopants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' In 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='9% Yb3+:CrI3, the average Cr3+ ion has only ~14% probability of having a neighboring Yb3+, and ~50% probability of having at least one Yb3+ within its first two cation shells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Energy must therefore migrate over at least a few lattice sites within the 4T2g lifetime to fully quench the Cr3+ emission as observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Figure 3a shows the anticipated electronic structure of Yb3+ in CrI3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' In the free ion, spin-orbit coupling splits the 2F term into 2F5/2 (excited) and 2F7/2 (ground) states by an amount ΔE = 7/2ζ, where ζ = 361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='8 meV is the free-ion spin-orbit coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='23 In crystals, each of these states is further split by the crystal field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Figure 3b shows circularly polarized PL spectra of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='9% Yb3+:CrI3 measured in a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content="5 T field applied parallel to the crystal's c axis (vide infra)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Three zero-phonon electronic origins are observed and assigned to the Γ8 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Γ6, Γ8, and Γ7 transitions anticipated from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' 3a using idealized Oh notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' The actual cation site symmetry in CrI3 is lower (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' 3a, right),24 but the expected low-symmetry splitting of the Γ8 origin is not clearly identifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' The Γ6 peak is broad with observable structure on its high-energy shoulder, thus making the precise energy of this origin unclear within ~20 cm-1 (~2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='5 meV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Analysis of these PL energies within the Angular Overlap Model (AOM)25 reproduces the 2F7/2 splittings well, 7 predicting a 2F5/2 splitting of ~34 meV and splittings of the two Γ8 levels by <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='5 meV each (see SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Additional satellite features are observed ~127 cm-1 (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='7 meV) below the Γ8 and Γ7 electronic origins and assigned as phonon sidebands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Raman spectra show a totally symmetric lattice breathing mode of CrI3 at this energy (ν = 127 cm-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='26 A striking aspect of this Yb3+:CrI3 PL is its very low energy relative to other Yb3+ PL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' This energy is primarily determined by spin-orbit coupling (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Yb3+ spin-orbit coupling can be reduced from that in the free ion by covalent expansion of the f-electron wavefunctions (nephelauxetic effect),27,28 but f-orbital covalency in trivalent lanthanides is typically very small and this effect is usually considered negligible at ambient pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' A survey of Yb3+-doped crystals shows that the energy gap between Yb3+ 2F5/2 and 2F7/2 barycenters remains very near the free-ion value of ΔE ~ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='266 eV across doped oxide, fluoride, chloride, bromide, sulfide, and phosphide lattices (see SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='29-33 We note that we have been unable to find any reports of PL from other Yb3+-doped iodide crystals, perhaps because Yb3+ is easily reduced to Yb2+ under common iodide crystal-growth conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Yb3+:CrI3 deviates from this typical behavior substantially: ΔE is only ~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content='163 eV, or ~9% smaller than in the free ion, representing the smallest spin-orbit coupling yet reported for Yb3+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Covalency in Yb3+:CrI3 is certainly enhanced by the large ionic radius and polarizability of the iodides, but this consideration alone likely cannot explain the anomaly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' The atomic spin-orbit coupling of I is also much greater than those of other common ligands for Yb3+, and should contribute to the spectroscopic spin-orbit splitting via covalency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Furthermore, the large ionic radius of Yb3+ compared to Cr3+ means that Yb3+ experiences an internal pressure imposed by the surrounding lattice, which may also increase covalency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' Importantly, Yb3+-I- covalency is essential for strong Yb3+-Cr3+ superexchange coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' 8 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFKT4oBgHgl3EQf8i5H/content/2301.11949v1.pdf'} +page_content=' (a) Splitting of the Yb3+ free-ion 2F term due to spin-orbit (ζ) and crystal-field (Oh, 0, (3) a concentration for each tile type i, ci, and +(4) a standard tile displacement rate constant k. The as- +sociated set of assemblies A consists of finite arrays of tile +types (or empty). Given a specific assembly, we say that +a specific toehold on a specific tile is closed if the cor- +responding toehold on the neighboring tile has the same +(i.e. matching) bond type (i.e. they form a bond), while +we say that it is open otherwise. The bond energy E(A) +of an assembly A is the sum � −Eb over all closed toe- +holds in the assembly, while the free energy G(A) of the +assembly is its bond energy plus the sum � ln ci/c0 over +all tiles in the assembly, where c0 is the reference con- +centration (e.g. 1 M). Given these, we associate a formal +chemical reaction network (CRN) with reactions +A + ti +k +−→ A′ + tj +where A is an assembly with tile tj at some position x, A′ +is the same assembly but with ti instead at that same po- +sition x, and ti is a valid displacement: on all sides where +tj has a neighbor, ti forms a matching bond with (at +3 + +A +tiles that can displace the center tile: +tiles that cannot displace the center tile: +neutral +downhill +neutral +but no west toehold +uphill +downhill +but no north toehold +warning +a +b +c +d +x +w +y +z +a +b +c +d +x +w +y +z +a +b +c +d +C +D +B +irreversible +strand displacement +reversible +strand displacement +Figure 2: +(A) Abstract tile displacement model. (B) Irreversible and reversible strand displacement. (C) DNA origami +tile implementation. (D) Single-stranded tile implementation. +least one) open toehold, and the total number of match- +ing bonds increases or stays the same (i.e. the assembly’s +bond energy decreases or stays the same). When all reac- +tions are reversible, which implies that the bond energy +of the assembly never changes, the CRN satisfies detailed +balance with respect to the assembly bond energy, with +monomer tiles having zero energy. +We consider standard stochastic kinetics according +to +Gillespie +simulation +with +chemostatted +constant +monomer tile concentration [25, 26]. For an initial state +containing a single assembly, this results in a finite +continuous-time Markov chain (CTMC) where the set of +states are all assemblies reachable via tile displacement +reactions, and the transition A → A′ involving invading +tile ti, as above, will have rate k × ci. If all reactions +are reversible, this CTMC will satisfy detailed balance +with respect to the assembly free energy, such that the +equilibrium probability of assembly A is +p(A) = 1 +Z e−G(A) +with +Z = +� +A′ +e−G(A′) +where the partition function sum Z is taken with respect +to all assemblies reachable by tile displacement. +A tile displacement system simulation is considered un- +reliable if at any time there is an energetically neutral or +downhill tile replacement that does not form at least one +new toehold with each neighboring tile. In this case, the +simulation issues a warning, as illustrated in figure 2A. +We briefly consider possible experimental implementa- +tions of single-assembly tile displacement systems. Fig- +ure 2C shows the motivating DNA origami tile scheme, +using a geometrically-symmetrical tile design modeled af- +ter those used in several prior experimental works [27, 17, +3, 28, 29]. More speculatively, in figure 2D we envision +an implementation that makes use of topologically two- +dimensional arrays of single-stranded tiles [30, 31], which +have been shown to tolerate a wide variety of structural +variations (including single-stranded regions as we would +require for toeholds) and permitting strand displacement +reactions that remove tiles from the array [32, 33]. How- +ever, single-strand tile reactions analogous to the four- +toehold reversible tile displacement reactions required +here have not been experimentally demonstrated. +Re- +gardless of whether considering DNA origami tiles or +single-stranded tiles – or something else – a major ob- +stacle to any experimental implementation would be the +creation of the initial array with a desired initial pattern. +One possible avenue – still difficult – would be to initially +assemble a uniquely-addressed DNA origami array [17] +or single-stranded tile array [31], use that array to geo- +metrically organize the desired pattern of non-uniquely- +addressed tiles needed for tile displacement behaviors, +and then via photocleavable bonds or other mechanisms, +remove and dispose of the uniquely-addressed array. But +for now, we will assume that arbitrary initial assemblies +can be synthesized. +3. WIRES, GATES, AND CIRCUITS +To get a feel for how tile displacement systems can be +programmed, we begin with the most basic task: signal +transmission. As shown in figure 3A, this can be accom- +plished using a single tile type (“wire”) that is used in +the initial assembly to indicate where the wire is, plus a +single tile type (“signal”) that carries the signal x. Two +additional tile types (“top” and “bottom”) are used to +provide neighboring tiles for the wire, as in general the +4 + +A +B +C +signal +wire +bottom +top +reversible transmission of signal x ⇌ x: +neutral +irreversible transmission of signal x → x: +downhill +signal +wire +bottom +top +neutral +reversible transmission of signals w + x ⇌ y + z: +signal w +wire (S-N) +wire (W-E) +signal x +wire cross +signal y +signal z +gate +downhill +neutral +but no south toehold +warning +irreversible transmission of signals w + x → y + z: +D +warning +neutral +but no south toehold +❷ position (3,3) +❸ position (4,3) +❶ position (4,3) +signal w +signal x +wire cross +signal y +signal z +gate +wire (S-N) +wire (W-E) +Figure 3: +(A) Reversible wire. All toeholds are strength 1 except for toehold “−”, which is inert, i.e. strength 0. The +top and bottom tiles have toeholds that are not shown, such that in the assembly the unlabeled sides are bound to each +other via matching closed toeholds. In assemblies, closed toeholds are shown with light grey labels and a solid dark grey bar +indicates their bond. For open toeholds, black or light grey is used to highlight relevant locations for the tile displacement +reaction of interest, but have no formal meaning. In the monomer tile that is a reactant of the indicated reaction, solid dark +grey bars indicate where new toehold bonds will be formed. Here and in later figures, only the forward reaction is shown +for any reversible reactions (i.e. the assembly and monomer tile that are the products of the indicated reaction do not have +their relevant toeholds highlighted for the backward reaction). (B) Irreversible wire. (C) Reversible wire cross. The initial +assembly shown here illustrates the moment when both reversible signals arrive at the wire cross location. At this time, a +reversible bond-energy neutral reaction can occur that inserts the gate tile in the central location, enabling reversible signals +y and z to propagate on the output wires. (D) An unreliable irreversible wire cross that has two possible types of warnings. +5 + +wire will be embedded within a larger assembly. Each +tile displacement reaction is neutral with respect to the +bond energy, so when both the wire and signal tiles are +at the same concentration, every tile displacement occurs +at the same rate, and the signal transmission performs an +unbiased random walk. Thus the expected time for signal +transmission along a wire of length N is O(N 2). +Faster signal transmission is possible if each tile dis- +placement step is irreversible, which can be accomplished +if new toehold bonds are formed such that the bond en- +ergy change is downhill. Shown in figure 3B, the wire is +as before, but now the signal tile has an additional toe- +hold. Thus, tile displacement reactions are energetically +downhill, forming one net additional bond with each re- +action step, and the expected time for signal transmission +is now O(N). +When a horizontal and a vertical wire meet, we can +perform a computational step. Figure 3C shows two re- +versible wires, one carrying signal x and the other carry- +ing signal w, meeting at a “wire cross” tile in the center. +At this location, reversible tile displacement by a “gate +tile” can effect the w + x ⇀ +↽ y + z reaction. +Because +the initial wire cross tile has four closed toeholds, tile dis- +placement by the gate tile must form all four new toehold +bonds, and thus tile displacement here prior to arrival of +both the x and w signals would be energetically unfa- +vorable and would not occur. This gate design is robust +and flexible: it is straightforward to design more powerful +variants. For example, the horizontal wire can carry one +of two signals, 0 or 1, the vertical wire also can carry 0 +or 1, and there are now four gate tiles, one for each input +combination, with output signals that effectively compute +the logic function of interest. +Specifically, to compute +NAND and output using the same signal varieties, we +would use four gate tiles that replace (w, x, y, z) respec- +tively by (0, 0, 1, 1), (0, 1, 1, 1), (1, 0, 1, 1), and (1, 1, 0, 0). +Can we similarly perform logic gate operations using +irreversible wires, thus making computation faster? Un- +fortunately, the above schemes no longer work in this case, +as illustrated in figure 3D. The problem is that now, prior +to arrival of the second signal, an energetically neutral tile +displacement is possible at the gate position that simply +ignores the missing input wire. +Indeed, if the vertical +wire is meant to be capable of carrying two signals (here +w or y), then an energetically neutral tile displacement +could analogously flip the signal content. Thus, this tile +set and gate design is deemed unreliable and would issue +warnings in our simulator. The lesson is that irreversible +reactions will form an additional toehold when operating +as intended, and this presents the possibility that a sim- +ilar context that differs by just one open toehold, where +the reaction is not intended to occur, will be energetically +neutral and lead to an error. +So does this mean that linear-time binary signal trans- +mission and circuit computation is impossible with tile +displacement systems? Thankfully, no. The trick is that +while the leading wavefront of signal propagation and +computation still must be reversible, in order to reliably +discriminate single-toehold differences, it can be safe to +irreversibly latch a decision in a context where all neigh- +boring tile contain the same information, so differences +between a 0 signal and a 1 signal by necessity involve +two toeholds. Now, if the irreversible tile displacement +involves the formation of one extra toehold in the correct +context, in the incorrect context it would have to ignore +two toeholds and thus would be uphill. +This principle +is illustrated in the design shown in figure 4A, where the +latch tile can irreversibly insert itself into a three-tile-long +segment of signal-carrying wire. Exactly where the latch +tile inserts does not matter; the signal ratchets forward +either way. +We are now ready to take these designs for wires and +gates, and combine them to construct feedforward logic +circuits that compute in time linear with the depth of the +circuit (as laid out in an array). There is, however, one +more problem to solve if we want to build circuits that +utilize multiple types of logic gates (e.g. +XOR, AND, +OR, NAND, NOR, WIRECROSS, and others). When an +invading gate tile (e.g. “XOR 10” shown in figure 4B) +displaces the initial gate tile (e.g. +“XOR”), it makes +bonds with toeholds on the neighboring four tiles but +not with the displaced tile itself – therefore, information +about which function should be computed must be con- +tained in the neighboring tiles, and not just the gate tiles. +We achieve this goal by using a gate-specific toehold in +the initial gate tile, which directs the incorporation of a +translator tile in the final position of each wire, as shown +in figure 4B. Now the translator tile contains information +about which logic function should be computed. Thus, an +arbitrary number of gate types may coexist in the same +system. +Simulations of two feedforward circuit computations +are shown in figure 5. Using just a systolic array of XOR +gates, a collection of parity outputs (involving different +subsets of the inputs) are produced, incidentally creating +a Sierpinski triangle pattern within the completed wires +and gates. +The second circuit makes use of four gate +types: a NOR gate that produces the same signal on both +output wires, a NAND/XOR gate that produces NAND +to the north and XOR to the east, a WIRECROSS that +sends it south input to the north and its west input to the +east, and a WIREPASS that sends its south input to the +east and its west input to the north. The positions and +identities of the gates are laid out in the initial tile array. +Computation of an N ×N circuit will take expected time +O(N). +The tile system with eastward and northward latching +binary wires and the five types of gate functions discussed +above consists of 66 tile types altogether. For a circuit +that can be laid out effectively in this format, an area of +O(N 2) tiles can support N 2 gates. Arbitrary feedforward +circuits with N gates can be implemented in O(N 2) area +using a standard crossbar array architecture (for exam- +ple see [34]) and a new FANOUT gate that copies one +6 + +A +B +wire +bottom +top +signal 0 +latch 0 +signal 1 +latch 1 +irreversible transmission of signals 0 and 1: +translator +0 ⇌ 0⊕ (W-E) +translator +1 ⇌ 1⊕ (W-E) +translator +0 ⇌ 0⊕ (S-N) +translator +1 ⇌ 1⊕ (S-N) +XOR +XOR 00 +XOR 01 +XOR 10 +XOR 11 +❶ +❷ +❸ +❹ +Figure 4: +(A) Irreversible wire with no warnings. The next effect of two separate, independent tile displacement steps is +shown for each wire. Importantly, no unreliable tile displacement reactions are possible. (B) An XOR gate. The gate itself +is entirely reversible; latch steps in the input and output wires are sufficient for ensuring net progress within a circuit. +input and ignores the other (using 75 tiles types if the +new gate is just added, or 57 tile types if the redundant +NAND/XOR and XOR gates are removed). +Can we do better than just feedforward circuits? It is +clear from inspection that our latching binary-signal wires +can transmit information in either direction, depending +on where the signal first arrives, and it is straightforward +to implement gates that receive inputs from any two sides +and produce outputs on the two other sides, so we can ar- +range for signals to go around in cycles. Furthermore, the +7 + +A +B +T = 0 +T = 1000 +T = 2000 +T = 3000 +T = 0 +T = 1000 +T = 2000 +T = 3000 +Figure 5: +(A) A 9 by 9 array of XOR gates. Black tiles are “caps” that terminate the output wires. (B) A 18-input 18-output +logic circuit composed of XOR, NAND, and NOR gates. Two types of wire routing are implemented with a WIRECROSS +tile that sends it south input to the north and its west input to the east, and a WIREPASS tile that sends its south input to +the east and its west input to the north. The simulation time T measures the number of tile displacement reactions that have +occurred, rather than the real time in the Gillespie simulation. +tile displacement model in principle allows displacement +to occur an arbitrary number of times in a given location. +As a trivial example, the reversible wire of figure 3A can +endlessly perform a random walk, back and forth forever. +This raises the prospect of a tile displacement system of +size N simulating a recurrent (iterated, feedback) circuit +of size O(N), which can perform computations of length +2O(N) – exponentially more than what a feedforward cir- +cuit of the same size can do. This is to say, with respect +to the size of the initial array and tile set, our existing +construction can solve PTIME problems, while a reach +goal would be to solve PSPACE problems like recur- +rent circuits can. Unfortunately, this is not compatible +with the use of latching wires to ensure linear-time signal +propagation: an area-N tile array initially has at most +O(N) open toeholds, and thus at most O(N) irreversible +8 + +tile displacement steps can take place before the system +comes to a standstill – or more precisely, until it must +henceforth rely exclusively on reversible steps. +4. 1D CA SPACE-TIME HISTORIES +Boiling down what we learned about circuits to its re- +versible essence, we can re-implement the above compu- +tations using fewer tile types, more compact layouts with +just one tile per logic gate, and power for driving the com- +putation forward coming from concentration differences +rather than from irreversible toehold formation. +We start by providing generalized construction for sim- +ulating the space-time history of one-dimensional block +cellular automata (1D BCA) that is very similar to +their simulation by algorithmic self-assembly of DNA +tiles [35, 2]. The instantaneous state of a 1D BCA is just +a one-dimensional array of symbols from a given alphabet +A, and in each time step the entire array is synchronously +updated by applying a rule (x, y) → (f(x, y), g(x, y)) to +a partitioning of the array into pairs, where f and g are +functions that define the BCA and the parity of the par- +tition alternates on each time step. The size of the ar- +ray may be infinite, finite, or expanding, with given ini- +tial state and boundary conditions (typically a finite core +then periodic). Our tile displacement system construc- +tion, shown in figure 6A, makes use of 2 + 2N + N 2 tile +types for a 1D BCA with an alphabet of size N. The +initial array uses 1 tile in the lower-left corner, N tiles to +define input boundary conditions to be fed in at each time +step from the left, N tiles to define the input boundary +conditions to be fed in at each time step from the bottom, +and 1 tile type filling in the remaining “blank” uncom- +puted region of the array. The remaining N 2 tile types +encode every input/output case for the update rule. For +example, a binary alphabet (N = 2) will result in 10 tile +types (figure 6B). The nth synchronous update of the 1D +BCA will be encoded in the nth diagonal of the tile ar- +ray. Similar to the gate tiles in the circuit construction, +displacement must match all four open toehold positions, +else it will be energetically unfavorable. This can only +happen when both the tile to the left and the tile below +have already updated, thus ensuring that the computed +information is based on the correct information from the +preceding diagonal. +Because our model insists that any tile that can be +displaced in a simulation must have a non-zero concen- +tration as a monomer in solution, every reaction will +be reversible. However, by chemostatting the blank tile +at a lower concentration than the rule tiles, each dis- +placement reaction can be biased forward by some factor +r = crule/cblank. From detailed balance of the CRN and +CTMC, this ensures that the equilibrium probability of +the rule-tile containing assembly is r times higher than +that of the blank-tile containing assembly. Although the +system will never get irreversibly locked into a final out- +put assembly state, the complete assembly with all rule +tiles in place will be rm times more likely than an assem- +bly with m blank tiles still present, which we consider +“good enough”. Note that if a final irreversible step is +desired to lock in place the completed computation, this +is also possible by adapting the techniques used in the +circuit construction, just at the upper right corner. +Comparing the circuit construction of figure 5A to the +cellular automaton space-time history construction in fig- +ure 6BC, both of which compute parallel systolic arrays +of XOR gates, we see that for the same size array, the +cellular automaton approach computes roughly 9 times +more gates. It also uses just 10 tile types, compared to +30 for the circuit construction (if the tiles used for logic +gates other than XOR are omitted). +However, our cellular automaton construction, by its +very nature as a cellular automaton, receives information +only in the initial 1D boundary conditions, and thus an +assembly cannot specify a two-dimensional layout for the +circuit that will be computed by tile displacement. A sim- +ple modification of the ideas resolves this apparent limita- +tion: we generalize the construction to cellular automaton +transformers whose cell update now depends both on the +current state (x, y) ∈ A1 and a time-and-space-dependent +input pattern (p, q) ∈ A0, as shown in figure 6D. Instead +of an initial array containing uniform blank tiles, the ini- +tial array will contain a layout of “pattern” tiles that each +encode the information p that the gate below it will need +to read, as well as the information q that the tile to its +left will need to read. +If A1 is size N and A0 is size +M, then there are 2N input tiles, M 2 pattern tiles, and +N 2M 2 rule tiles. Each reversible tile displacement reac- +tion now must match four variable pieces of information, +in two pattern toeholds and two state toeholds. As shown +in figure 6EF, laying out exactly the same circuit as in +figure 5B now requires 9 times less space, uses just 39 tile +types (N = 2 state bit values plus a terminator, M = 5 +logic functions, but not all combinations are needed) in- +stead of 57, and, with concentration bias again, computes +significantly faster. +Both these constructions exhibit strong similarities to +computation via algorithmic growth during self-assembly +of tiles – in the first case, 2D tiles growing a 2D structure +from a 1D boundary [35, 2], and in the second case, 3D +tiles growing an additional layer on top of a patterned +2D initial assembly [36]. A significant difference is that +rather than growing in size, the tile displacement system +always remains the same size; rather than each tile attach- +ment requiring new bond energy to counteract the lost en- +tropy due to localization of the tile, the tile displacement +system remains neutral with respect to bond energy be- +cause each incoming tile is balanced by an outgoing tile. +Thus, rather than finding suitable operating conditions +by balancing temperature (controlling the bond energies) +against tile concentrations (which simultaneously affect +the kinetics), in tile displacement we balance concentra- +tion against concentration (which permits similar bias +at different speeds and temperatures). +These benefits +reflect similar observations about the increased robust- +9 + +B +C +input 0 (N) +input 1 (N) +input 0 (E) +input 1 (E) +blank +XOR00 +XOR10 +XOR01 +XOR11 +E +T = 0 +T = 1000 +T = 2000 +T = 3000 +NOR00 +NOR10 +NOR01 +NOR11 +XOR-NAND00 +XOR-NAND10 +XOR-NAND01 +XOR-NAND11 +XOR01 +XOR01 +XOR10 +diagonal-wire01 +XOR-NAND11 +XOR-NAND10 +A +1D block cellular automaton update rule: +(������������, ������������) → (������������ ������������, ������������ , ������������(������������, ������������)) +corner +input (N) +input (E) +blank +rule +1D block cellular automaton transformer update rule: +state (������������, ������������) pattern (������������, ������������) → state (������������, ������������) +corner +input (N) +input (E) +pattern +rule +D +F +T = 0 +T = 30 +T = 60 +T = 90 +corner +Figure 6: +(A) General case implementation of 1D block cellular automaton. Here a and b, written in roman font, denote +specific toeholds. In contrast, x, y, f and g are variables and thus shown in italics. There will be a separate rule tile for each +possible pair x, y ∈ A, with f and g being dependent on x and y, and similarly for the input tiles. (B) An example 1D block +cellular automaton that computes the same function as the circuit shown in Fig. 5A. (C) Simulation snapshots. (D) General +case implementation of 1D block cellular automaton transformer. (E) An example 1D block cellular automaton transformer +that computes the same function as the circuit shown in Fig. 5B. (F) Simulation snapshots. +10 + +ness of strand displacement and toehold exchange com- +pared to direct hybridization of complementary oligonu- +cleotides [37, 38]. Seen more generally, tile displacement +systems involve reconfiguration of a constant-sized assem- +bly via local propagation of information, which is remi- +niscent of the distinction between crystal growth from +monomers in dilute solution (the case generally assumed +in algorithmic self-assembly of DNA tiles) versus crys- +tallization from the melt (wherein the initial state is a +disorganized constant-density liquid of monomers, within +which crystalline order locally propagates during crystal +growth). +Have we identified new concepts for tile displacement +systems that allow us to perform more computation in a +limited space? Powering computation forward via con- +centration bias in reversible reactions has given rise to +compact constructions that naturally avoid the unrelia- +bility warnings that plagued our initial wire and circuit +constructions, but the computational power still remains +PTIME. One way of looking at this is that the free en- +ergy of the assembly, G(A) decreases every time a higher- +concentration tile replaces a lower-concentration tile, yet +the minimum (most favorable) free energy occurs if all +tiles in the array are highest-concentration tiles. That is +to say, the free energy is bounded below, and if each for- +ward computational step is biased by a minimum amount, +there are a bounded number of such steps that can occur +before the computation is done. The situation is not so +different from the limitation we encountered when power- +ing computation by new toehold formation in irreversible +displacement steps. Is this limitation to PTIME a feature +of tile displacement systems in general, or is it particular +to the lack of imagination in the constructions we have +presented so far? +5. 2D CA IN-PLACE EXECUTION +We can get some ideas from the notion of a cellular au- +tomaton transformer, which reads a 2D pattern as a wave +of activity passes over it, leaving a new pattern in its +wake. Suppose that the new pattern can be read by a sec- +ond wave, corresponding to a second cellular automaton +transformer using a new set of rule tiles. For example, +the initial pattern might use toehold alphabet A0, the +first cellular automaton transformer uses states in alpha- +bet A1 and writes a new pattern using alphabet A2 by +utilizing the two locations that, in figure 6D, have useless +inert “−” toeholds. Then, the second cellular automaton +transformer can read A2, store its transient state in A3, +and write a third pattern using A4. To drive the com- +putation forward, the first transformer’s rule tiles should +have a higher concentration than the pattern tiles, and +the second transformer’s rule tiles should have a higher +concentration than the first transformer’s rule tiles. This +idea could be extended to K waves, each with its own +set of rule tiles. This would improve upon the previous +constructions, in which each location in the array expe- +riences just net one forward tile displacement step – at +that location, either one has the initial tile, or the final +tile. Whereas, in an implementation of a multiple-wave +cellular automaton transformer, each location would go +through a sequence of changes, one for each wave. +In +a sense, we achieve K-fold more computation within the +same assembly area. This is somewhat analogous to freez- +ing cellular automata, which are restricted to change a +cell’s state a limited number of times [39]. +There are two problems here, as you have probably +already noticed. +First, if the concentration ratio from +wave to wave is r, then a K-wave computation requires +a ratio of rK between the lowest-concentration tiles and +the highest-concentration tiles. That quickly becomes im- +practical, and theoretically unappealing. +Second, each +wave requires a new set of tiles – yet for PSPACE com- +putations we would require an exponential number of tile +updates and thus a comparable number of waves. So this +idea doesn’t get us where we want to go. +To keep a constant number of tile types while allowing +an unbounded number of tile displacement steps per site, +perhaps we could have a small number K of waves, but +have wave K output its new pattern using alphabet A0 so +that the tiles of wave 1 can read it – thus allowing iterated +computation, such as binary counters and perhaps univer- +sal space-bounded algorithms. This is indeed the essence +of the construction we’ll arrive at, but it comes at a cost: +for wave 1 tiles to displace wave K tiles, they cannot be +at a lower concentration, which basically implies that all +rules tiles must be at the same concentration, and we have +no concentration bias pushing the computation forward. +(This conclusion is not specific to periodic waves of cel- +lular automata transformers; it follows in general that if +we want to implement a computation that may update a +given site an unknown and unbounded number of times, +then every tile type may at some point be an incoming +tile and at other times be the outgoing tile, so the con- +centrations of all rule tiles must be equal.) If we have +already accepted that our designs should exclusively use +bond-energy neutral tile displacement, then in fact the +bond energy and free energy of our assembly will remain +constant over time – we are truly dealing with reversible +computation. +Thankfully, reversible computation is by +no means impossible [14, 40]. +Our approach will be to exhibit a surprisingly natural +correspondence between certain tile displacement systems +and the well-studied class of two-dimensional block cel- +lular automata (2D BCA) that arose in the study of re- +versible computation by discrete models of ballistic phys- +ical dynamics [20, 21, 22]. The 2D BCA model is a natu- +ral generalization of the 1D BCA discussed above: rather +than partitioning a 1D array into pairs of cells that get +synchronously rewritten with alternating partition par- +ity on alternate time steps, we now partition a 2D array +into 2 × 2 blocks of cells that get synchronously rewrit- +ten with alternating partition parity on alternating time +steps (compare figure 7A with figure 7E). The formalism +11 + +A +( ⃗������������, ⃖������������) → (������������ ⊕ ������������, ������������ ⊕ ������������) +������������ = 4 +0 +0 +������������ = 3 +1 +1 +1 +1 +������������ = 2 +1 +1 +0 +0 +1 +1 +������������ = 1 +0 +0 +1 +1 +1 +1 +0 +0 +������������ = 0 +0 +0 +0 +1 +1 +0 +0 +0 +B +(������������, ������������) → (������������ ⊕ ������������, ������������ ⊕ ������������) +10 steps +0 +1 +1 +0 +0 +1 +1 +0 +9 steps +0 +1 +1 +1 +1 +1 +1 +0 +8 steps +0 +1 +1 +0 +1 +1 +1 +0 +⋮ +2 steps +0 +0 +1 +1 +1 +0 +0 +0 +1 steps +0 +0 +0 +1 +1 +0 +0 +0 +0 steps +0 +0 +0 +1 +1 +0 +0 +0 +time sheet 0 +time sheet 8 +C +������������ = 25 +������������ = 0 +������������ = 500 +������������ = 0 +D +E +→ +F +������������ +������������ +↘ ↙ +↗ ↖ +������������ +������������ +↖ +↗ +������������ ������������ +������������ ������������ +↙ +↘ +bottom +view +top +view +side +view += +21 tile attachment steps +time sheet 21 +������������ = 0 +������������ = 1 +������������ = 1 +������������ = 2 +Figure 7: +(A) Execution of a synchronous 1D block cellular automaton. (B) Asynchronous 2D tile self-assembly that +simulates the computation in (A). (C) and (D) Simulations of two example 2D block cellular automata: Billiard Ball Model (C) +and Critters (D). (E) Execution of a synchronous 2D block cellular automaton. (F) Asynchronous 3D tile self-assembly that +simulates the computation in (E). +allows the rewrite rules to be arbitrary functions +f +�� +a +b +d +c +�� += +� +w +x +z +y +� +but if the rewrite function is a bijection, then the 2D BCA +is logically reversible in the sense that iterating with f −1 +instead of with f will bring the simulation backwards in +time. The most famous 2D BCA rule, the Billiard Ball +Model (BBM), is logically reversible, rotationally and +mirror symmetric, conserves the total number of 1s, can +directly simulate reversible circuits, and with an infinite +periodic initial state can simulate universal Turing ma- +chines [20]. Example simulations of two binary-state 2D +BCA, the BBM and “Critters”, are shown in figure 7CD. +With larger alphabets, 2D BCA can simulate arbitrary +classical cellular automata and Turing machines, either +of the irreversible or reversible variety. (Generalizations +to using blocks larger than 2 × 2 is also natural, but will +not be considered here.) +There are three obstacles to implementing arbitrary 2D +BCA as tile displacement systems, and we will solve them +all. The first is that tile displacement reactions are asyn- +chronous (occurring at random locations and in random +orders) while 2D BCA require synchronous updates of the +entire array (and fail utterly if the same update function +is applied asynchronously with no other modifications). +The second is that the mechanics of tile displacement +must be designed to avoid irreversible steps that close +too many toeholds at once. +And the third obstacle is +that with exclusively reversible reactions and no concen- +tration bias, there must be some other way to drive the +system forward if we don’t want to wait forever. +For the first challenge, we adapt prior methods for im- +buing asynchronous cellular automata with locally syn- +chronizing mechanisms [41, 42, 43]. The specific approach +used here generalizes the approach used for simulation of +1D cellular block automata space-time histories in the +previous section. +Figure 7A gives an example of a 1D +BCA, with boxes highlighting the partitioning into pairs +with alternate parity on each synchronous time step. Fig- +ure 7B shows the same computation interpreted as 2D +tile self-assembly where, starting from the 5 tiles at the +bottom that encode the 8 input bits as well as their par- +titioning, rule tiles attach whenever they can match two +sides of existing tiles in the assembly, thus asynchronously +growing the space-time history. We have augmented the +tiles with arrows that point to where incoming tiles could +attach; thus, in the initial assembly of 5 tiles, the sites +12 + +where tile can attach are exactly those locations where ar- +rows are pointing inward toward the incoming tile. A cut +through the assembly’s space-time diagram corresponds a +particular moment during the asynchronous self-assembly +process – we show a cut after 0 tile additions (yellow) +and another after 8 tile additions (orange). We call these +“time sheets” because at different horizontal (x) posi- +tions, they are at different heights (t), and thus reading +out the binary (black/white) states along a time sheet +path correspond to states at different time steps of the +underlying synchronous cellular automaton. Nonetheless, +the time-sheet state information, augmented with the rel- +evant arrows, is all that is needed to correctly complete +the computation using an asynchronous update rule that +executes only when arrows point toward each other, oth- +erwise leaving the cells untouched. This process exactly +mimics the self-assembly of the deterministic space-time +history, despite its non-deterministic order of execution. +There is an exactly analogous arrow-augmented asyn- +chronous update rule for 2D BCA. Rather than square +tiles, we now have truncated octahedra as “tiles”, but +the self-assembling structure is again a space-time history +of the correct synchronous cellular automaton computa- +tion. Tiles may attach when they match four hexagonal +faces of existing tiles in the assembly. (The small square +faces are inert.) Again, if we imagine arrows orthogonal +to the hexagonal faces of tiles, pointing out of the tile, +then valid sites for attachment of a new tile correspond +exactly to situations where all four arrows on the match- +ing faces are pointing toward each other. +The growth +front for a give stage of assembly again corresponds to +a (now two dimensional) time sheet, and we can write +out the states of each exposed hexagonal face in a two +dimensional array along with the orientation of its corre- +sponding arrow. It is now a simple observation that the +asynchronous addition of a tile corresponds exactly to an +asynchronous update of a 2 × 2 block with four inward- +pointing arrows, resulting in updates of the four cells and +reversing all four arrows. Another way of thinking of it +is that after a block asynchronously updates, it will not +be able to update again until all four overlapping 2 × 2 +blocks have first updated and flipped the arrows back. +Thus, the arrow-augmented asynchronous updating cor- +responds exactly to synchronous parallel updating with +alternating-parity partitioning into blocks. +The second challenge is to implement this type of asyn- +chronous block cellular automaton updating rule using +tile displacement. Our construction, shown in figure 8, +introduces additional complexities due to the fact that +all tile displacement reactions are physically reversible, +even if the 2D BCA logic update rules are irreversible, +combined with the need to ensure that when one toe- +hold is closed, the neighboring toehold must be opened – +thus we must be able to guarantee a mismatch. Tripling +the cell state alphabet by adding α, β, γ markers solves +both problems. For each 2D BCA update case, we make +three tiles, one inputting α-symbols and outputting β- +symbols, another inputting β-symbols and outputting +γ-symbols, and the third inputting γ-symbols and out- +putting α-symbols. When an α → β tile inserts into the +array, that simulates a forward-time asynchronous up- +date. The swapping of which toehold is open and which +is closed reflects the flipping orientation of arrows in the +asynchronous cellular automaton; we can read the arrows +from a tile array by looking at the open toeholds and +drawing the arrow from α to β, from β to γ, or from γ +to α. Boundary conditions for finite arrays must also be +handled, using the same principles. +Each side of a tile encodes the state of a specific cell +in the 2D BCA (at a particular time mod 3, as per α- +β-γ of the open toehold), and thus the grid of simulated +BCA cells is oriented at a 45◦ angle relative to the array +of tiles. +State being encoded on the sides of tiles also +facilitates that each tile displacement step corresponds to +an update of a whole 2 × 2 block, and the fact that two +tiles share the same side location reflects that each cell in +a 2D BCA can be updated either by an odd-parity block +or an even-parity block. +The final challenge concerns how to drive the compu- +tation forward. Let us first consider reversible 2D BCA +rules. In this case, after any forward tile displacement +step, there is exactly one monomer tile type that can re- +verse the reaction: the tile that was just displaced. What +this means is that the full state space of the tile displace- +ment system’s CTMC is essentially linear; though fat and +fuzzy, it has the same thickness both arbitrarily far into +the future and arbitrarily far into the past. The thickness +has to do with all the possible contours of the time sheet +for a given average time. Thus we can say that the state +space of the tile displacement system consists exclusively +of correct reachable states of the computation; for a re- +versible 2D BCA simulating a compact recurrent circuit +for solving a PSPACE problem, the tile displacement sys- +tems’s state space will also be exponentially long and will +reach the same correct conclusion. Stochastic Gillespie +simulation of the tile displacement CRN will result in an +unbiased random walk back and forth along this fuzzy- +linear state space. (Every assembly in this reachable state +space has the same energy.) However, unlike a standard +reversible Turing machine with Brownian dynamics [14], +whose state spaces is strictly a linear graph so the ex- +pected random walk hitting time for reaching the end of +an T step computation is O(T 2), the time sheet diffuses +much more slowly. As a rough estimate for an N × N +tile array that requires N 2 forward updates to move the +time sheet 1 net synchronous update step into the future +under ideal circumstances, the same N 2 updates if half +forward and half backward will be expected to net move +the time sheet N steps either forward or backward, which +corresponds to just 1/N equivalent synchronous update +steps. This being just a polynomial inefficiency, perhaps +we should not be too concerned. +More interesting is what happens if the 2D BCA rules +are irreversible. This means there are multiple cases for +13 + +A +B +❶ +❷ +❸ +❹ +2D block cellular automaton update rule: +(������������, ������������, ������������, ������������) → (������������, ������������, ������������, ������������) +rule (time sheet γ → α) +rule (time sheet β → γ) +rule (time sheet α → β) +Figure 8: +(A) General case implementation of 2D block cellular automaton. (B) Example updates in the Billiard Ball +Model. +the 2×2 block input that map to the same output. There- +fore the state space for the tile displacement system will +be exponentially branched in the backwards-in-time di- +rection (as pictured by Bennett in figure 10 of his re- +view paper [14], but thicker and fuzzier). Consequently +Brownian dynamics will tend to be entropically biased +toward where there are more states, and the system will +run backwards. Can this entropic driving force be used to +encourage a system to perform a desired computation by +designing a system whose reverse dynamics are what we +want? Attempting to do so would be risky, and probably +futile, because the 2D block update rule being irreversible +means that there are some states that have no local pre- +decessor, and backward progress will get stuck as such +14 + +local configurations are encountered. +A better way to exploit an entropic driving force is to +have nondeterministic, stochastic forward update rules +added to an otherwise-reversible system. +For exam- +ple, consider a tile displacement simulation of the BBM +model, with boundary tiles designed to implement a re- +flecting boundary (as they must for the system to remain +reversible). If we design a special boundary tile that can +either reflect a ball or (in the forward direction) produce +a new ball out of nothing, then we obtain a new system +that is still entirely reversible in the sense that there ex- +ists (at least one) possible applicable block update in all +circumstances, so the system cannot get stuck either in +the forward direction or the reverse direction. With uni- +form tile concentrations, all assembly states will still be +isoenergetic. But started with an empty N ×N array, for- +ward updates of the special tile will about half the time +produce a new ball, which will entropically drive the sys- +tem to a density such that forward production of balls is +balanced by the reverse reaction, the absorption of balls +into the special tile. At this point, which will be O(N 2) +synchronous time steps in the future, the time sheet will +stop advancing on average. Another way of looking at +it is that with all reactions being neutral, equilibrium +will reflect equipartition among all reachable states, and +the combinatorially greatest number of states will have a +number of ball near the optimal density – so, that’s what +we are likely to observe. And the only way to get there is +to run the time sheet forward enough to emit that number +of balls. +A gas-filled BBM simulation is not of great use by itself, +but we can make use of it by also placing a circuit in the +array, and drawing a 2-cell-thick wall around it. In the +BBM model, balls bounce off walls, and walls are stable. +Thus, despite random stochastic gas entering the areas of +the array outside the box, the circuit will remain perfectly +isolated from the gas. But due to the time-sheet coupling +enforced by the asynchronous arrow rules, the time sheet +that is being driven forward by the expansion of the gas +will simultaneously drive the circuit forward. +Unfortunately, for an array of area N 2, we will only +drive the computation forward by O(N 2) steps – this is +no better than the PTIME computational power of the +original circuit construction. Essentially, in a small con- +fined space, our circuit “heats up” and stops working. +To run it for a long time, we need a larger space into +which we can release the simulated heat. For example, +if we are willing to entertain a half-infinite-plane array +for tile displacement, we can draw a BBM wall down the +middle, release gas on one side, and let the other side +simulate an interesting recurrent circuit. Now, although +the array is infinite (or very large) in direct proportion +to how much computation we want to do, we can say +that we have confined the interesting part of the compu- +tation – the circuit itself – to a very small area relative +to the potentially exponentially long computation. This +isn’t PSPACE computation in terms of the size of the +array, but rather in terms of the size of the part of the +array that we care about. Similar constructions can be +used to drive forward computation not just for other re- +versible 2D BCA rules, but even for irreversible rules: the +cellular automaton alphabet can be expanded to encode +an inert “wall”, time sheets within the walled region and +outside of it remain coupled, and sufficient entropy must +be generated by stochastic rules outside the wall, to be +dissipated into a sufficiently larger area. +6. DISCUSSION +Tile displacement within arrays of square DNA origami +tiles was discovered accidentally [3]. While some aspects +of the formal model, such as the four-sided generaliza- +tion of toehold exchange, were invented for mathemat- +ical elegance rather than detailed realism, they are not +too far flung from what has been experimentally demon- +strated and characterized. So it is quite delightful that +within the design space for tile displacement systems, we +find natural implementations for feedforward circuits and +one-dimensional cellular automata that compute in lin- +ear time, powered by irreversible toehold formation or +by concentration gradients. Even more delightful is that +attempts to squeeze out more computational power per +area seemed almost inevitably to lead us consider physical +constraints such as energy, reversibility, and asynchrony +– which in turn lead to classical two-dimensional cellular +automata models that arose in early studies of the physics +of computation [14, 20]. +Our strongest result (despite +weak time efficiency) is that a tile displacement array of +size N can reversibly simulate a recurrent reversible cir- +cuit (via the Billiard Ball Model cellular automaton) for +an arbitrary number of steps. In other words, the reach- +ability question for tile displacement is PSPACE com- +plete – a result strongly reminiscent of Thachuk & Con- +don’s beautiful PSPACE-hardness result for CRNs and +DSDs [44]. +We believe that there remains a lot undiscovered within +the tile displacement design space. For example, while +our constructions showed that the asynchronous tile dis- +placement model can simulate synchronous cellular au- +tomata, the needed flipping-arrow mechanism for local +synchronization seems almost built-in to the tile displace- +ment model in the form of open and closed toeholds for +toehold exchange, and it’s not obvious how to directly +simulate asynchronous cellular automaton models such as +reversible surface CRNs [45, 34, 43]. We might also ask +whether using information within the branch migration +domains rather than just in toeholds – or whether having +even more toeholds and branch migration domains on a +tile’s sides – could have advantages either theoretically +or experimentally. Further, the most interesting systems +demonstrated experimentally in the initial work on tile +displacement [3] involved systems of interacting multi-tile +arrays, rather than a single array and a monomer. How do +our single-assembly results fit into that larger picture? Fi- +nally, might tile displacement systems be combined with +15 + +other molecular mechanisms to solve our problems driving +the computation forward – for example, an oscillator [46] +that periodically activates and deactivates the α, β, and +γ monomer tiles in sequence. +When first discovered, the tile displacement mecha- +nism seemed most closely related to strand displace- +ment mechanisms, only two dimensional. However, as we +investigated the capabilities of single-assembly tile dis- +placement, many parallels to passive tile assembly [2] +became prominent. +Tile displacement systems appear +to combine the principles of DNA strand displacement +and self-assembly in different ways than hairpin-based +programmable self-assembly [47], signal-passing tile self- +assembly [48, 49], CRN-controlled tile assembly [50, 51], +and other models we are aware of. Comparing the ben- +efits, drawbacks, and relationships between these models +may help uncover a more unified way of thinking about +programmable molecular systems. +And even if tile displacement systems, as explored the- +oretically here, never become useful experimentally, we +hope that it was interesting and perhaps inspiring to look +long and deep at a simple mechanism until intricate pat- +terns emerge. +ACKNOWLEDGEMENTS +The authors thank William Poole and Ho-Lin Chen for +useful discussions. This work was partially supported by +NSF awards 2008589 and 1813550. +[1] John E Savage. Models of computation, volume 136. Addison- +Wesley Reading, MA, 1998. +[2] David Doty. Theory of algorithmic self-assembly. 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Integrating DNA strand-displacement circuitry with +DNA tile self-assembly. +Nature Communications, 4(1):1–10, +2013. +[51] Nicholas Schiefer and Erik Winfree. +Universal computation +and optimal construction in the chemical reaction network- +controlled tile assembly model. +In DNA Computing and +Molecular Programming (LNCS volume 9211), pages 34–54. +Springer, 2015. +17 + diff --git a/6NAzT4oBgHgl3EQf9_7h/content/tmp_files/load_file.txt b/6NAzT4oBgHgl3EQf9_7h/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dedc1ba448e3b955d63eb9569c9297fb5fc8cd73 --- /dev/null +++ b/6NAzT4oBgHgl3EQf9_7h/content/tmp_files/load_file.txt @@ -0,0 +1,694 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf,len=693 +page_content='Two-dimensional tile displacement can simulate cellular automata Erik Winfree1,2,3⋆ and Lulu Qian1,2,3⋆ 1Bioengineering, 2Computer Science, 3Computation and Neural Systems California Institute of Technology, Pasadena, CA 91125, USA ⋆e-mail: winfree@caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='edu, luluqian@caltech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='edu Tile displacement is a newly-recognized mechanism in DNA nanotechnology that exploits principles anal- ogous to toehold-mediated strand displacement but within the context of self-assembled DNA origami tile arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Here, we formulate an abstract model of tile displacement for the simplest case: individual assemblies interacting with monomer tiles in solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' We give several constructions for programmable computation by tile displacement, from circuits to cellular automata, that vary in how they use energy (or not) to drive the system forward (or not), how much space and how many tile types they require, and whether their com- putational power is limited to PTIME or PSPACE with respect to the size of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' In particular, we show that tile displacement systems are Turing universal and can simulate arbitrary two-dimensional syn- chronous block cellular automata, where each transition rule for updating the state of a 2× 2 neighborhood is implemented by just a single tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Keywords: DNA origami, tile displacement, cellular automata, reversible computation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' INTRODUCTION A guiding principle in theoretical computer science has been “mechanism-to-model” exploration of connections between physical implementation and computational ca- pabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' For example, what can be computed by sys- tems of AND gates and OR gates is strictly less than what can be computed by systems of NOR gates, which in turn is less than what can be computed by finite state machines coupled with an unbounded memory tape [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Likewise, molecular programming theory aims to understand how fundamental molecular mechanism can be used to build systems, and how the choice of mechanism determines the range of what can be built.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' An example would be the self-assembly of molecular structures by programmable cooperative binding, which can reliably grow structures that cannot grow reliably via non-cooperative binding [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' When a new molecular mechanism is discovered, it is of interest to understand the nature – the limitations and capabilities – of systems that exploit that mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Doing so entails formulation of an abstract model that captures the essential features of the mechanism, which can then be rigorously analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Since its invention two decades ago [4], toehold- mediated DNA strand displacement has been a central mechanism for programming dynamical function in DNA nanotechnology [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' As shown in figure 1A, a stable complex of two strands can be reconfigured such that an invading strand replaces the original partner via a branch migration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The single-stranded portion of the original complex – known as the toehold because that is where the invading strand initiates contact – is critical for the displacement: an invader that does not match and bind to the toehold may be a million-fold slower to per- form the displacement, and thus neglected as “leak”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' In abstract models that consider networks of more complex (but still non-pseudoknotted) DNA molecules that inter- act in solution using toehold-mediated strand displace- ment reactions – including both the irreversible mecha- nism shown here and a reversible variant known as “toe- hold exchange” – have been shown capable of simulating arbitrary formal chemical reaction network dynamics and even Turing-universal computation [7, 8, 9, 10, 11, 12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' However, a limitation of these results is that they are in- trinsically distributed computations, where state is en- coded within a collection of molecules in solution, and therefore a single test tube can perform only one com- putation at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' More complex molecular mecha- nisms, such as the hypothetical polymer-modifying en- zymes envisioned by Bennett [14], are in principle capable of performing independent Turing-universal computation in parallel in the same test tube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The mechanism of “tile displacement”, shown in fig- ure 1B, was recently discovered during investigations into why the self-assembly of DNA origami tiles [15, 16, 17] failed to become kinetically trapped in undesired inter- mediates that the naive theory predicted [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' There is a strong analogy to toehold-mediated strand displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Beyond using components that are two orders of magni- tude larger than the individual strands involved in strand displacement, the tile displacement mechanism has sev- eral distinct features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' (1) Nucleotides are on multiple he- lices that are oriented orthogonally to the axis of branch 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='01929v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='ET] 5 Jan 2023 A B C invader tile released tile branch migration game board player 1 player 2 \uf072 ꓳ \uf072 ꓳ \uf072 \uf072 ꓳ \uf072 ꓳ \uf072 ꓳ \uf072 ꓳ \uf072 \uf072 ꓳ \uf072 ꓳ ꓳ \uf072 \uf072 ꓳ \uf072 ꓳ ꓳ \uf072 \uf072 \uf072 \uf072 (1,3) ꓳ (1,1) \uf072 (3,1) ꓳ (2,2) \uf072 (3,3) ꓳ (2,3) \uf072 (3,2) day 0 day 1 day 2 day 3 day 4 day 5 day 6 day 7 invader strand released strand branch migration Figure 1: (A) Strand displacement mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' For scale, the DNA molecules are roughly 2 nm in diameter and 7 nm long.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' (B) Tile displacement mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' These hypothetical 10-helix DNA origami tiles are smaller than the 22-helix square tiles from [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' (C) A tic-tac-toe game implemented using tile displacement (adapted from reference [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Scale bar for atomic force microscopy (AFM) images is 100 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' migration, rather than being on a single helix that is par- allel to (identical to) the axis of branch migration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' (2) Tile-tile binding may be due to multiple helix-end stack- ing bonds [18] in addition to (or instead of) being due to base-pair formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' (3) Toehold and branch migration domain specificity can be encoded both by tile geometry and by the sequences in multiple very short (1 or 2 nt) sticky ends, rather than being exclusively encoded by nu- cleotide sequences within a single helix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' (4) The released tile will be less flexible than a single-stranded oligonu- cleotide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' (5) Rather than having just one “side” and ini- tiating displacement via a single toehold, tiles may have many (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' four) sides and may initiate displacement via cooperative action of multiple toeholds, as highlighted by the replacement of the central tile of a 3 × 3 tic-tac-toe game board [3] shown in figure 1C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Despite these dif- ferences, it remains that tile displacement is highly sen- sitive to toehold and branch migration sequences, such that the kinetics of tile displacement without a matching toehold may be orders of magnitude slower and similarly negligible as “leak”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Indeed, systems of interacting tile monomers and tile assemblies were shown to be reconfig- urable by toehold-mediated tile displacement [3], and the same or similar constructs ought to be sufficient to im- plement more complex information-processing networks following, for example, the seesaw motif for circuits [19] or the two-domain scheme for formal chemical reaction network dynamics [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Here we are interested in whether the tile displacement mechanism enabled new ways of programming dynamical behaviors, beyond simply replicating strand displacement on a larger scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The ability to perform displacement within a two-dimensional array being an especially novel feature of tile displacement, we ask whether – unlike ex- isting strand displacement constructions – reconfiguration of a single tile assembly in a constant soup of monomer tiles might be sufficient for substantial computation, in which case parallel computation could be achieved with each tile assembly performing an independent computa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' We present three results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' First, with a feedforward Boolean circuit laid out on the initial array, there is a tile set that, via displacement, propagates signals along wires and executes the specified logic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' This system is powered by the energy of toehold formation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' the final state is in an energy minimum and cannot be reused.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Second, as a sim- plification and generalization of the first construction, any one-dimensional cellular automaton can be directly trans- lated into a set of tiles such that a wave of tile displace- ment converts an assembly, initially empty but for the input, into the space-time history of the cellular automa- ton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' This system is powered by a concentration difference between the monomer tiles that are invading over those that are displaced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The above two constructions displace each tile in the original array at most once, using energy that is linear in the area used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The third construction ad- dresses whether iterated computation can be performed in-place, which requires replacing the tile at a given loca- tion an unbounded number of times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Remarkably, using locally reversible asynchronous tile displacement, we can simulate arbitrary synchronous block cellular automata that use the 2 × 2 Margolus neighborhood, including his globally reversible Billiard Ball Model that is known to be Turing universal by simulation of infinite or finite re- current Boolean circuits [20, 21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' A key issue is how to bias the computation forward;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' we show that it is enough to include a large empty part of the array into which en- tropy is injected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' This paper does not aim for novel advances in molecu- 2 全全lar programming that make technical applications closer to reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' We are sharing these observations mainly be- cause we find them to be beautiful and surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Tile displacement may indeed be useful for reconfiguration of adaptive molecular systems, but for most implementation goals that are merely computational, there are more di- rect and more reliable ways to achieve them using other mechanisms in DNA nanotechnology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' However, it is re- markable that a molecular mechanism accidentally dis- covered in the laboratory gives rise to a theoretical model with such natural and direct connections to an esoteric but well-studied model of computation that arose in the study of the fundamental physics and ballistic motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' We hope you will see through our imperfect figures and clumsy explanations to see the poetry within the con- cepts [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' TILE DISPLACEMENT MODEL The abstract model developed in this work, which we call the Single-Assembly Tile Displacement (SATiDi) model, defines the behavior of a single tile assembly within a sea of monomer tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' There are a number of assumptions that must hold in order for the model to be experimentally plausible, while also allowing its definition to be fairly clean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The concentration of multi-tile assemblies is sufficiently low (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' there is exactly one) that they do not interact with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Monomers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Binding between two monomer tiles is suf- ficiently weak (at the given temperature and con- centrations) that any dimers are fleeting and their presence can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Tiles within an assembly (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' with four neighbors) are sufficiently strongly attached that they will not dissociate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' tiles on the boundaries and corners, with only two or three neighbors, may have special binding interactions that make them as stable as the internal tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' No growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' With a single-side attachment being unsta- ble for dimers, similarly new tiles may not attach by a single side to a facet of a multi-tile assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' When the assembly is rectangular, as will be exclusively considered here, that means the number of tiles in an assembly will never change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Full toeholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' For consistent tile displacement kinetics, we require that the displacement process on each side has its own mediating toehold, so a tile that is bound to four neighbors will be displaced by a tile that forms a toehold on each of the four sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' While fewer toeholds may be sufficient for displacement, it is all too plausible that their kinetics would be ir- regular;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' our simulator will issue a warning whenever such a displacement possibility is encountered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Energetics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Tile displacement reactions must be either energetically neutral or energetically downhill, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' the number of toeholds formed is either the same as or more than the number of toeholds broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Uniform design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Each side of every tile will consist of a first toehold domain, a branch migration do- main, and a second toehold domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' We will as- sume that the branch migration domains are distinct on the north, east, west, and south such that they force tile to maintain a specific orientation (although non-oriented versions of the model could be formu- lated when non-oriented tiles are desirable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Because branch migration domains cannot be changed by tile displacement, they will not be formally represented or accounted for in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The model is illustrated in figure 2A, where both a valid neutral tile displacement and a valid downhill tile displacement are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Neutral displacement can be though of as generalizing the “toehold exchange” mech- anism from strand displacement [24]: formation of me- diating toehold ensures fast kinetics, while dissociation of prior toeholds both ensures that the reaction is en- ergetically neutral and opens up those toeholds for use in subsequent steps, as shown in figure 2B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Toehold ex- change in tile displacement was demonstrated experimen- tally [3], although not in the exact geometric configura- tion required here;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' tuning of toehold strength (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' via temperature) would be required to ensure that the dis- sociation step (which may involve breaking four toeholds simultaneously) is sufficiently fast while still being effec- tive for mediating the reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Formally, a SATiDi system is defined by (1) a finite set of square tile types S, each of which specifies an or- dered pair of bond types (toeholds) for each of the four sides, (2) a bond strength function for each bond type b, Eb > 0, (3) a concentration for each tile type i, ci, and (4) a standard tile displacement rate constant k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The as- sociated set of assemblies A consists of finite arrays of tile types (or empty).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Given a specific assembly, we say that a specific toehold on a specific tile is closed if the cor- responding toehold on the neighboring tile has the same (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' matching) bond type (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' they form a bond), while we say that it is open otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The bond energy E(A) of an assembly A is the sum � −Eb over all closed toe- holds in the assembly, while the free energy G(A) of the assembly is its bond energy plus the sum � ln ci/c0 over all tiles in the assembly, where c0 is the reference con- centration (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' 1 M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Given these,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' we associate a formal chemical reaction network (CRN) with reactions A + ti k −→ A′ + tj where A is an assembly with tile tj at some position x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' A′ is the same assembly but with ti instead at that same po- sition x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' and ti is a valid displacement: on all sides where tj has a neighbor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' ti forms a matching bond with (at 3 A tiles that can displace the center tile: tiles that cannot displace the center tile: neutral downhill neutral but no west toehold uphill downhill but no north toehold warning a b c d x w y z a b c d x w y z a b c d C D B irreversible strand displacement reversible strand displacement Figure 2: (A) Abstract tile displacement model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' (B) Irreversible and reversible strand displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' (C) DNA origami tile implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' (D) Single-stranded tile implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' least one) open toehold, and the total number of match- ing bonds increases or stays the same (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' the assembly’s bond energy decreases or stays the same).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' When all reac- tions are reversible, which implies that the bond energy of the assembly never changes, the CRN satisfies detailed balance with respect to the assembly bond energy, with monomer tiles having zero energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' We consider standard stochastic kinetics according to Gillespie simulation with chemostatted constant monomer tile concentration [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' For an initial state containing a single assembly, this results in a finite continuous-time Markov chain (CTMC) where the set of states are all assemblies reachable via tile displacement reactions, and the transition A → A′ involving invading tile ti, as above, will have rate k × ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' If all reactions are reversible, this CTMC will satisfy detailed balance with respect to the assembly free energy, such that the equilibrium probability of assembly A is p(A) = 1 Z e−G(A) with Z = � A′ e−G(A′) where the partition function sum Z is taken with respect to all assemblies reachable by tile displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' A tile displacement system simulation is considered un- reliable if at any time there is an energetically neutral or downhill tile replacement that does not form at least one new toehold with each neighboring tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' In this case, the simulation issues a warning, as illustrated in figure 2A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' We briefly consider possible experimental implementa- tions of single-assembly tile displacement systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Fig- ure 2C shows the motivating DNA origami tile scheme, using a geometrically-symmetrical tile design modeled af- ter those used in several prior experimental works [27, 17, 3, 28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' More speculatively, in figure 2D we envision an implementation that makes use of topologically two- dimensional arrays of single-stranded tiles [30, 31], which have been shown to tolerate a wide variety of structural variations (including single-stranded regions as we would require for toeholds) and permitting strand displacement reactions that remove tiles from the array [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' How- ever, single-strand tile reactions analogous to the four- toehold reversible tile displacement reactions required here have not been experimentally demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Re- gardless of whether considering DNA origami tiles or single-stranded tiles – or something else – a major ob- stacle to any experimental implementation would be the creation of the initial array with a desired initial pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' One possible avenue – still difficult – would be to initially assemble a uniquely-addressed DNA origami array [17] or single-stranded tile array [31], use that array to geo- metrically organize the desired pattern of non-uniquely- addressed tiles needed for tile displacement behaviors, and then via photocleavable bonds or other mechanisms, remove and dispose of the uniquely-addressed array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' But for now, we will assume that arbitrary initial assemblies can be synthesized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' WIRES, GATES, AND CIRCUITS To get a feel for how tile displacement systems can be programmed, we begin with the most basic task: signal transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' As shown in figure 3A, this can be accom- plished using a single tile type (“wire”) that is used in the initial assembly to indicate where the wire is, plus a single tile type (“signal”) that carries the signal x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Two additional tile types (“top” and “bottom”) are used to provide neighboring tiles for the wire,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' as in general the 4 A B C signal wire bottom top reversible transmission of signal x ⇌ x: neutral irreversible transmission of signal x → x: downhill signal wire bottom top neutral reversible transmission of signals w + x ⇌ y + z: signal w wire (S-N) wire (W-E) signal x wire cross signal y signal z gate downhill neutral but no south toehold warning irreversible transmission of signals w + x → y + z: D warning neutral but no south toehold ❷ position (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='3) ❸ position (4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='3) ❶ position (4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='3) signal w signal x wire cross signal y signal z gate wire (S-N) wire (W-E) Figure 3: (A) Reversible wire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' All toeholds are strength 1 except for toehold “−”, which is inert, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' strength 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The top and bottom tiles have toeholds that are not shown, such that in the assembly the unlabeled sides are bound to each other via matching closed toeholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' In assemblies, closed toeholds are shown with light grey labels and a solid dark grey bar indicates their bond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' For open toeholds, black or light grey is used to highlight relevant locations for the tile displacement reaction of interest, but have no formal meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' In the monomer tile that is a reactant of the indicated reaction, solid dark grey bars indicate where new toehold bonds will be formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Here and in later figures, only the forward reaction is shown for any reversible reactions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' the assembly and monomer tile that are the products of the indicated reaction do not have their relevant toeholds highlighted for the backward reaction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' (B) Irreversible wire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' (C) Reversible wire cross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The initial assembly shown here illustrates the moment when both reversible signals arrive at the wire cross location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' At this time, a reversible bond-energy neutral reaction can occur that inserts the gate tile in the central location, enabling reversible signals y and z to propagate on the output wires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' (D) An unreliable irreversible wire cross that has two possible types of warnings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' 5 wire will be embedded within a larger assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Each tile displacement reaction is neutral with respect to the bond energy, so when both the wire and signal tiles are at the same concentration, every tile displacement occurs at the same rate, and the signal transmission performs an unbiased random walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Thus the expected time for signal transmission along a wire of length N is O(N 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Faster signal transmission is possible if each tile dis- placement step is irreversible, which can be accomplished if new toehold bonds are formed such that the bond en- ergy change is downhill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Shown in figure 3B, the wire is as before, but now the signal tile has an additional toe- hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Thus, tile displacement reactions are energetically downhill, forming one net additional bond with each re- action step, and the expected time for signal transmission is now O(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' When a horizontal and a vertical wire meet, we can perform a computational step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Figure 3C shows two re- versible wires, one carrying signal x and the other carry- ing signal w, meeting at a “wire cross” tile in the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' At this location, reversible tile displacement by a “gate tile” can effect the w + x ⇀ ↽ y + z reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Because the initial wire cross tile has four closed toeholds, tile dis- placement by the gate tile must form all four new toehold bonds, and thus tile displacement here prior to arrival of both the x and w signals would be energetically unfa- vorable and would not occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' This gate design is robust and flexible: it is straightforward to design more powerful variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' For example, the horizontal wire can carry one of two signals, 0 or 1, the vertical wire also can carry 0 or 1, and there are now four gate tiles, one for each input combination, with output signals that effectively compute the logic function of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Specifically, to compute NAND and output using the same signal varieties, we would use four gate tiles that replace (w, x, y, z) respec- tively by (0, 0, 1, 1), (0, 1, 1, 1), (1, 0, 1, 1), and (1, 1, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Can we similarly perform logic gate operations using irreversible wires, thus making computation faster?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Un- fortunately, the above schemes no longer work in this case, as illustrated in figure 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The problem is that now, prior to arrival of the second signal, an energetically neutral tile displacement is possible at the gate position that simply ignores the missing input wire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Indeed, if the vertical wire is meant to be capable of carrying two signals (here w or y), then an energetically neutral tile displacement could analogously flip the signal content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Thus, this tile set and gate design is deemed unreliable and would issue warnings in our simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The lesson is that irreversible reactions will form an additional toehold when operating as intended, and this presents the possibility that a sim- ilar context that differs by just one open toehold, where the reaction is not intended to occur, will be energetically neutral and lead to an error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' So does this mean that linear-time binary signal trans- mission and circuit computation is impossible with tile displacement systems?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Thankfully, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The trick is that while the leading wavefront of signal propagation and computation still must be reversible, in order to reliably discriminate single-toehold differences, it can be safe to irreversibly latch a decision in a context where all neigh- boring tile contain the same information, so differences between a 0 signal and a 1 signal by necessity involve two toeholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Now, if the irreversible tile displacement involves the formation of one extra toehold in the correct context, in the incorrect context it would have to ignore two toeholds and thus would be uphill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' This principle is illustrated in the design shown in figure 4A, where the latch tile can irreversibly insert itself into a three-tile-long segment of signal-carrying wire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Exactly where the latch tile inserts does not matter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' the signal ratchets forward either way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' We are now ready to take these designs for wires and gates, and combine them to construct feedforward logic circuits that compute in time linear with the depth of the circuit (as laid out in an array).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' There is, however, one more problem to solve if we want to build circuits that utilize multiple types of logic gates (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' XOR, AND, OR, NAND, NOR, WIRECROSS, and others).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' When an invading gate tile (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' “XOR 10” shown in figure 4B) displaces the initial gate tile (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' “XOR”), it makes bonds with toeholds on the neighboring four tiles but not with the displaced tile itself – therefore, information about which function should be computed must be con- tained in the neighboring tiles, and not just the gate tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' We achieve this goal by using a gate-specific toehold in the initial gate tile, which directs the incorporation of a translator tile in the final position of each wire, as shown in figure 4B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Now the translator tile contains information about which logic function should be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Thus, an arbitrary number of gate types may coexist in the same system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Simulations of two feedforward circuit computations are shown in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Using just a systolic array of XOR gates, a collection of parity outputs (involving different subsets of the inputs) are produced, incidentally creating a Sierpinski triangle pattern within the completed wires and gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The second circuit makes use of four gate types: a NOR gate that produces the same signal on both output wires, a NAND/XOR gate that produces NAND to the north and XOR to the east, a WIRECROSS that sends it south input to the north and its west input to the east, and a WIREPASS that sends its south input to the east and its west input to the north.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The positions and identities of the gates are laid out in the initial tile array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Computation of an N ×N circuit will take expected time O(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The tile system with eastward and northward latching binary wires and the five types of gate functions discussed above consists of 66 tile types altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' For a circuit that can be laid out effectively in this format, an area of O(N 2) tiles can support N 2 gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Arbitrary feedforward ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='circuits with N gates can be implemented in O(N 2) area ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='using a standard crossbar array architecture (for exam- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='ple see [34]) and a new FANOUT gate that copies one ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='wire ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='bottom ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='top ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='signal 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='latch 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='signal 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='latch 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='irreversible transmission of signals 0 and 1: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='translator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='0 ⇌ 0⊕ (W-E) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='translator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='1 ⇌ 1⊕ (W-E) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='translator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='0 ⇌ 0⊕ (S-N) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='translator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='1 ⇌ 1⊕ (S-N) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='XOR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='XOR 00 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='XOR 01 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='XOR 10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='XOR 11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='❶ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='❷ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='❸ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='❹ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='Figure 4: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='(A) Irreversible wire with no warnings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The next effect of two separate, independent tile displacement steps is shown for each wire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Importantly, no unreliable tile displacement reactions are possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' (B) An XOR gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The gate itself is entirely reversible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' latch steps in the input and output wires are sufficient for ensuring net progress within a circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' input and ignores the other (using 75 tiles types if the new gate is just added, or 57 tile types if the redundant NAND/XOR and XOR gates are removed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Can we do better than just feedforward circuits?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' It is clear from inspection that our latching binary-signal wires can transmit information in either direction, depending on where the signal first arrives, and it is straightforward to implement gates that receive inputs from any two sides and produce outputs on the two other sides, so we can ar- range for signals to go around in cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Furthermore, the 7 A B T = 0 T = 1000 T = 2000 T = 3000 T = 0 T = 1000 T = 2000 T = 3000 Figure 5: (A) A 9 by 9 array of XOR gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Black tiles are “caps” that terminate the output wires.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' (B) A 18-input 18-output logic circuit composed of XOR, NAND, and NOR gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Two types of wire routing are implemented with a WIRECROSS tile that sends it south input to the north and its west input to the east, and a WIREPASS tile that sends its south input to the east and its west input to the north.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The simulation time T measures the number of tile displacement reactions that have occurred, rather than the real time in the Gillespie simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' tile displacement model in principle allows displacement to occur an arbitrary number of times in a given location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' As a trivial example, the reversible wire of figure 3A can endlessly perform a random walk, back and forth forever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' This raises the prospect of a tile displacement system of size N simulating a recurrent (iterated, feedback) circuit of size O(N), which can perform computations of length 2O(N) – exponentially more than what a feedforward cir- cuit of the same size can do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' This is to say, with respect to the size of the initial array and tile set, our existing construction can solve PTIME problems, while a reach goal would be to solve PSPACE problems like recur- rent circuits can.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Unfortunately, this is not compatible with the use of latching wires to ensure linear-time signal propagation: an area-N tile array initially has at most O(N) open toeholds, and thus at most O(N) irreversible 8 tile displacement steps can take place before the system comes to a standstill – or more precisely, until it must henceforth rely exclusively on reversible steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' 1D CA SPACE-TIME HISTORIES Boiling down what we learned about circuits to its re- versible essence, we can re-implement the above compu- tations using fewer tile types, more compact layouts with just one tile per logic gate, and power for driving the com- putation forward coming from concentration differences rather than from irreversible toehold formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' We start by providing generalized construction for sim- ulating the space-time history of one-dimensional block cellular automata (1D BCA) that is very similar to their simulation by algorithmic self-assembly of DNA tiles [35, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The instantaneous state of a 1D BCA is just a one-dimensional array of symbols from a given alphabet A, and in each time step the entire array is synchronously updated by applying a rule (x, y) → (f(x, y), g(x, y)) to a partitioning of the array into pairs, where f and g are functions that define the BCA and the parity of the par- tition alternates on each time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The size of the ar- ray may be infinite, finite, or expanding, with given ini- tial state and boundary conditions (typically a finite core then periodic).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Our tile displacement system construc- tion, shown in figure 6A, makes use of 2 + 2N + N 2 tile types for a 1D BCA with an alphabet of size N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The initial array uses 1 tile in the lower-left corner, N tiles to define input boundary conditions to be fed in at each time step from the left, N tiles to define the input boundary conditions to be fed in at each time step from the bottom, and 1 tile type filling in the remaining “blank” uncom- puted region of the array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The remaining N 2 tile types encode every input/output case for the update rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' For example, a binary alphabet (N = 2) will result in 10 tile types (figure 6B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The nth synchronous update of the 1D BCA will be encoded in the nth diagonal of the tile ar- ray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Similar to the gate tiles in the circuit construction, displacement must match all four open toehold positions, else it will be energetically unfavorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' This can only happen when both the tile to the left and the tile below have already updated, thus ensuring that the computed information is based on the correct information from the preceding diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Because our model insists that any tile that can be displaced in a simulation must have a non-zero concen- tration as a monomer in solution, every reaction will be reversible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' However, by chemostatting the blank tile at a lower concentration than the rule tiles, each dis- placement reaction can be biased forward by some factor r = crule/cblank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' From detailed balance of the CRN and CTMC, this ensures that the equilibrium probability of the rule-tile containing assembly is r times higher than that of the blank-tile containing assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Although the system will never get irreversibly locked into a final out- put assembly state, the complete assembly with all rule tiles in place will be rm times more likely than an assem- bly with m blank tiles still present, which we consider “good enough”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Note that if a final irreversible step is desired to lock in place the completed computation, this is also possible by adapting the techniques used in the circuit construction, just at the upper right corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Comparing the circuit construction of figure 5A to the cellular automaton space-time history construction in fig- ure 6BC, both of which compute parallel systolic arrays of XOR gates, we see that for the same size array, the cellular automaton approach computes roughly 9 times more gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' It also uses just 10 tile types, compared to 30 for the circuit construction (if the tiles used for logic gates other than XOR are omitted).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' However, our cellular automaton construction, by its very nature as a cellular automaton, receives information only in the initial 1D boundary conditions, and thus an assembly cannot specify a two-dimensional layout for the circuit that will be computed by tile displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' A sim- ple modification of the ideas resolves this apparent limita- tion: we generalize the construction to cellular automaton transformers whose cell update now depends both on the current state (x, y) ∈ A1 and a time-and-space-dependent input pattern (p, q) ∈ A0, as shown in figure 6D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Instead of an initial array containing uniform blank tiles, the ini- tial array will contain a layout of “pattern” tiles that each encode the information p that the gate below it will need to read, as well as the information q that the tile to its left will need to read.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' If A1 is size N and A0 is size M, then there are 2N input tiles, M 2 pattern tiles, and N 2M 2 rule tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Each reversible tile displacement reac- tion now must match four variable pieces of information, in two pattern toeholds and two state toeholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' As shown in figure 6EF, laying out exactly the same circuit as in figure 5B now requires 9 times less space, uses just 39 tile types (N = 2 state bit values plus a terminator, M = 5 logic functions, but not all combinations are needed) in- stead of 57, and, with concentration bias again, computes significantly faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Both these constructions exhibit strong similarities to computation via algorithmic growth during self-assembly of tiles – in the first case, 2D tiles growing a 2D structure from a 1D boundary [35, 2], and in the second case, 3D tiles growing an additional layer on top of a patterned 2D initial assembly [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' A significant difference is that rather than growing in size, the tile displacement system always remains the same size;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' rather than each tile attach- ment requiring new bond energy to counteract the lost en- tropy due to localization of the tile, the tile displacement system remains neutral with respect to bond energy be- cause each incoming tile is balanced by an outgoing tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Thus, rather than finding suitable operating conditions by balancing temperature (controlling the bond energies) against tile concentrations (which simultaneously affect the kinetics), in tile displacement we balance concentra- tion against concentration (which permits similar bias at different speeds and temperatures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' These benefits reflect similar observations about the increased robust- 9 B C input 0 (N) input 1 (N) input 0 (E) input 1 (E) blank XOR00 XOR10 XOR01 XOR11 E T = 0 T = 1000 T = 2000 T = 3000 NOR00 NOR10 NOR01 NOR11 XOR-NAND00 XOR-NAND10 XOR-NAND01 XOR-NAND11 XOR01 XOR01 XOR10 diagonal-wire01 XOR-NAND11 XOR-NAND10 A 1D block cellular automaton update rule: (������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' ������������) → (������������ ������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' ������������ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' ������������(������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' ������������)) corner input (N) input (E) blank rule 1D block cellular automaton transformer update rule: state (������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' ������������) pattern (������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' ������������) → state (������������,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' ������������) corner input (N) input (E) pattern rule D F T = 0 T = 30 T = 60 T = 90 corner Figure 6: (A) General case implementation of 1D block cellular automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Here a and b, written in roman font, denote specific toeholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' In contrast, x, y, f and g are variables and thus shown in italics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' There will be a separate rule tile for each possible pair x, y ∈ A, with f and g being dependent on x and y, and similarly for the input tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' (B) An example 1D block cellular automaton that computes the same function as the circuit shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' 5A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' (C) Simulation snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' (D) General case implementation of 1D block cellular automaton transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' (E) An example 1D block cellular automaton transformer that computes the same function as the circuit shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' 5B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' (F) Simulation snapshots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' 10 ness of strand displacement and toehold exchange com- pared to direct hybridization of complementary oligonu- cleotides [37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Seen more generally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' tile displacement systems involve reconfiguration of a constant-sized assem- bly via local propagation of information,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' which is remi- niscent of the distinction between crystal growth from monomers in dilute solution (the case generally assumed in algorithmic self-assembly of DNA tiles) versus crys- tallization from the melt (wherein the initial state is a disorganized constant-density liquid of monomers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' within which crystalline order locally propagates during crystal growth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Have we identified new concepts for tile displacement systems that allow us to perform more computation in a limited space?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Powering computation forward via con- centration bias in reversible reactions has given rise to compact constructions that naturally avoid the unrelia- bility warnings that plagued our initial wire and circuit constructions, but the computational power still remains PTIME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' One way of looking at this is that the free en- ergy of the assembly, G(A) decreases every time a higher- concentration tile replaces a lower-concentration tile, yet the minimum (most favorable) free energy occurs if all tiles in the array are highest-concentration tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' That is to say, the free energy is bounded below, and if each for- ward computational step is biased by a minimum amount, there are a bounded number of such steps that can occur before the computation is done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The situation is not so different from the limitation we encountered when power- ing computation by new toehold formation in irreversible displacement steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Is this limitation to PTIME a feature of tile displacement systems in general, or is it particular to the lack of imagination in the constructions we have presented so far?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' 2D CA IN-PLACE EXECUTION We can get some ideas from the notion of a cellular au- tomaton transformer, which reads a 2D pattern as a wave of activity passes over it, leaving a new pattern in its wake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Suppose that the new pattern can be read by a sec- ond wave, corresponding to a second cellular automaton transformer using a new set of rule tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' For example, the initial pattern might use toehold alphabet A0, the first cellular automaton transformer uses states in alpha- bet A1 and writes a new pattern using alphabet A2 by utilizing the two locations that, in figure 6D, have useless inert “−” toeholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Then, the second cellular automaton transformer can read A2, store its transient state in A3, and write a third pattern using A4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' To drive the com- putation forward, the first transformer’s rule tiles should have a higher concentration than the pattern tiles, and the second transformer’s rule tiles should have a higher concentration than the first transformer’s rule tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' This idea could be extended to K waves, each with its own set of rule tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' This would improve upon the previous constructions, in which each location in the array expe- riences just net one forward tile displacement step – at that location, either one has the initial tile, or the final tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Whereas, in an implementation of a multiple-wave cellular automaton transformer, each location would go through a sequence of changes, one for each wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' In a sense, we achieve K-fold more computation within the same assembly area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' This is somewhat analogous to freez- ing cellular automata, which are restricted to change a cell’s state a limited number of times [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' There are two problems here, as you have probably already noticed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' First, if the concentration ratio from wave to wave is r, then a K-wave computation requires a ratio of rK between the lowest-concentration tiles and the highest-concentration tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' That quickly becomes im- practical, and theoretically unappealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Second, each wave requires a new set of tiles – yet for PSPACE com- putations we would require an exponential number of tile updates and thus a comparable number of waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' So this idea doesn’t get us where we want to go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' To keep a constant number of tile types while allowing an unbounded number of tile displacement steps per site, perhaps we could have a small number K of waves, but have wave K output its new pattern using alphabet A0 so that the tiles of wave 1 can read it – thus allowing iterated computation, such as binary counters and perhaps univer- sal space-bounded algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' This is indeed the essence of the construction we’ll arrive at, but it comes at a cost: for wave 1 tiles to displace wave K tiles, they cannot be at a lower concentration, which basically implies that all rules tiles must be at the same concentration, and we have no concentration bias pushing the computation forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' (This conclusion is not specific to periodic waves of cel- lular automata transformers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' it follows in general that if we want to implement a computation that may update a given site an unknown and unbounded number of times, then every tile type may at some point be an incoming tile and at other times be the outgoing tile, so the con- centrations of all rule tiles must be equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=') If we have already accepted that our designs should exclusively use bond-energy neutral tile displacement, then in fact the bond energy and free energy of our assembly will remain constant over time – we are truly dealing with reversible computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Thankfully, reversible computation is by no means impossible [14, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Our approach will be to exhibit a surprisingly natural correspondence between certain tile displacement systems and the well-studied class of two-dimensional block cel- lular automata (2D BCA) that arose in the study of re- versible computation by discrete models of ballistic phys- ical dynamics [20, 21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The 2D BCA model is a natu- ral generalization of the 1D BCA discussed above: rather than partitioning a 1D array into pairs of cells that get synchronously rewritten with alternating partition par- ity on alternate time steps, we now partition a 2D array into 2 × 2 blocks of cells that get synchronously rewrit- ten with alternating partition parity on alternating time steps (compare figure 7A with figure 7E).' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='0 steps ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='time sheet 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='time sheet 8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='������������ = 25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='������������ = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='������������ = 500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='������������ = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='D ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='→ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='F ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='↘ ↙ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='↗ ↖ ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='↙ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='↘ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='bottom ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='view ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='top ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='view ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='side ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='view ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='21 tile attachment steps ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='time sheet 21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='������������ = 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='������������ = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='������������ = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='������������ = 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='Figure 7: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content='(A) Execution of a synchronous 1D block cellular automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' (B) Asynchronous 2D tile self-assembly that simulates the computation in (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' (C) and (D) Simulations of two example 2D block cellular automata: Billiard Ball Model (C) and Critters (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' (E) Execution of a synchronous 2D block cellular automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' (F) Asynchronous 3D tile self-assembly that simulates the computation in (E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' allows the rewrite rules to be arbitrary functions f �� a b d c �� = � w x z y � but if the rewrite function is a bijection, then the 2D BCA is logically reversible in the sense that iterating with f −1 instead of with f will bring the simulation backwards in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The most famous 2D BCA rule, the Billiard Ball Model (BBM), is logically reversible, rotationally and mirror symmetric, conserves the total number of 1s, can directly simulate reversible circuits, and with an infinite periodic initial state can simulate universal Turing ma- chines [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Example simulations of two binary-state 2D BCA, the BBM and “Critters”, are shown in figure 7CD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' With larger alphabets, 2D BCA can simulate arbitrary classical cellular automata and Turing machines, either of the irreversible or reversible variety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' (Generalizations to using blocks larger than 2 × 2 is also natural, but will not be considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=') There are three obstacles to implementing arbitrary 2D BCA as tile displacement systems, and we will solve them all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The first is that tile displacement reactions are asyn- chronous (occurring at random locations and in random orders) while 2D BCA require synchronous updates of the entire array (and fail utterly if the same update function is applied asynchronously with no other modifications).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The second is that the mechanics of tile displacement must be designed to avoid irreversible steps that close too many toeholds at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' And the third obstacle is that with exclusively reversible reactions and no concen- tration bias, there must be some other way to drive the system forward if we don’t want to wait forever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' For the first challenge, we adapt prior methods for im- buing asynchronous cellular automata with locally syn- chronizing mechanisms [41, 42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The specific approach used here generalizes the approach used for simulation of 1D cellular block automata space-time histories in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Figure 7A gives an example of a 1D BCA, with boxes highlighting the partitioning into pairs with alternate parity on each synchronous time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Fig- ure 7B shows the same computation interpreted as 2D tile self-assembly where, starting from the 5 tiles at the bottom that encode the 8 input bits as well as their par- titioning, rule tiles attach whenever they can match two sides of existing tiles in the assembly, thus asynchronously growing the space-time history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' We have augmented the tiles with arrows that point to where incoming tiles could attach;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' thus, in the initial assembly of 5 tiles, the sites 12 where tile can attach are exactly those locations where ar- rows are pointing inward toward the incoming tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' A cut through the assembly’s space-time diagram corresponds a particular moment during the asynchronous self-assembly process – we show a cut after 0 tile additions (yellow) and another after 8 tile additions (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' We call these “time sheets” because at different horizontal (x) posi- tions, they are at different heights (t), and thus reading out the binary (black/white) states along a time sheet path correspond to states at different time steps of the underlying synchronous cellular automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Nonetheless, the time-sheet state information, augmented with the rel- evant arrows, is all that is needed to correctly complete the computation using an asynchronous update rule that executes only when arrows point toward each other, oth- erwise leaving the cells untouched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' This process exactly mimics the self-assembly of the deterministic space-time history, despite its non-deterministic order of execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' There is an exactly analogous arrow-augmented asyn- chronous update rule for 2D BCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Rather than square tiles, we now have truncated octahedra as “tiles”, but the self-assembling structure is again a space-time history of the correct synchronous cellular automaton computa- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Tiles may attach when they match four hexagonal faces of existing tiles in the assembly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' (The small square faces are inert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=') Again, if we imagine arrows orthogonal to the hexagonal faces of tiles, pointing out of the tile, then valid sites for attachment of a new tile correspond exactly to situations where all four arrows on the match- ing faces are pointing toward each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The growth front for a give stage of assembly again corresponds to a (now two dimensional) time sheet, and we can write out the states of each exposed hexagonal face in a two dimensional array along with the orientation of its corre- sponding arrow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' It is now a simple observation that the asynchronous addition of a tile corresponds exactly to an asynchronous update of a 2 × 2 block with four inward- pointing arrows, resulting in updates of the four cells and reversing all four arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Another way of thinking of it is that after a block asynchronously updates, it will not be able to update again until all four overlapping 2 × 2 blocks have first updated and flipped the arrows back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Thus, the arrow-augmented asynchronous updating cor- responds exactly to synchronous parallel updating with alternating-parity partitioning into blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The second challenge is to implement this type of asyn- chronous block cellular automaton updating rule using tile displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Our construction, shown in figure 8, introduces additional complexities due to the fact that all tile displacement reactions are physically reversible, even if the 2D BCA logic update rules are irreversible, combined with the need to ensure that when one toe- hold is closed, the neighboring toehold must be opened – thus we must be able to guarantee a mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Tripling the cell state alphabet by adding α, β, γ markers solves both problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' For each 2D BCA update case, we make three tiles, one inputting α-symbols and outputting β- symbols, another inputting β-symbols and outputting γ-symbols, and the third inputting γ-symbols and out- putting α-symbols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' When an α → β tile inserts into the array, that simulates a forward-time asynchronous up- date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The swapping of which toehold is open and which is closed reflects the flipping orientation of arrows in the asynchronous cellular automaton;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' we can read the arrows from a tile array by looking at the open toeholds and drawing the arrow from α to β, from β to γ, or from γ to α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Boundary conditions for finite arrays must also be handled, using the same principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Each side of a tile encodes the state of a specific cell in the 2D BCA (at a particular time mod 3, as per α- β-γ of the open toehold), and thus the grid of simulated BCA cells is oriented at a 45◦ angle relative to the array of tiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' State being encoded on the sides of tiles also facilitates that each tile displacement step corresponds to an update of a whole 2 × 2 block, and the fact that two tiles share the same side location reflects that each cell in a 2D BCA can be updated either by an odd-parity block or an even-parity block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The final challenge concerns how to drive the compu- tation forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Let us first consider reversible 2D BCA rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' In this case, after any forward tile displacement step, there is exactly one monomer tile type that can re- verse the reaction: the tile that was just displaced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' What this means is that the full state space of the tile displace- ment system’s CTMC is essentially linear;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' though fat and fuzzy, it has the same thickness both arbitrarily far into the future and arbitrarily far into the past.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' The thickness has to do with all the possible contours of the time sheet for a given average time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Thus we can say that the state space of the tile displacement system consists exclusively of correct reachable states of the computation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' for a re- versible 2D BCA simulating a compact recurrent circuit for solving a PSPACE problem, the tile displacement sys- tems’s state space will also be exponentially long and will reach the same correct conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Stochastic Gillespie simulation of the tile displacement CRN will result in an unbiased random walk back and forth along this fuzzy- linear state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' (Every assembly in this reachable state space has the same energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=') However, unlike a standard reversible Turing machine with Brownian dynamics [14], whose state spaces is strictly a linear graph so the ex- pected random walk hitting time for reaching the end of an T step computation is O(T 2), the time sheet diffuses much more slowly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' As a rough estimate for an N × N tile array that requires N 2 forward updates to move the time sheet 1 net synchronous update step into the future under ideal circumstances, the same N 2 updates if half forward and half backward will be expected to net move the time sheet N steps either forward or backward, which corresponds to just 1/N equivalent synchronous update steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' This being just a polynomial inefficiency, perhaps we should not be too concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' More interesting is what happens if the 2D BCA rules are irreversible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' This means there are multiple cases for 13 A B ❶ ❷ ❸ ❹ 2D block cellular automaton update rule: (������������, ������������, ������������, ������������) → (������������, ������������, ������������, ������������) rule (time sheet γ → α) rule (time sheet β → γ) rule (time sheet α → β) Figure 8: (A) General case implementation of 2D block cellular automaton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' (B) Example updates in the Billiard Ball Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' the 2×2 block input that map to the same output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' There- fore the state space for the tile displacement system will be exponentially branched in the backwards-in-time di- rection (as pictured by Bennett in figure 10 of his re- view paper [14], but thicker and fuzzier).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Consequently Brownian dynamics will tend to be entropically biased toward where there are more states, and the system will run backwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Can this entropic driving force be used to encourage a system to perform a desired computation by designing a system whose reverse dynamics are what we want?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Attempting to do so would be risky, and probably futile, because the 2D block update rule being irreversible means that there are some states that have no local pre- decessor, and backward progress will get stuck as such 14 local configurations are encountered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' A better way to exploit an entropic driving force is to have nondeterministic, stochastic forward update rules added to an otherwise-reversible system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' For exam- ple, consider a tile displacement simulation of the BBM model, with boundary tiles designed to implement a re- flecting boundary (as they must for the system to remain reversible).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' If we design a special boundary tile that can either reflect a ball or (in the forward direction) produce a new ball out of nothing, then we obtain a new system that is still entirely reversible in the sense that there ex- ists (at least one) possible applicable block update in all circumstances, so the system cannot get stuck either in the forward direction or the reverse direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' With uni- form tile concentrations, all assembly states will still be isoenergetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' But started with an empty N ×N array, for- ward updates of the special tile will about half the time produce a new ball, which will entropically drive the sys- tem to a density such that forward production of balls is balanced by the reverse reaction, the absorption of balls into the special tile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' At this point, which will be O(N 2) synchronous time steps in the future, the time sheet will stop advancing on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Another way of looking at it is that with all reactions being neutral, equilibrium will reflect equipartition among all reachable states, and the combinatorially greatest number of states will have a number of ball near the optimal density – so, that’s what we are likely to observe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' And the only way to get there is to run the time sheet forward enough to emit that number of balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' A gas-filled BBM simulation is not of great use by itself, but we can make use of it by also placing a circuit in the array, and drawing a 2-cell-thick wall around it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' In the BBM model, balls bounce off walls, and walls are stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Thus, despite random stochastic gas entering the areas of the array outside the box, the circuit will remain perfectly isolated from the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' But due to the time-sheet coupling enforced by the asynchronous arrow rules, the time sheet that is being driven forward by the expansion of the gas will simultaneously drive the circuit forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Unfortunately, for an array of area N 2, we will only drive the computation forward by O(N 2) steps – this is no better than the PTIME computational power of the original circuit construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Essentially, in a small con- fined space, our circuit “heats up” and stops working.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' To run it for a long time, we need a larger space into which we can release the simulated heat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' For example, if we are willing to entertain a half-infinite-plane array for tile displacement, we can draw a BBM wall down the middle, release gas on one side, and let the other side simulate an interesting recurrent circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Now, although the array is infinite (or very large) in direct proportion to how much computation we want to do, we can say that we have confined the interesting part of the compu- tation – the circuit itself – to a very small area relative to the potentially exponentially long computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' This isn’t PSPACE computation in terms of the size of the array, but rather in terms of the size of the part of the array that we care about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Similar constructions can be used to drive forward computation not just for other re- versible 2D BCA rules, but even for irreversible rules: the cellular automaton alphabet can be expanded to encode an inert “wall”, time sheets within the walled region and outside of it remain coupled, and sufficient entropy must be generated by stochastic rules outside the wall, to be dissipated into a sufficiently larger area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' DISCUSSION Tile displacement within arrays of square DNA origami tiles was discovered accidentally [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' While some aspects of the formal model, such as the four-sided generaliza- tion of toehold exchange, were invented for mathemat- ical elegance rather than detailed realism, they are not too far flung from what has been experimentally demon- strated and characterized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' So it is quite delightful that within the design space for tile displacement systems, we find natural implementations for feedforward circuits and one-dimensional cellular automata that compute in lin- ear time, powered by irreversible toehold formation or by concentration gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Even more delightful is that attempts to squeeze out more computational power per area seemed almost inevitably to lead us consider physical constraints such as energy, reversibility, and asynchrony – which in turn lead to classical two-dimensional cellular automata models that arose in early studies of the physics of computation [14, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Our strongest result (despite weak time efficiency) is that a tile displacement array of size N can reversibly simulate a recurrent reversible cir- cuit (via the Billiard Ball Model cellular automaton) for an arbitrary number of steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' In other words, the reach- ability question for tile displacement is PSPACE com- plete – a result strongly reminiscent of Thachuk & Con- don’s beautiful PSPACE-hardness result for CRNs and DSDs [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' We believe that there remains a lot undiscovered within the tile displacement design space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' For example, while our constructions showed that the asynchronous tile dis- placement model can simulate synchronous cellular au- tomata, the needed flipping-arrow mechanism for local synchronization seems almost built-in to the tile displace- ment model in the form of open and closed toeholds for toehold exchange, and it’s not obvious how to directly simulate asynchronous cellular automaton models such as reversible surface CRNs [45, 34, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' We might also ask whether using information within the branch migration domains rather than just in toeholds – or whether having even more toeholds and branch migration domains on a tile’s sides – could have advantages either theoretically or experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Further, the most interesting systems demonstrated experimentally in the initial work on tile displacement [3] involved systems of interacting multi-tile arrays, rather than a single array and a monomer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' How do our single-assembly results fit into that larger picture?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Fi- nally, might tile displacement systems be combined with 15 other molecular mechanisms to solve our problems driving the computation forward – for example, an oscillator [46] that periodically activates and deactivates the α, β, and γ monomer tiles in sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' When first discovered, the tile displacement mecha- nism seemed most closely related to strand displace- ment mechanisms, only two dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' However, as we investigated the capabilities of single-assembly tile dis- placement, many parallels to passive tile assembly [2] became prominent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Tile displacement systems appear to combine the principles of DNA strand displacement and self-assembly in different ways than hairpin-based programmable self-assembly [47], signal-passing tile self- assembly [48, 49], CRN-controlled tile assembly [50, 51], and other models we are aware of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Comparing the ben- efits, drawbacks, and relationships between these models may help uncover a more unified way of thinking about programmable molecular systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' And even if tile displacement systems, as explored the- oretically here, never become useful experimentally, we hope that it was interesting and perhaps inspiring to look long and deep at a simple mechanism until intricate pat- terns emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The authors thank William Poole and Ho-Lin Chen for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' This work was partially supported by NSF awards 2008589 and 1813550.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' [1] John E Savage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NAzT4oBgHgl3EQf9_7h/content/2301.01929v1.pdf'} +page_content=' Models of computation, volume 136.' 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(2022). An Efficient Hybrid Classification Approach for COVID-19 Based on +Harris Hawks Optimization and Salp Swarm Optimization. International Journal of Online and Biomedi- +cal Engineering (iJOE), 18(13), pp. 113–130. https://doi.org/10.3991/ijoe.v18i13.33195 + +1 + +An efficient hybrid classification approach for +COVID-19 based on Harris Hawks Optimiza- +tion and Salp Swarm Optimization + +Abubakr Issa, University of Technology, Baghdad, iraq +Yossra Ali, University of Technology, Baghdad, Iraq +Tarik Rashid, University of Kurdistan Hewler, KRG, Iraq + +Abstract— Feature selection can be defined as one of the pre-processing +steps that decrease the dimensionality of a dataset by identifying the most signif- +icant attributes while also boosting the accuracy of classification. For solv- +ing feature selection problems, this study presents a hybrid binary version of the +Harris Hawks Optimization algorithm (HHO) and Salp Swarm Optimization +(SSA) (HHOSSA) for Covid-19 classification. The proposed (HHOSSA) pre- +sents a strategy for improving the basic HHO's performance using the Salp algo- +rithm's power to select the best fitness values. The HHOSSA was tested against +two well-known optimization algorithms, the Whale Optimization Algorithm +(WOA) and the Grey wolf optimizer (GWO), utilizing a total of 800 chest X-ray +images. A total of four performance metrics (Accuracy, Recall, Precision, +F1) were employed in the studies using three classifiers (Support vector machines +(SVMs), k-Nearest Neighbor (KNN), and Extreme Gradient Boosting +(XGBoost)). The proposed algorithm (HHOSSA) achieved 96% accuracy with +the SVM classifier, and 98% accuracy with two classifiers, XGboost and KNN. + + +Keywords—— Feature selection, Hybrid Swarm intelligence, classification, +Covid-19, medical image +1 +Introduction + +Medical image processing can be defined as one of the most significant areas in +medical science, and it has a substantial effect on visualization applications. Also, med- +ical image processing has a broad range of applications in medical diagnoses (treating +and investigating diseases) and medical sciences (such as physiological and anatomi- +cal studies). Medical physics, medical engineering, biology, and optics are some of the + +Issa, A., Ali, Y., & Rashid , T. (2022). An Efficient Hybrid Classification Approach for COVID-19 Based on +Harris Hawks Optimization and Salp Swarm Optimization. International Journal of Online and Biomedi- +cal Engineering (iJOE), 18(13), pp. 113–130. https://doi.org/10.3991/ijoe.v18i13.33195 + +2 + +fields of science that make up this medical science. With the discovery of X-rays, Wil- +liam Roentgen initiated the first efforts at contemporary medical imaging. Coronavirus +(COVID-19), also known as SARS-Corona Virus-2, is a virus that results in causing +severe acute respiratory syndrome (SARS-CoV2), is a viral infection that first occurred +in Wuhan at the end of 2019. Due to such an outbreak, COVID-19 became a pandemic, +threatening human lives and wreaking havoc on the economy. Therefore, many stud- +ies have been launched in an attempt to identify a way to restrict mortality and spread. +Those researches include the suggested treatment strategy, the screening method for +early-stage patients, and the evaluation of different phases and recovery of treated pa- +tients. In hospitals, imaging techniques like chest X-rays are commonly utilized for +detecting the severity and existence of COVID-19 pneumonia [1][2]. For improving +the suggested system's training, X-ray images are often maintained in a medical data- +base for subsequent investigation by multiple research organizations. Low contrast, +noise, blurs, and faded colors are frequent problems, and images should be pre-pro- +cessed to enhance quality by reducing noise. +The second stage is image segmentation, which depends on some attributes includ- +ing color, texture, and depth measurements. The type of image and characteristics of +the problem (disease) are chosen to determine which segmentation technique is used. +The identification and extraction of features is the third stage. As the number of features +that have been extracted from the image grows, the accuracy of classification decreases. +In the classification vision, we can call it the curse of dimensionality. Feature optimi- +zation is a viable option for dealing with this issue.[3] +The 4th stage is the feature selection that has been obtained from the known proper- +ties using robust Optimization algorithms for better disease identifications from the +medical images. The image was classified using one of the classifiers. Feature selection +is a step in the preprocessing process that tries to increase the relevancy of obtained +data by deleting irrelevant characteristics and choosing just relevant or useful variables +[5]. Feature selection comprises reviewing feature subsets, employing certain search +approaches to locate the best feature subset, assessing the chosen features, stopping cri- +teria, and subset validation in general.[6] +There are three types of feature selection classifiers: wrapper schemes, filer +schemes, and embedding schemes. The filter method, in contrast to the wrapper +scheme, which is characterized by good classification accuracy and low speed, is rapid +but inaccurate. The embedded system is preferred in the case when handling a certain +model [7]. Filter techniques use the qualities of training data to assess the quality of +features. Those approaches do not employ machine learning algorithms. Before choos- +ing features with the highest score, filter methods usually take into account the score of +all features. At the same time, other filtering approaches favor features with the greatest +score per iteration [8]. Other well-known methods, like the correlation-based feature +selection approach in [9] as well as dimensionality reduction methods and NNs in [10], +can greatly decrease computational load and system complexity. Filter approaches +overlook the performance regarding the chosen characteristics despite their speed and +low computational cost [11]. + +Issa, A., Ali, Y., & Rashid , T. (2022). An Efficient Hybrid Classification Approach for COVID-19 Based on +Harris Hawks Optimization and Salp Swarm Optimization. International Journal of Online and Biomedi- +cal Engineering (iJOE), 18(13), pp. 113–130. https://doi.org/10.3991/ijoe.v18i13.33195 + +3 + +Wrapper approaches utilize an evaluation algorithm to assess the specified features’ +quality. SVMs, Decision trees (DTs), KNN, Naïve Bayesian (NB), linear discriminant +analysis (LDA), local neighborhood structure preserving embedding (LNSPE), artifi- +cial neural networks (ANNs), and local geometrical structure Fisher analysis +(LGSFA) are some of the major wrapper’s methods utilized for feature selection. In +almost all cases, wrapper approaches outperform filter ones. Meta-heuristic algorithms +are more advanced search algorithms that result from the evolution and expansion of +feature selection. For instance, ongoing research to increase the performance regard- +ing evolutionary algorithms (EA) like Genetic Algorithms (GAs) and Swarm Intelli- +gence (SI) like Particle Swarm Optimization (PSO), Artificial Bee Colony (ABC), and +Ant Colony Optimization (ACO) are underway. Grasshopper Optimization Algorithm +(GOA), Grey Wolf Optimizer (GWO), Butterfly Optimization Algorithm (BOA), Har- +ris Hawks Optimization (HHO) Whale Optimization Algorithm (WOA), and Ant Lion +Optimization (ALO) are examples of recent algorithms. Metaheuristic algorithms are +classified according to their exploration and exploitation phases into single solution +based (i.e., Tabu Search (TS) and Simulated annealing (SA)) or population size based +(in other words, GA, ACO, and PSO). The key contributions of this research are listed +below: + +• Suggest an effective hybrid classification method for COVID-19 with the use of +the hybrid swarm algorithms (HHO, SSA). +This novel hybrid algorithm must improve resource consumption and performance, +as well as storage capacity, reducing processing time. +• With the use of multiple classifiers (KNN, SVM, XGboost), test the sug- +gested (HHOSSA) algorithm on datasets containing some positive negative COVID-19 +chest X-ray scan images. +• Individual, hybridized predictor models and state-of-the-art techniques (WOA, +GWO) are compared in terms of performance. + +The sections of this paper are organized as follows: Section 2 provides a concise +summary of some of the most related works. Section 3 discusses methodology. In sec- +tion 4 we described in detail our proposed approach. Tools are illustrated in section 5. +Performance evaluation is described in section 6. Results and discussion are included +in section 7. Finally, the conclusions and future works are stated in section 8. +2 +Related Works +Many studies have employed hybrid algorithms to handle a range of challenges re- +cently. Hybrid algorithms have received a lot of attention lately, notably in feature se- +lection optimization. Low-level hybrid algorithms and high-level hybrid algorithms are +the 2 forms of hybrid algorithms. There are 2 types of hybridization schemes in high- +level hybrid algorithms: high-level teamwork hybridization (HTH) and high-level rely- +on hybridization (HRH). The self-contained meta-heuristics have been carried out in + +Issa, A., Ali, Y., & Rashid , T. (2022). An Efficient Hybrid Classification Approach for COVID-19 Based on +Harris Hawks Optimization and Salp Swarm Optimization. International Journal of Online and Biomedi- +cal Engineering (iJOE), 18(13), pp. 113–130. https://doi.org/10.3991/ijoe.v18i13.33195 + +4 + +order in HRH, whereas in the HTH, one algorithm assists the other by supplying infor- +mation via cooperative search. Low-level hybridization has been separated into two +types: low-level teamwork hybrid (LTH) and low-level rely-on hybrid (LRH), both of +which contain one meta-heuristic algorithm [12]. In the feature selection field, it has +been observed that hybrid algorithms surpass native algorithms concerning perfor- +mance. In the year 2004, the search process has been controlled by merging local search +approaches with a GA algorithm, which was the first time a hybrid metaheuristics ap- +proach was utilized in feature selection. A combination with the EGA filter has +been provided in a wrapper technique for text categorization [13]. A hybrid ap- +proach for feature selection has lately been created in various metaheuristic algorithms. +In [13], the Binary Grey Wolf algorithm was combined with the Harris Hawks algo- +rithm to create an excellent balance between exploitation and exploration to prevent +local optimum solutions and increase solution precision. Harris Hawks was hybridized +in [14] using Bitwise operations and Simulated Annealing for supporting the HHO al- +gorithm's exploitation capacity and getting out of local optima. In [15], the Salp swarm +algorithm was used to modify teaching–learning based optimization. This integration +gives TLBO more flexibility in the exploration of population and achieving variety +while also allowing it to swiftly attain the optimal value. They combined the Salp +swarm algorithm with the Particle swarm algorithm in [16], in which the SSA was uti- +lized for updating the salps positions and the PSO was utilized otherwise. This hybrid- +ization was utilized for the improvement of the exploration and exploitation of the Salp +swarm algorithm. +3 +Methodology +3.1 Harris Hawks optimization algorithm +HHO can be defined as one of the swarm metaheuristic algorithms inspired by Har- +ris Hawks' hunting behavior of "seven kills" or "surprise pounce." Based on the prey's +fleeing behavior nature, hunting duration can range from some seconds to many hours. +The modeling algorithm of HHO is split into 2 parts (exploitation and exploration). +Harris' hawks have been employed as candidate solutions in the HHO algorithm, with +the best candidate solution reflecting the desired or optimum prey in each stage [17]. +The first phase pertains to the process of perching and detection of the prey. The algo- +rithm simulates Harris' hawks' perching methods in 2 separate scenarios. Harris' hawks +are assumed to perch in various locations during their group home range in the first +scenario. In Eq (1), q=0.50 models that condition. + +X1 +⃗⃗⃗⃗ (t+1)= { +𝑋𝑟𝑎𝑛𝑑(𝑡) − 𝑟1|𝑋𝑟𝑎𝑛𝑑(𝑡) − 2𝑟2𝑋(𝑡)|, 𝑞 ≥ 0.50 +(𝑋𝑟𝑎𝑏𝑏𝑖𝑡(𝑡) − 𝑋𝑚(𝑡)) − 𝑟3(𝐿𝐵 + 𝑟4(𝑈𝐵 − 𝐿𝐵)), 𝑞 < 0.50( 1 ) + +While the other likelihood is that Harris' hawks would perch on positions near other +swarm members and prey. This condition has been introduced in Eq1 for q < 0.50: + +Issa, A., Ali, Y., & Rashid , T. (2022). An Efficient Hybrid Classification Approach for COVID-19 Based on +Harris Hawks Optimization and Salp Swarm Optimization. International Journal of Online and Biomedi- +cal Engineering (iJOE), 18(13), pp. 113–130. https://doi.org/10.3991/ijoe.v18i13.33195 + +5 + +where X1 +⃗⃗⃗⃗ (t+1) is Hawks' position vector, t represents the following iteration, +𝑋𝑟𝑎𝑛𝑑(𝑡) is a hawk that has been chosen at random from the current population, +𝑋(𝑡) represents the position vector of hawks, r1, r2, r3, r4, and q represent random +numbers in the range of (0,1), Xrabbit(t) represents rabbit position, Xm denotes the aver- +age position of the current population of the hawks, lower and upper bounds for gener- +ating random locations inside the Hawks' stadium are Lb and Ub, respectively. + +While in the phase of exploitation, the Harris' hawks attack prey which has been +identified in the preceding step. The algorithm has 4 different possibilities for modeling +various attacking styles that have been utilized by Harris' hawks. +While r denotes the probability of prey escaping, successful escape has been donated +by r < 0.50, whereas r ≥ 0.50 denotes failure to escape. Depending upon the prey's +chances of escaping (r), hawks will use either soft or hard besieges to catch prey. The +algorithm's parameter E has been utilized for the determination of the type of attacking +besieges. If the prey is unable to escape when r ≥ 0.50 hard besiege happens when |E| +< 0.50 and soft besiege takes the place in the case where |E|≥ 0.50 The mathematical +Modelling of soft besiege has been represented by Eqs (2) through (3), and hard besiege +has been shown by Eq (4): + +𝑿(𝒕 + 𝟏)=∆X(t) –E|JxXrabbit (t) –X(t)| ( 2 ) +∆(t) =Xrabbit (t) –X(t) ( 3 ) +X( t+1) =Xrabbit(t) –E|∆X(t)| ( 4 ) + +In the case of successful escaping of the prey (r<0.50), soft besiege with a progres- +sive rapid dive take is applied in the case where |E|≥ 0.50 as shown in Eq (5), Eq (7), +Eq(8) while Hard besiege with the progressive fast dive occurs in a case where |E|≥ +0.50 as shown in Eqs (6), (7), and (8): + +𝒀 = 𝑿𝒓𝒂𝒃𝒃𝒊𝒕(𝒕) − 𝑬|𝑱 ∗ 𝑿𝒓𝒂𝒃𝒃𝒊𝒕(𝒕) − 𝑿(𝒕)| ( 5 ) +𝒀 = 𝑿𝒓𝒂𝒃𝒃𝒊𝒕(𝒕) − 𝑬|𝑱 ∗ 𝑿𝒓𝒂𝒃𝒃𝒊𝒕(𝒕) − 𝑿𝒎(𝒕)| , 𝑿𝒎(𝒕) = +𝟏 +𝑵 ∗ ∑ +𝑿𝒊 +𝑵 +𝒊=𝟏 +(𝒕) ( 6 ) +𝒁 = 𝒀 + 𝑺 × 𝑳𝑭(𝑫) ( 7 ) + +𝑿(𝒕 + 𝟏) = {𝒀, 𝒊𝒇 𝒇(𝒀) < 𝑭(𝑿(𝒕)) +𝒁, 𝒊𝒇 𝒇(𝒁) < 𝑭(𝑿(𝒕)) ( 8 ) + +D represents the problem dimension and S represents the random vector by 1xD size +and LF represents the function of levy flight, estimated with the use of Eq. (9): + +𝑳𝑭(𝒙) = 𝟎. 𝟎𝟏 × +𝒖 ×𝛔 +|𝒗| +𝟏 +𝜷 + , 𝛔 = ( +(𝜞(𝟏+𝜷) ×𝒔𝒊𝒏 (𝝅𝜷/𝟐) +𝜞(𝟏+𝜷 +𝟐 )×𝜷×𝟐(𝜷−𝟏 +𝟐 ) ) ( 9 ) + +The energy of a rabbit is modeled as 𝑬 = 𝟐𝑬𝟎 (𝟏 − +𝒕 +𝑻) ( 10 ) +Where E represents the prey’s escaping energy, T represents the maximal number of +iterations, and Eo represents its initial energy state. + +Issa, A., Ali, Y., & Rashid , T. (2022). An Efficient Hybrid Classification Approach for COVID-19 Based on +Harris Hawks Optimization and Salp Swarm Optimization. International Journal of Online and Biomedi- +cal Engineering (iJOE), 18(13), pp. 113–130. https://doi.org/10.3991/ijoe.v18i13.33195 + +6 + + +3.2 Salp swarm optimization algorithm + +SSA can be defined as a swarm metaheuristic algorithm [18] that was created for +solving various optimization problems. It was inspired by the activity of Salps in na- +ture; salps are a type of jellyfish with tissues comparable to jellyfish and a high water +percentage in their moving behavior and weights [19]. They move by contracting their +bodies and shifting positions by pumping water through them. The salp chain describes +the swarming behavior of salps in the ocean. By allowing for faster and more harmonic +changes, this behavior could benefit salps in foraging and better movement. [18] Salp +chains were theoretically modeled and after that tested in optimization problems as a +result of this characteristic [16]. The algorithm starts its work by dividing the generated +population into 2 parts (which are: leader and followers ( where the leader leads the salp +chain and the remaining salps play the role of followers. A salp uses the food source as +a target in an n-dimensional search space. The following equation has been used to +update the leader's position: + +𝑿𝒋 +𝟏 = { +𝑭𝒋 + 𝒓𝟏 ((𝑽𝒎𝒂𝒙𝒋 − 𝑽𝒎𝒊𝒏𝒋)𝒓𝟐 + 𝑽𝒎𝒊𝒏𝒊) , 𝒓𝟑 ≥ 𝟎 +𝑭𝒋 − 𝒓𝟏 ((𝑽𝒎𝒂𝒙𝒋 − 𝑽𝒎𝒊𝒏𝒋)𝒓𝟐 + 𝑽𝒎𝒊𝒏𝒊) , 𝒓𝟑 < 𝟎 +} ( 11 ) + +Where 𝑋𝑗 +1 represent the position of leader in the jth dimension and Fj is food's loca- +tion. The upper is represented by 𝑉𝑚𝑎𝑥𝑗 and the lower bounds that have been denoted +by 𝑉𝑚𝑖𝑛𝑗. The search space is maintained using the 2 random variables 𝑟2 & 𝑟3 in the +range [0, 1]. + +The parameter 𝑟1 is also an important control parameter in the process of exploration +and exploitation and it is calculated by using Eq (12). + +𝒓𝟏 = 𝟐𝒆(−𝟒𝒕 +𝑵 )𝟐 ( 12 ) + +Where t represents the current iteration and N denotes the maximum amount of iter- +ations. In a case where the position of the leader has been changed, Eq (13) is used to +change the followers' position: + +𝑿𝒋 +𝒊 = +𝟏 +𝟐 (𝑿𝒋 +𝟏 − 𝑿𝒋 +𝒊−𝟏) ( 13 ) + +Where 𝐗𝐣 +𝐢 denotes the ith follower's position in the jth dimension, and the value +of I must be > 1. + + +Issa, A., Ali, Y., & Rashid , T. (2022). An Efficient Hybrid Classification Approach for COVID-19 Based on +Harris Hawks Optimization and Salp Swarm Optimization. International Journal of Online and Biomedi- +cal Engineering (iJOE), 18(13), pp. 113–130. https://doi.org/10.3991/ijoe.v18i13.33195 + +7 + +4 +The proposed approach +Despite its simple structure and fast convergence rate, the HHO algorithm is not +without flaws. However, in the domain of feature selection optimization, the algorithm +may encounter a balancing problem between the exploration and exploitation phases, +resulting in a local optimum. Problems can arise during the feature selection process +when dealing with the high-dimensional feature set. In general, the HHO algorithm +optimization power depends on the best optimal solution selected based on the best +fitness value. In this paper, we present a strategy for improving the basic HHO's per- +formance using the Salp algorithm's power to select the best solution. +4.1 The structure of HHOSSA +The proposed hybrid algorithm HHOSSA contains many stages: Initialization and +binarization function, Best fitness selection, and Evaluation. +4.2 Initialization and binarization function +In this phase, the HHO algorithm generates a random initial population X that con- +tains k Hawks which is every k represents a new solution this vector of d dimension of +features and using binary representations of (0 and 1) to represent the selected features +where every feature that selected will represent by 1 and every refused feature will +represent by 0 by using of the following binarization function: + +𝑿 +𝒃𝒊𝒏𝒂𝒓𝒚={𝟏 𝒊𝒇 𝒙>𝒕𝒉𝒓𝒆_𝒗𝒂𝒍 +𝟎 𝒊𝒇 𝒙<𝒕𝒉𝒓𝒆_𝒗𝒂𝒍 𝒘𝒉𝒆𝒓𝒆 𝒕𝒉𝒓𝒆_𝒗𝒂𝒍=𝟎.𝟓 ( 14 ) +4.3 Best fitness selection +In basic HHO the position vectors Xrand and Xrabbit are responsible for the explo- +ration step that has been characterized by Eq1, which is critical for balancing the ex- +ploitation and exploration phases. Position vectors with higher significance speed up +global exploration, while those with lower significance speed up exploitation. As a re- +sult, an appropriate Xrand and Xrabbit selection should be made to achieve a stable +balance between local exploitation and global exploration [20]. In this phase, the SSA +algorithm will be used to find a better solution where the SSA algorithm finds the new +fitness and if the new one is better than the one that has been found by the HHO algo- +rithm so the new one will be replaced and the Xrabbit will be changed also otherwise, +the HHO solution remains unchanged. +The goal of feature selection is to reduce the number of features and classification +error rate, i.e., through the removal of the redundant and irrelevant features and keeping +the relevant ones only, classification accuracy is improved. The KNN classifier was +used in this study because it is simple to evaluate the fitness function Eq (15), which +was used, expresses the fitness function that was used. + +Issa, A., Ali, Y., & Rashid , T. (2022). An Efficient Hybrid Classification Approach for COVID-19 Based on +Harris Hawks Optimization and Salp Swarm Optimization. International Journal of Online and Biomedi- +cal Engineering (iJOE), 18(13), pp. 113–130. https://doi.org/10.3991/ijoe.v18i13.33195 + +8 + +𝑭𝒊𝒕𝒏𝒆𝒔𝒔 = 𝒂 ∗ 𝒄𝒍𝒂𝒔𝒔𝒆𝒓𝒓 + 𝒃 ∗ ( +𝒇𝒔𝒆𝒍 +𝒇𝒎𝒂𝒙) (15) +Where a =0.9 is constant for controlling the accuracy, b=[0.1, a] random number +enhances the accuracy,classerr is the rate of classification error and 𝒇𝒔𝒆𝒍 represents the +number of the selected feature and 𝒇𝒎𝒂𝒙 represents the total amount of features. + +Algorithm1 Pseudo-Code of HHOSSA Algorithm +Input: H population size, T iteration number, ub=1, lb=0, thre_val=0.5, +levy_beta=1.5 +Output: Best selected features vector + +Randomly initialize of population H random hawks xi (i=1,2,3,….., H) +Compute the fitness value of every one of the hawks Fhho +Xrabbit = best solution found + +While (the stop condition isn’t met) do +Compute the fitness values of the hawks +Set Xrabbit as rabbit location (i.e. optimal location) +For (each hawk (Xi)) do + Update (Eo , J) + if (|E| ≥ 1) then + Update location vector according to Eq1 + if (|E| < 1) then + if (r ≥0.50 & |E| ≥ 0.50 ) then + Update location vector through utilizing Eq. (2) + else if (r ≥0.50 & |E| < 0.50 ) then + Update location vector through utilizing Eq. (4) + else if (r <0.50 & |E| < 0.50 ) then + Update location vector through utilizing Eq. (8) + else if (r <0.50 & |E| < 0.50 ) then + Update location vector through utilizing Eq. (8) +Apply the SSA algorithm to find the best fitness Fssa using Eq. (15) +If (Fssa < Fhho ) +Update (Xrabbit, Xrand ) +End if + +End While + + + + + + +Issa, A., Ali, Y., & Rashid , T. (2022). An Efficient Hybrid Classification Approach for COVID-19 Based on +Harris Hawks Optimization and Salp Swarm Optimization. International Journal of Online and Biomedi- +cal Engineering (iJOE), 18(13), pp. 113–130. https://doi.org/10.3991/ijoe.v18i13.33195 + +9 + + + + Figure 1: Structure of the proposed HHOSSA algorithm +Start +Randomly initialize of +population H +Calculate Fhho and Xrabbit +Initialize of E0 and +update E1 +Stopping_condition +met ? +Yes +No +Apply SSA algorithm +to found Fssa +Fssa< Fhho +Update Xrabbit and +Xrand +Keep Xrabbit and +Xrand +Evalute the selected feature with +FS wrapper method by using +KNN classifier +Yes +No +Features Extraction +Split the dataset into +training and testing +Preprocessing +Stop + +Issa, A., Ali, Y., & Rashid , T. (2022). An Efficient Hybrid Classification Approach for COVID-19 Based on +Harris Hawks Optimization and Salp Swarm Optimization. International Journal of Online and Biomedi- +cal Engineering (iJOE), 18(13), pp. 113–130. https://doi.org/10.3991/ijoe.v18i13.33195 + +10 + +5 +Tools +5.1 Dataset +We are working with a dataset of 800 chest X-ray images obtained from [21-25]. +This dataset consists of 400 chest X-ray images with confirmed COVID-19 infection, +and 400 chest X-ray images of normal condition. This dataset images come with PNG +file format and grey level scale and all images are resized to 200 × 200 pixels. +5.2 Classifiers +The main goal of classification is to categorize new samples that haven't been labeled +for a particular class. However, we must first train the classifier for it to recognize the +characteristics of the data, as well as the relationship between attribute values and the +class label. Three classifiers are used in the methodology presented in this paper. The +first one K‑nearest neighbor classifier and it’s used for the reasons of its straightforward +implementation, with only one parameter K denoting the number of neighbors, which +makes it more useful for identifying the best subset of attributes [26]. The second one +is the SVM classifier which is a well-known constructive learning technique that is +formalized by a separating hyperplane. Making a nonlinear transformation of the orig- +inal input set to the high-dimensional set of features, where the optimum separating +hyperplane may be found, can lead to a solution [27]. The third classifier is Extreme +Gradient Boosting (XGBoost) which is a machine learning method that has been used +for solving supervised learning problems. It has excellent scalability and a fast running +speed, making it a popular machine-learning method [28]. + +6 +Performance evaluation +The metrics of evaluation that are used to measure classification performance in this +study are accuracy, precision, recall, and F1 as defined below: + +𝑨𝒄𝒄𝒖𝒓𝒂𝒄𝒚 = +𝑻𝑷+𝑻𝑵 +𝑻𝑷+𝑻𝑵+𝑭𝑷+𝑭𝑵 ( 16 ) + +𝒑𝒓𝒆𝒄𝒊𝒔𝒊𝒐𝒏 = +𝑻𝑷 +𝑻𝑷+𝑭𝑷 ( 17 ) + +𝒓𝒆𝒄𝒂𝒍𝒍 = +𝑻𝑷 +𝑻𝑷+𝑭𝑵 ( 18 ) + +𝑭𝟏 = 𝟐 × +𝑺𝒑𝒆𝒄𝒊𝒇𝒊𝒄𝒊𝒕𝒚 × 𝑹𝒆𝒄𝒂𝒍𝒍 +𝑺𝒑𝒆𝒄𝒊𝒇𝒊𝒄𝒊𝒕𝒚+ 𝑹𝒆𝒄𝒂𝒍𝒍 (19), 𝒔𝒑𝒆𝒄𝒊𝒇𝒊𝒄𝒊𝒕𝒚 = +𝑻𝑵 +𝑻𝑵+𝑭𝑷 (20) + + +Issa, A., Ali, Y., & Rashid , T. (2022). An Efficient Hybrid Classification Approach for COVID-19 Based on +Harris Hawks Optimization and Salp Swarm Optimization. International Journal of Online and Biomedi- +cal Engineering (iJOE), 18(13), pp. 113–130. https://doi.org/10.3991/ijoe.v18i13.33195 + +11 + +In which "TP" (true positives) denotes positive COVID-19 images which the classi- +fier accurately labeled, and "TN" (i.e. true negatives) corresponds to the nega- +tives COVID-19 images that have been successfully labeled by the classifier. False +positives (FP) are positive COVID-19 images mislabeled as negative, whereas false +negatives (FN) are negative COVID-19 images that have been incorrectly identified as +positive COVID-19 images [29]. +7 +Results and discussion +A total of 800 X-ray images (400 covid-19 and 400 normal) have been collected +from the digital database and utilized for testing the efficacy of the suggested hybrid +approach, which utilized two state-of-art algorithms (SSA, HHO) for feature selection +to improve the classification of the covid-19 infection with the use of automatic AI +techniques and showed a high level of classification accuracy following testing and +training. The dataset was divided into two sections: 20% for validation and testing and +80% for training. Table 2 demonstrates that the suggested hybrid method has a high +accuracy percentage based on the classifiers utilized. The parameter setting for the sug- +gested methodology has been listed in Table 1. + +Table1: Parameter values for used methods +Methods +Parameter values + +HHOSSA algorithm + +Feature size: 126 +Population size: 30 +Number of iterations for HHO:100 +Number of iterations for SSA:20 +Ub:1 +Lb:0 +Thre_val:0.5 +Beta:1.5 +Random variables a and b: 0.9, [0.1, a ] + +KNN classifier + +K=5 +Classes count:2 +No.of training set:224 + +SVM classifier + +Classes count:2 +No.of training set:224 + +XGboost classifier + +Classes count:2 +No.of training set:224 + +Table 2: Performance of HHOSSA over three classifiers KNN, SVM, and XGboost. +Classifier +Accuracy +Precision +Recall +F1 +KNN +98.21428571428571 +0.97 +0.99 +0.98 + +Issa, A., Ali, Y., & Rashid , T. (2022). An Efficient Hybrid Classification Approach for COVID-19 Based on +Harris Hawks Optimization and Salp Swarm Optimization. International Journal of Online and Biomedi- +cal Engineering (iJOE), 18(13), pp. 113–130. https://doi.org/10.3991/ijoe.v18i13.33195 + +12 + +SVM +96.42857142857143 +0.96 +0.96 +0.96 +XGboost +98.21428571428571 +0.99 +0.96 +0.98 + +7.1 Comparative study +The suggested system's performance was assessed utilizing a variety of modern op- +timization methods (GWO, WOA). Table (3) shows the performance of the HHO algo- +rithm used for feature selection and gets 94%,89%, and 94% over three classifiers +KNN, SVM, and XGboost, while Table(4) shows the performance of the SSA algo- +rithm used for feature selection and gets 96%,80%,94% over three classifiers KNN, +SVM, XGboost, Table (5) shows the performance of GWO algorithm used for feature +selection and gets 96%,82%,92% over three classifiers KNN, SVM, XGboost, While +Table (6) shows the performance of WOA algorithm used for feature selection and gets +96%,86%,96% over three classifiers KNN, SVM, XGboost. + +Table 3: Performance of HHO over three classifiers KNN, SVM, and XGboost. +Classifier +Accuracy +Precision +Recall +F1 +KNN +94.64285714285714 +0.90 +0.99 +0.95 +SVM +89.28571428571429 +0.87 +0.93 +0.90 +XGboost +94.64285714285714 +0.93 +0.96 +0.95 + +Table 4: Performance of SSA over three classifiers KNN, SVM, and +XGboost. +Classifier +Accuracy +Precision +Recall +F1 +KNN +96.64285714285714 +0.93 +0.96 +0.95 +SVM +80.35714285714286 +0.81 +0.79 +0.80 +XGboost +94.64285714285714 +0.96 +0.93 +0.95 + +Table 5: Performance of GWO over three classifiers KNN, SVM, and XGboost. +Classifier +Accuracy +Precision +Recall +F1 +KNN +96.42857142857143 +0.93 +0.99 +0.97 +SVM +82.14285714285714 +0.74 +0.99 +0.85 +XGboost +92.85714285714286 +0.90 +0.96 +0.93 + + +Issa, A., Ali, Y., & Rashid , T. (2022). An Efficient Hybrid Classification Approach for COVID-19 Based on +Harris Hawks Optimization and Salp Swarm Optimization. International Journal of Online and Biomedi- +cal Engineering (iJOE), 18(13), pp. 113–130. https://doi.org/10.3991/ijoe.v18i13.33195 + +13 + +Table 6: Performance of WOA over three classifier KNN, SVM, and XGboost. +Classifier +Accuracy +Precision +Recall +F1 +KNN +94.64285714285714 +0.90 +0.99 +0.95 +SVM +89.28571428571429 +0.87 +0.93 +0.90 +XGboost +96.42857142857143 +0.99 +0.93 +0.96 + + +Figure 2: The accuracy, precision, recall, and the F1 values for all algorithms over the KNN +classifier + +0.84 +0.86 +0.88 +0.9 +0.92 +0.94 +0.96 +0.98 +1 +HHOSSA +HHO +SSA +WOA +GWO +Accuracy +Precision +Recall +F1 + +Issa, A., Ali, Y., & Rashid , T. (2022). An Efficient Hybrid Classification Approach for COVID-19 Based on +Harris Hawks Optimization and Salp Swarm Optimization. International Journal of Online and Biomedi- +cal Engineering (iJOE), 18(13), pp. 113–130. https://doi.org/10.3991/ijoe.v18i13.33195 + +14 + + +Figure 3: The accuracy, precision, recall, and the F1 values for all algorithms over the SVM +classifier + +Figure 4: The accuracy, precision, recall, and the F1 values for all algorithms over the +XGboost classifier + +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +HHOSSA +HHO +SSA +WOA +GWO +Accuracy +Precision +Recall +F1 +0.84 +0.86 +0.88 +0.9 +0.92 +0.94 +0.96 +0.98 +1 +HHOSSA +HHO +SSA +WOA +GWO +Accuracy +Precision +Recall +F1 + +Issa, A., Ali, Y., & Rashid , T. (2022). An Efficient Hybrid Classification Approach for COVID-19 Based on +Harris Hawks Optimization and Salp Swarm Optimization. International Journal of Online and Biomedi- +cal Engineering (iJOE), 18(13), pp. 113–130. https://doi.org/10.3991/ijoe.v18i13.33195 + +15 + +7.2 Software and Hardware Requirements +The proposed system operates by using a personal computer Lenovo that has speci- +fications such as Intel(R) Intel(R) Core(TM) i7-6500U @ 2.59 GHz for CPU, 8 GB +windows10 of RAM, and 64-bit Operating System. The proposed system is operated +by using python 10 languages with (Pycharm) IDE. Table (7) shows the processing +time of the proposed algorithm and stand-alone algorithms depending on the classifi- +cation processing time of the testing dataset. +Table 7: Processing time of proposed (HHOSSA), HHO, SSA. +Algorithm +Total processing time (seconds) +HHOSSA +1.0661 +HHO +0.9906 +SSA +1.1425 + +It should be noted that the hybrid algorithm's processing time for completing the clas- +sification process is less than the sum of the processing times for the Harris hawk and +Salp algorithms because the Salp algorithm's iterations are fewer than those of the +Harris hawk algorithm within the hybrid algorithm. However, this improved the clas- +sification process and accelerated performance without degrading the hybrid algo- +rithm's quality. + +8 +Conclusion and future works +The presented work presents a new hybrid swarm algorithm (referred to +as HHOSSA) that combines the SSA and HHO for selecting the best features subset to +improve the detection and classification of the COVID-19 virus with the use of chest +X-ray images. The novel method provided to improve the process of the feature section +and also for achieving the balance between exploitation and exploration of the HHO +algorithm with the use of the capability of SSA for finding the best features subset It is +noted that the processing time required to complete the classification process using the +hybrid algorithm is less than the sum of the processing time of the Harris hawk and +Salp algorithms because the number of iterations of the Salp algorithm is less than the +iterations of Harris hawk algorithm inside hybrid algorithm, However, this did not af- +fect the quality of the hybrid algorithm, but rather it increased the speed of performance +and improved the classification process. A total of 800 (400 covid-19 and 400 normal) +X-ray images are taken from the digital database to assess the HHOSSA's performance. +XGboost and KNN classifiers get 98% accuracy, whereas SVM classifiers score 96%. +We want to adapt the suggested technique to more applications in the future, including + +Issa, A., Ali, Y., & Rashid , T. (2022). An Efficient Hybrid Classification Approach for COVID-19 Based on +Harris Hawks Optimization and Salp Swarm Optimization. International Journal of Online and Biomedi- +cal Engineering (iJOE), 18(13), pp. 113–130. https://doi.org/10.3991/ijoe.v18i13.33195 + +16 + +signal processing and cloud computing task scheduling. Furthermore, the HHO algo- +rithm's searching power was used to construct a novel suggested algorithm in several +aspects. +9 Acknowledgment +The authors would like to thank the University of Technology, Baghdad, Iraq for their +continuous support for this research work. +10 +References +[1] Lakshmanarao, A., Raja Babu, M., & Srinivasa Ravi Kiran, T. (2021). An Efficient +Covid19 Epidemic Analysis and Prediction Model Using Machine Learning Algorithms. Inter- +national Journal of Online and Biomedical Engineering (iJOE), 17(11), pp. 176–184. +https://doi.org/10.3991/ijoe.v17i11.25209 + +[2] S. Cho, S. Lim, C. Kim, S.Wi, T. Kwon,W. S. Youn, S. H. Lee, B. S. Kang, and S. Cho, +``Enhancement of soft-tissue contrast in cone-beam CT using an anti-scatter grid with a sparse +sampling approach,'' Phys. Medica, vol. 70, pp. 1_9, Feb. 2020 + +[3] Allam, Mohan, and M. Nandhini. "A study on optimization techniques in feature selection +for medical image analysis." 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Repository, +“Cohen’s +GitHub +repository,” +Cohen’s +GitHub +repository, +2021. +https://github.com/ieee8023 + +[24] Italian Society of Medical and Interventional Radiology (SIRM), “Italian Society of Medical and Inter- +ventional Radiology (SIRM),” Italian Society of Medical and Interventional Radiology (SIRM), 2020. +https://sirm.org/la-radiologia-medica/ + +[25] Kaggle, “Kaggle’s chest X-ray images (Pneumonia) dataset,” Kaggle’s chest X-ray images (Pneumonia) +dataset, 2020. https://www.kaggle.com/datasets/paultimothymooney/chest-xray-pneumonia + +[26] Abdel-Basset, Mohamed, Weiping Ding, and Doaa El-Shahat. "A hybrid Harris Hawks optimization +algorithm with simulated annealing for feature selection." Artificial Intelligence Review54, no. 1 (2021): +593-637. + + [27] Fusilier, Donato Hernández, Manuel Montes-y-Gómez, Paolo Rosso, and Rafael Guzmán Cabrera. +"Detecting positive and negative deceptive opinions using PU-learning." Information processing & manage- +ment 51, no. 4 (2015): 433-443. + +[28] Chen, Tianqi, and Carlos Guestrin. "Xgboost: A scalable tree boosting system." In Proceedings of the +22nd acm sigkdd international conference on knowledge discovery and data mining, pp. 785-794. 2016. + +[29] Yasir, M. A., & Ali, Y. H. (2021). Review on Real Time Background Extraction: Models, Applications, +Environments, Challenges, and Evaluation Approaches. International Journal of Online and Biomedical En- +gineering (iJOE), 17(02), pp. 37–68. https://doi.org/10.3991/ijoe.v17i02.18013 + +11 Authors +Abubakr S. Issa received his bachelor’s degree in computer science department – +Artificial intelligence branch from the University of Technology (UOT) – Iraq 2014. +Since 2014, he is working as a programmer at the Information Technology Center, at +the University of Technology up till now. Meanwhile, he is an M.Sc candidate at the +University of Technology (UOT) – Iraq. +Assistant Professor Dr. Yossra Hussain Ali. She received her B.Sc, M.Sc, and Ph.D. +degrees in 1996, 2002, and 2006 respectively from Iraq, the University of Technology, +Department of Computer Sciences. She joined the University of Technology, Iraq in +1997. During her postgraduate studies, she worked on Computer Networks, Infor- +mation systems, Agent Programming and Image Processing as well as some experience +in Artificial Intelligent and Computer Data Security. She is a reviewer at many confer- +ences and journals and she supervised several undergraduate and postgraduate (PhD. +and MSc.) dissertations in Computer sciences. Yossra has many professional certifi- +cates and she has published in well-regarded journals (e-mail: yossra.h.ali@uotechnol- +ogy.edu.iq). +Tarik A. Rashid received his Ph.D. in Computer Science and Informatics degree from +the College of Engineering, Mathematical and Physical Sciences, University College +Dublin (UCD) in 2001–2006. He pursued his Post-Doctoral Fellow at the Computer +Science and Informatics School, College of Engineering, Mathematical and Physical +Sciences, University College Dublin (UCD) from 2006–2007. He joined the University + +Issa, A., Ali, Y., & Rashid , T. (2022). An Efficient Hybrid Classification Approach for COVID-19 Based on +Harris Hawks Optimization and Salp Swarm Optimization. International Journal of Online and Biomedi- +cal Engineering (iJOE), 18(13), pp. 113–130. https://doi.org/10.3991/ijoe.v18i13.33195 + +19 + +of Kurdistan Hewlêr (UKH) in 2017. He has also been included in the prestigious Stan- +ford University list of 2.% of the best world researchers for the years 2020 and 2022. + diff --git a/7NE4T4oBgHgl3EQf2Q0O/content/tmp_files/load_file.txt b/7NE4T4oBgHgl3EQf2Q0O/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..891de669f9c1e93d2a4389e90a8d41c429d2c414 --- /dev/null +++ b/7NE4T4oBgHgl3EQf2Q0O/content/tmp_files/load_file.txt @@ -0,0 +1,864 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf,len=863 +page_content='Issa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', Ali, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', & Rashid , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' An Efficient Hybrid Classification Approach for COVID-19 Based on Harris Hawks Optimization and Salp Swarm Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' International Journal of Online and Biomedi- cal Engineering (iJOE), 18(13), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' 113–130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='3991/ijoe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='v18i13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='33195 1 An efficient hybrid classification approach for COVID-19 based on Harris Hawks Optimiza- tion and Salp Swarm Optimization Abubakr Issa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' University of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Baghdad,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' iraq Yossra Ali,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' University of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Baghdad,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Iraq Tarik Rashid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' University of Kurdistan Hewler,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' KRG,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Iraq Abstract— Feature selection can be defined as one of the pre-processing steps that decrease the dimensionality of a dataset by identifying the most signif- icant attributes while also boosting the accuracy of classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' For solv- ing feature selection problems, this study presents a hybrid binary version of the Harris Hawks Optimization algorithm (HHO) and Salp Swarm Optimization (SSA) (HHOSSA) for Covid-19 classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=" The proposed (HHOSSA) pre- sents a strategy for improving the basic HHO's performance using the Salp algo- rithm's power to select the best fitness values." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' The HHOSSA was tested against two well-known optimization algorithms, the Whale Optimization Algorithm (WOA) and the Grey wolf optimizer (GWO), utilizing a total of 800 chest X-ray images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' A total of four performance metrics (Accuracy, Recall, Precision, F1) were employed in the studies using three classifiers (Support vector machines (SVMs), k-Nearest Neighbor (KNN), and Extreme Gradient Boosting (XGBoost)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' The proposed algorithm (HHOSSA) achieved 96% accuracy with the SVM classifier, and 98% accuracy with two classifiers, XGboost and KNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Keywords—— Feature selection, Hybrid Swarm intelligence, classification, Covid-19, medical image 1 Introduction Medical image processing can be defined as one of the most significant areas in medical science, and it has a substantial effect on visualization applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Also, med- ical image processing has a broad range of applications in medical diagnoses (treating and investigating diseases) and medical sciences (such as physiological and anatomi- cal studies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Medical physics, medical engineering, biology, and optics are some of the Issa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', Ali, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', & Rashid , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' An Efficient Hybrid Classification Approach for COVID-19 Based on Harris Hawks Optimization and Salp Swarm Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' International Journal of Online and Biomedi- cal Engineering (iJOE), 18(13), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' 113–130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='3991/ijoe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='v18i13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='33195 2 fields of science that make up this medical science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' With the discovery of X-rays, Wil- liam Roentgen initiated the first efforts at contemporary medical imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Coronavirus (COVID-19), also known as SARS-Corona Virus-2, is a virus that results in causing severe acute respiratory syndrome (SARS-CoV2), is a viral infection that first occurred in Wuhan at the end of 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Due to such an outbreak, COVID-19 became a pandemic, threatening human lives and wreaking havoc on the economy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Therefore, many stud- ies have been launched in an attempt to identify a way to restrict mortality and spread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Those researches include the suggested treatment strategy, the screening method for early-stage patients, and the evaluation of different phases and recovery of treated pa- tients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' In hospitals, imaging techniques like chest X-rays are commonly utilized for detecting the severity and existence of COVID-19 pneumonia [1][2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=" For improving the suggested system's training, X-ray images are often maintained in a medical data- base for subsequent investigation by multiple research organizations." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Low contrast, noise, blurs, and faded colors are frequent problems, and images should be pre-pro- cessed to enhance quality by reducing noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' The second stage is image segmentation, which depends on some attributes includ- ing color, texture, and depth measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' The type of image and characteristics of the problem (disease) are chosen to determine which segmentation technique is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' The identification and extraction of features is the third stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' As the number of features that have been extracted from the image grows, the accuracy of classification decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' In the classification vision, we can call it the curse of dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Feature optimi- zation is a viable option for dealing with this issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' [3] The 4th stage is the feature selection that has been obtained from the known proper- ties using robust Optimization algorithms for better disease identifications from the medical images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' The image was classified using one of the classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Feature selection is a step in the preprocessing process that tries to increase the relevancy of obtained data by deleting irrelevant characteristics and choosing just relevant or useful variables [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Feature selection comprises reviewing feature subsets, employing certain search approaches to locate the best feature subset, assessing the chosen features, stopping cri- teria, and subset validation in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' [6] There are three types of feature selection classifiers: wrapper schemes, filer schemes, and embedding schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' The filter method, in contrast to the wrapper scheme, which is characterized by good classification accuracy and low speed, is rapid but inaccurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' The embedded system is preferred in the case when handling a certain model [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Filter techniques use the qualities of training data to assess the quality of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Those approaches do not employ machine learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Before choos- ing features with the highest score, filter methods usually take into account the score of all features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' At the same time, other filtering approaches favor features with the greatest score per iteration [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Other well-known methods, like the correlation-based feature selection approach in [9] as well as dimensionality reduction methods and NNs in [10], can greatly decrease computational load and system complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Filter approaches overlook the performance regarding the chosen characteristics despite their speed and low computational cost [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Issa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', Ali, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', & Rashid , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' An Efficient Hybrid Classification Approach for COVID-19 Based on Harris Hawks Optimization and Salp Swarm Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' International Journal of Online and Biomedi- cal Engineering (iJOE), 18(13), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' 113–130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='3991/ijoe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='v18i13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='33195 3 Wrapper approaches utilize an evaluation algorithm to assess the specified features’ quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' SVMs, Decision trees (DTs), KNN, Naïve Bayesian (NB), linear discriminant analysis (LDA), local neighborhood structure preserving embedding (LNSPE), artifi- cial neural networks (ANNs), and local geometrical structure Fisher analysis (LGSFA) are some of the major wrapper’s methods utilized for feature selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' In almost all cases, wrapper approaches outperform filter ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Meta-heuristic algorithms are more advanced search algorithms that result from the evolution and expansion of feature selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' For instance, ongoing research to increase the performance regard- ing evolutionary algorithms (EA) like Genetic Algorithms (GAs) and Swarm Intelli- gence (SI) like Particle Swarm Optimization (PSO), Artificial Bee Colony (ABC), and Ant Colony Optimization (ACO) are underway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Grasshopper Optimization Algorithm (GOA), Grey Wolf Optimizer (GWO), Butterfly Optimization Algorithm (BOA), Har- ris Hawks Optimization (HHO) Whale Optimization Algorithm (WOA), and Ant Lion Optimization (ALO) are examples of recent algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Metaheuristic algorithms are classified according to their exploration and exploitation phases into single solution based (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', Tabu Search (TS) and Simulated annealing (SA)) or population size based (in other words, GA, ACO, and PSO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' The key contributions of this research are listed below: Suggest an effective hybrid classification method for COVID 19 with the use of the hybrid swarm algorithms (HHO, SSA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' This novel hybrid algorithm must improve resource consumption and performance, as well as storage capacity, reducing processing time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' With the use of multiple classifiers (KNN, SVM, XGboost), test the sug gested (HHOSSA) algorithm on datasets containing some positive negative COVID 19 chest X ray scan images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Individual, hybridized predictor models and state of the art techniques (WOA, GWO) are compared in terms of performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' The sections of this paper are organized as follows: Section 2 provides a concise summary of some of the most related works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Section 3 discusses methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' In sec- tion 4 we described in detail our proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Tools are illustrated in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Performance evaluation is described in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Results and discussion are included in section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Finally, the conclusions and future works are stated in section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' 2 Related Works Many studies have employed hybrid algorithms to handle a range of challenges re- cently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Hybrid algorithms have received a lot of attention lately, notably in feature se- lection optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Low-level hybrid algorithms and high-level hybrid algorithms are the 2 forms of hybrid algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' There are 2 types of hybridization schemes in high- level hybrid algorithms: high-level teamwork hybridization (HTH) and high-level rely- on hybridization (HRH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' The self-contained meta-heuristics have been carried out in Issa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', Ali, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', & Rashid , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' An Efficient Hybrid Classification Approach for COVID-19 Based on Harris Hawks Optimization and Salp Swarm Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' International Journal of Online and Biomedi- cal Engineering (iJOE), 18(13), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' 113–130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='3991/ijoe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='v18i13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='33195 4 order in HRH, whereas in the HTH, one algorithm assists the other by supplying infor- mation via cooperative search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Low-level hybridization has been separated into two types: low-level teamwork hybrid (LTH) and low-level rely-on hybrid (LRH), both of which contain one meta-heuristic algorithm [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' In the feature selection field, it has been observed that hybrid algorithms surpass native algorithms concerning perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' In the year 2004, the search process has been controlled by merging local search approaches with a GA algorithm, which was the first time a hybrid metaheuristics ap- proach was utilized in feature selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' A combination with the EGA filter has been provided in a wrapper technique for text categorization [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' A hybrid ap- proach for feature selection has lately been created in various metaheuristic algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' In [13], the Binary Grey Wolf algorithm was combined with the Harris Hawks algo- rithm to create an excellent balance between exploitation and exploration to prevent local optimum solutions and increase solution precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=" Harris Hawks was hybridized in [14] using Bitwise operations and Simulated Annealing for supporting the HHO al- gorithm's exploitation capacity and getting out of local optima." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' In [15], the Salp swarm algorithm was used to modify teaching–learning based optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' This integration gives TLBO more flexibility in the exploration of population and achieving variety while also allowing it to swiftly attain the optimal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' They combined the Salp swarm algorithm with the Particle swarm algorithm in [16], in which the SSA was uti- lized for updating the salps positions and the PSO was utilized otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' This hybrid- ization was utilized for the improvement of the exploration and exploitation of the Salp swarm algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' 3 Methodology 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='1 Harris Hawks optimization algorithm HHO can be defined as one of the swarm metaheuristic algorithms inspired by Har- ris Hawks\' hunting behavior of "seven kills" or "surprise pounce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='" Based on the prey\'s fleeing behavior nature, hunting duration can range from some seconds to many hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' The modeling algorithm of HHO is split into 2 parts (exploitation and exploration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=" Harris' hawks have been employed as candidate solutions in the HHO algorithm, with the best candidate solution reflecting the desired or optimum prey in each stage [17]." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' The first phase pertains to the process of perching and detection of the prey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=" The algo- rithm simulates Harris' hawks' perching methods in 2 separate scenarios." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=" Harris' hawks are assumed to perch in various locations during their group home range in the first scenario." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' In Eq (1), q=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='50 models that condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' X1 ⃗⃗⃗⃗ (t+1)= { 𝑋𝑟𝑎𝑛𝑑(𝑡) − 𝑟1|𝑋𝑟𝑎𝑛𝑑(𝑡) − 2𝑟2𝑋(𝑡)|, 𝑞 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='50 (𝑋𝑟𝑎𝑏𝑏𝑖𝑡(𝑡) − 𝑋𝑚(𝑡)) − 𝑟3(𝐿𝐵 + 𝑟4(𝑈𝐵 − 𝐿𝐵)), 𝑞 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content="50( 1 ) While the other likelihood is that Harris' hawks would perch on positions near other swarm members and prey." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' This condition has been introduced in Eq1 for q < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='50: Issa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', Ali, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', & Rashid , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' An Efficient Hybrid Classification Approach for COVID-19 Based on Harris Hawks Optimization and Salp Swarm Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' International Journal of Online and Biomedi- cal Engineering (iJOE), 18(13), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' 113–130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='3991/ijoe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='v18i13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content="33195 5 where X1 ⃗⃗⃗⃗ (t+1) is Hawks' position vector," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' t represents the following iteration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' 𝑋𝑟𝑎𝑛𝑑(𝑡) is a hawk that has been chosen at random from the current population,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' 𝑋(𝑡) represents the position vector of hawks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' r1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' r2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' r3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' r4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' and q represent random numbers in the range of (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Xrabbit(t) represents rabbit position,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Xm denotes the aver- age position of the current population of the hawks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=" lower and upper bounds for gener- ating random locations inside the Hawks' stadium are Lb and Ub," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=" While in the phase of exploitation, the Harris' hawks attack prey which has been identified in the preceding step." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=" The algorithm has 4 different possibilities for modeling various attacking styles that have been utilized by Harris' hawks." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' While r denotes the probability of prey escaping, successful escape has been donated by r < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='50, whereas r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='50 denotes failure to escape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=" Depending upon the prey's chances of escaping (r), hawks will use either soft or hard besieges to catch prey." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=" The algorithm's parameter E has been utilized for the determination of the type of attacking besieges." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' If the prey is unable to escape when r ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='50 hard besiege happens when |E| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='50 and soft besiege takes the place in the case where |E|≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='50 The mathematical Modelling of soft besiege has been represented by Eqs (2) through (3), and hard besiege has been shown by Eq (4): 𝑿(𝒕 + 𝟏)=∆X(t) –E|JxXrabbit (t) –X(t)| ( 2 ) ∆(t) =Xrabbit (t) –X(t) ( 3 ) X( t+1) =Xrabbit(t) –E|∆X(t)| ( 4 ) In the case of successful escaping of the prey (r<0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='50), soft besiege with a progres- sive rapid dive take is applied in the case where |E|≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='50 as shown in Eq (5), Eq (7), Eq(8) while Hard besiege with the progressive fast dive occurs in a case where |E|≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='50 as shown in Eqs (6), (7), and (8): 𝒀 = 𝑿𝒓𝒂𝒃𝒃𝒊𝒕(𝒕) − 𝑬|𝑱 ∗ 𝑿𝒓𝒂𝒃𝒃𝒊𝒕(𝒕) − 𝑿(𝒕)| ( 5 ) 𝒀 = 𝑿𝒓𝒂𝒃𝒃𝒊𝒕(𝒕) − 𝑬|𝑱 ∗ 𝑿𝒓𝒂𝒃𝒃𝒊𝒕(𝒕) − 𝑿𝒎(𝒕)| , 𝑿𝒎(𝒕) = 𝟏 𝑵 ∗ ∑ 𝑿𝒊 𝑵 𝒊=𝟏 (𝒕) ( 6 ) 𝒁 = 𝒀 + 𝑺 × 𝑳𝑭(𝑫) ( 7 ) 𝑿(𝒕 + 𝟏) = {𝒀, 𝒊𝒇 𝒇(𝒀) < 𝑭(𝑿(𝒕)) 𝒁, 𝒊𝒇 𝒇(𝒁) < 𝑭(𝑿(𝒕)) ( 8 ) D represents the problem dimension and S represents the random vector by 1xD size and LF represents the function of levy flight, estimated with the use of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' (9): 𝑳𝑭(𝒙) = 𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' 𝟎𝟏 × 𝒖 ×𝛔 |𝒗| 𝟏 𝜷 , 𝛔 = ( (𝜞(𝟏+𝜷) ×𝒔𝒊𝒏 (𝝅𝜷/𝟐) 𝜞(𝟏+𝜷 𝟐 )×𝜷×𝟐(𝜷−𝟏 𝟐 ) ) ( 9 ) The energy of a rabbit is modeled as 𝑬 = 𝟐𝑬𝟎 (𝟏 − 𝒕 𝑻) ( 10 ) Where E represents the prey’s escaping energy, T represents the maximal number of iterations, and Eo represents its initial energy state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Issa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', Ali, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', & Rashid , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' An Efficient Hybrid Classification Approach for COVID-19 Based on Harris Hawks Optimization and Salp Swarm Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' International Journal of Online and Biomedi- cal Engineering (iJOE), 18(13), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' 113–130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='3991/ijoe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='v18i13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='33195 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='2 Salp swarm optimization algorithm SSA can be defined as a swarm metaheuristic algorithm [18] that was created for solving various optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' It was inspired by the activity of Salps in na- ture;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' salps are a type of jellyfish with tissues comparable to jellyfish and a high water percentage in their moving behavior and weights [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' They move by contracting their bodies and shifting positions by pumping water through them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' The salp chain describes the swarming behavior of salps in the ocean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' By allowing for faster and more harmonic changes, this behavior could benefit salps in foraging and better movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' [18] Salp chains were theoretically modeled and after that tested in optimization problems as a result of this characteristic [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' The algorithm starts its work by dividing the generated population into 2 parts (which are: leader and followers ( where the leader leads the salp chain and the remaining salps play the role of followers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' A salp uses the food source as a target in an n-dimensional search space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=" The following equation has been used to update the leader's position: 𝑿𝒋 𝟏 = { 𝑭𝒋 + 𝒓𝟏 ((𝑽𝒎𝒂𝒙𝒋 − 𝑽𝒎𝒊𝒏𝒋)𝒓𝟐 + 𝑽𝒎𝒊𝒏𝒊) , 𝒓𝟑 ≥ 𝟎 𝑭𝒋 − 𝒓𝟏 ((𝑽𝒎𝒂𝒙𝒋 − 𝑽𝒎𝒊𝒏𝒋)𝒓𝟐 + 𝑽𝒎𝒊𝒏𝒊) , 𝒓𝟑 < 𝟎 } ( 11 ) Where 𝑋𝑗 1 represent the position of leader in the jth dimension and Fj is food's loca- tion." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' The upper is represented by 𝑉𝑚𝑎𝑥𝑗 and the lower bounds that have been denoted by 𝑉𝑚𝑖𝑛𝑗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' The search space is maintained using the 2 random variables 𝑟2 & 𝑟3 in the range [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' The parameter 𝑟1 is also an important control parameter in the process of exploration and exploitation and it is calculated by using Eq (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' 𝒓𝟏 = 𝟐𝒆(−𝟒𝒕 𝑵 )𝟐 ( 12 ) Where t represents the current iteration and N denotes the maximum amount of iter- ations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=" In a case where the position of the leader has been changed, Eq (13) is used to change the followers' position: 𝑿𝒋 𝒊 = 𝟏 𝟐 (𝑿𝒋 𝟏 − 𝑿𝒋 𝒊−𝟏) ( 13 ) Where 𝐗𝐣 𝐢 denotes the ith follower's position in the jth dimension, and the value of I must be > 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Issa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', Ali, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', & Rashid , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' An Efficient Hybrid Classification Approach for COVID-19 Based on Harris Hawks Optimization and Salp Swarm Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' International Journal of Online and Biomedi- cal Engineering (iJOE), 18(13), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' 113–130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='3991/ijoe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='v18i13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='33195 7 4 The proposed approach Despite its simple structure and fast convergence rate, the HHO algorithm is not without flaws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' However, in the domain of feature selection optimization, the algorithm may encounter a balancing problem between the exploration and exploitation phases, resulting in a local optimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Problems can arise during the feature selection process when dealing with the high-dimensional feature set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' In general, the HHO algorithm optimization power depends on the best optimal solution selected based on the best fitness value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=" In this paper, we present a strategy for improving the basic HHO's per- formance using the Salp algorithm's power to select the best solution." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='1 The structure of HHOSSA The proposed hybrid algorithm HHOSSA contains many stages: Initialization and binarization function, Best fitness selection, and Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='2 Initialization and binarization function In this phase,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' the HHO algorithm generates a random initial population X that con- tains k Hawks which is every k represents a new solution this vector of d dimension of features and using binary representations of (0 and 1) to represent the selected features where every feature that selected will represent by 1 and every refused feature will represent by 0 by using of the following binarization function: 𝑿 𝒃𝒊𝒏𝒂𝒓𝒚={𝟏 𝒊𝒇 𝒙>𝒕𝒉𝒓𝒆_𝒗𝒂𝒍 𝟎 𝒊𝒇 𝒙<𝒕𝒉𝒓𝒆_𝒗𝒂𝒍 𝒘𝒉𝒆𝒓𝒆 𝒕𝒉𝒓𝒆_𝒗𝒂𝒍=𝟎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='𝟓 ( 14 ) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='3 Best fitness selection In basic HHO the position vectors Xrand and Xrabbit are responsible for the explo- ration step that has been characterized by Eq1, which is critical for balancing the ex- ploitation and exploration phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Position vectors with higher significance speed up global exploration, while those with lower significance speed up exploitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' As a re- sult, an appropriate Xrand and Xrabbit selection should be made to achieve a stable balance between local exploitation and global exploration [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' In this phase, the SSA algorithm will be used to find a better solution where the SSA algorithm finds the new fitness and if the new one is better than the one that has been found by the HHO algo- rithm so the new one will be replaced and the Xrabbit will be changed also otherwise, the HHO solution remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' The goal of feature selection is to reduce the number of features and classification error rate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', through the removal of the redundant and irrelevant features and keeping the relevant ones only, classification accuracy is improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' The KNN classifier was used in this study because it is simple to evaluate the fitness function Eq (15), which was used, expresses the fitness function that was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Issa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', Ali, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', & Rashid , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' An Efficient Hybrid Classification Approach for COVID-19 Based on Harris Hawks Optimization and Salp Swarm Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' International Journal of Online and Biomedi- cal Engineering (iJOE), 18(13), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' 113–130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='3991/ijoe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='v18i13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='33195 8 𝑭𝒊𝒕𝒏𝒆𝒔𝒔 = 𝒂 ∗ 𝒄𝒍𝒂𝒔𝒔𝒆𝒓𝒓 + 𝒃 ∗ ( 𝒇𝒔𝒆𝒍 𝒇𝒎𝒂𝒙) (15) Where a =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='9 is constant for controlling the accuracy, b=[0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='1, a] random number enhances the accuracy,classerr is the rate of classification error and 𝒇𝒔𝒆𝒍 represents the number of the selected feature and 𝒇𝒎𝒂𝒙 represents the total amount of features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Algorithm1 Pseudo-Code of HHOSSA Algorithm Input: H population size, T iteration number, ub=1, lb=0, thre_val=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='5, levy_beta=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='5 Output: Best selected features vector Randomly initialize of population H random hawks xi (i=1,2,3,….' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='., H) Compute the fitness value of every one of the hawks Fhho Xrabbit = best solution found While (the stop condition isn’t met) do Compute the fitness values of the hawks Set Xrabbit as rabbit location (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' optimal location) For (each hawk (Xi)) do Update (Eo , J) if (|E| ≥ 1) then Update location vector according to Eq1 if (|E| < 1) then if (r ≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='50 & |E| ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='50 ) then Update location vector through utilizing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' (2) else if (r ≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='50 & |E| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='50 ) then Update location vector through utilizing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' (4) else if (r <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='50 & |E| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='50 ) then Update location vector through utilizing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' (8) else if (r <0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='50 & |E| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='50 ) then Update location vector through utilizing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' (8) Apply the SSA algorithm to find the best fitness Fssa using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' (15) If (Fssa < Fhho ) Update (Xrabbit, Xrand ) End if End While Issa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', Ali, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', & Rashid , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' An Efficient Hybrid Classification Approach for COVID-19 Based on Harris Hawks Optimization and Salp Swarm Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' International Journal of Online and Biomedi- cal Engineering (iJOE), 18(13), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' 113–130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='3991/ijoe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='v18i13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='33195 9 Figure 1: Structure of the proposed HHOSSA algorithm Start Randomly initialize of population H Calculate Fhho and Xrabbit Initialize of E0 and update E1 Stopping_condition met ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Yes No Apply SSA algorithm to found Fssa Fssa< Fhho Update Xrabbit and Xrand Keep Xrabbit and Xrand Evalute the selected feature with FS wrapper method by using KNN classifier Yes No Features Extraction Split the dataset into training and testing Preprocessing Stop Issa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', Ali, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', & Rashid , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' An Efficient Hybrid Classification Approach for COVID-19 Based on Harris Hawks Optimization and Salp Swarm Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' International Journal of Online and Biomedi- cal Engineering (iJOE), 18(13), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' 113–130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='3991/ijoe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='v18i13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='33195 10 5 Tools 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='1 Dataset We are working with a dataset of 800 chest X-ray images obtained from [21-25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' This dataset consists of 400 chest X-ray images with confirmed COVID-19 infection, and 400 chest X-ray images of normal condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' This dataset images come with PNG file format and grey level scale and all images are resized to 200 × 200 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content="2 Classifiers The main goal of classification is to categorize new samples that haven't been labeled for a particular class." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' However, we must first train the classifier for it to recognize the characteristics of the data, as well as the relationship between attribute values and the class label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Three classifiers are used in the methodology presented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' The first one K‑nearest neighbor classifier and it’s used for the reasons of its straightforward implementation, with only one parameter K denoting the number of neighbors, which makes it more useful for identifying the best subset of attributes [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' The second one is the SVM classifier which is a well-known constructive learning technique that is formalized by a separating hyperplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Making a nonlinear transformation of the orig- inal input set to the high-dimensional set of features, where the optimum separating hyperplane may be found, can lead to a solution [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' The third classifier is Extreme Gradient Boosting (XGBoost) which is a machine learning method that has been used for solving supervised learning problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' It has excellent scalability and a fast running speed, making it a popular machine-learning method [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' 6 Performance evaluation The metrics of evaluation that are used to measure classification performance in this study are accuracy, precision, recall, and F1 as defined below: 𝑨𝒄𝒄𝒖𝒓𝒂𝒄𝒚 = 𝑻𝑷+𝑻𝑵 𝑻𝑷+𝑻𝑵+𝑭𝑷+𝑭𝑵 ( 16 ) 𝒑𝒓𝒆𝒄𝒊𝒔𝒊𝒐𝒏 = 𝑻𝑷 𝑻𝑷+𝑭𝑷 ( 17 ) 𝒓𝒆𝒄𝒂𝒍𝒍 = 𝑻𝑷 𝑻𝑷+𝑭𝑵 ( 18 ) 𝑭𝟏 = 𝟐 × 𝑺𝒑𝒆𝒄𝒊𝒇𝒊𝒄𝒊𝒕𝒚 × 𝑹𝒆𝒄𝒂𝒍𝒍 𝑺𝒑𝒆𝒄𝒊𝒇𝒊𝒄𝒊𝒕𝒚+ 𝑹𝒆𝒄𝒂𝒍𝒍 (19), 𝒔𝒑𝒆𝒄𝒊𝒇𝒊𝒄𝒊𝒕𝒚 = 𝑻𝑵 𝑻𝑵+𝑭𝑷 (20) Issa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', Ali, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', & Rashid , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' An Efficient Hybrid Classification Approach for COVID-19 Based on Harris Hawks Optimization and Salp Swarm Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' International Journal of Online and Biomedi- cal Engineering (iJOE), 18(13), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' 113–130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='3991/ijoe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='v18i13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='33195 11 In which "TP" (true positives) denotes positive COVID-19 images which the classi- fier accurately labeled, and "TN" (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' true negatives) corresponds to the nega- tives COVID-19 images that have been successfully labeled by the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' False positives (FP) are positive COVID-19 images mislabeled as negative, whereas false negatives (FN) are negative COVID-19 images that have been incorrectly identified as positive COVID-19 images [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' 7 Results and discussion A total of 800 X-ray images (400 covid-19 and 400 normal) have been collected from the digital database and utilized for testing the efficacy of the suggested hybrid approach, which utilized two state-of-art algorithms (SSA, HHO) for feature selection to improve the classification of the covid-19 infection with the use of automatic AI techniques and showed a high level of classification accuracy following testing and training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' The dataset was divided into two sections: 20% for validation and testing and 80% for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Table 2 demonstrates that the suggested hybrid method has a high accuracy percentage based on the classifiers utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' The parameter setting for the sug- gested methodology has been listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Table1: Parameter values for used methods Methods Parameter values HHOSSA algorithm Feature size: 126 Population size: 30 Number of iterations for HHO:100 Number of iterations for SSA:20 Ub:1 Lb:0 Thre_val:0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='5 Beta:1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='5 Random variables a and b: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='9, [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='1, a ] KNN classifier K=5 Classes count:2 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='of training set:224 SVM classifier Classes count:2 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='of training set:224 XGboost classifier Classes count:2 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='of training set:224 Table 2: Performance of HHOSSA over three classifiers KNN, SVM, and XGboost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Classifier Accuracy Precision Recall F1 KNN 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='21428571428571 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='98 Issa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', Ali, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', & Rashid , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' An Efficient Hybrid Classification Approach for COVID-19 Based on Harris Hawks Optimization and Salp Swarm Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' International Journal of Online and Biomedi- cal Engineering (iJOE), 18(13), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' 113–130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='3991/ijoe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='v18i13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='33195 12 SVM 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='42857142857143 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='96 XGboost 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='21428571428571 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='98 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content="1 Comparative study The suggested system's performance was assessed utilizing a variety of modern op- timization methods (GWO, WOA)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Table (3) shows the performance of the HHO algo- rithm used for feature selection and gets 94%,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='89%,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' and 94% over three classifiers KNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' SVM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' and XGboost,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' while Table(4) shows the performance of the SSA algo- rithm used for feature selection and gets 96%,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='80%,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='94% over three classifiers KNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' SVM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' XGboost,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Table (5) shows the performance of GWO algorithm used for feature selection and gets 96%,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='82%,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='92% over three classifiers KNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' SVM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' XGboost,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' While Table (6) shows the performance of WOA algorithm used for feature selection and gets 96%,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='86%,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='96% over three classifiers KNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' SVM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' XGboost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Table 3: Performance of HHO over three classifiers KNN, SVM, and XGboost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Classifier Accuracy Precision Recall F1 KNN 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='64285714285714 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='95 SVM 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='28571428571429 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='90 XGboost 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='64285714285714 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='95 Table 4: Performance of SSA over three classifiers KNN, SVM, and XGboost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Classifier Accuracy Precision Recall F1 KNN 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='64285714285714 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='95 SVM 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='35714285714286 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='81 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='80 XGboost 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='64285714285714 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='95 Table 5: Performance of GWO over three classifiers KNN, SVM, and XGboost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Classifier Accuracy Precision Recall F1 KNN 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='42857142857143 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='93 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='97 SVM 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='14285714285714 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='85 XGboost 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='85714285714286 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='93 Issa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', Ali, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', & Rashid , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' An Efficient Hybrid Classification Approach for COVID-19 Based on Harris Hawks Optimization and Salp Swarm Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' International Journal of Online and Biomedi- cal Engineering (iJOE), 18(13), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' 113–130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' https://doi.' metadata={'source': 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+page_content=', Ali, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', & Rashid , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' An Efficient Hybrid Classification Approach for COVID-19 Based on Harris Hawks Optimization and Salp Swarm Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' International Journal of Online and Biomedi- cal Engineering (iJOE), 18(13), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' 113–130.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='2 HHOSSA HHO SSA WOA GWO Accuracy Precision Recall F1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='84 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='92 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='98 1 HHOSSA HHO SSA WOA GWO Accuracy Precision Recall F1 Issa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', Ali, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', & Rashid , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' An Efficient Hybrid Classification Approach for COVID-19 Based on Harris Hawks Optimization and Salp Swarm Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' International Journal of Online and Biomedi- cal Engineering (iJOE), 18(13), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' 113–130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='3991/ijoe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='v18i13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='33195 15 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='2 Software and Hardware Requirements The proposed system operates by using a personal computer Lenovo that has speci- fications such as Intel(R) Intel(R) Core(TM) i7-6500U @ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='59 GHz for CPU, 8 GB windows10 of RAM, and 64-bit Operating System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' The proposed system is operated by using python 10 languages with (Pycharm) IDE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Table (7) shows the processing time of the proposed algorithm and stand-alone algorithms depending on the classifi- cation processing time of the testing dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Table 7: Processing time of proposed (HHOSSA), HHO, SSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Algorithm Total processing time (seconds) HHOSSA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='0661 HHO 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='9906 SSA 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content="1425 It should be noted that the hybrid algorithm's processing time for completing the clas- sification process is less than the sum of the processing times for the Harris hawk and Salp algorithms because the Salp algorithm's iterations are fewer than those of the Harris hawk algorithm within the hybrid algorithm." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=" However, this improved the clas- sification process and accelerated performance without degrading the hybrid algo- rithm's quality." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' 8 Conclusion and future works The presented work presents a new hybrid swarm algorithm (referred to as HHOSSA) that combines the SSA and HHO for selecting the best features subset to improve the detection and classification of the COVID-19 virus with the use of chest X-ray images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='The ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='hybrid ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='algorithm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' However,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' this did not af- fect the quality of the hybrid algorithm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' but rather it increased the speed of performance and improved the classification process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=" A total of 800 (400 covid-19 and 400 normal) X-ray images are taken from the digital database to assess the HHOSSA's performance." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' XGboost and KNN classifiers get 98% accuracy, whereas SVM classifiers score 96%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' We want to adapt the suggested technique to more applications in the future, including Issa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', Ali, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', & Rashid , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' An Efficient Hybrid Classification Approach for COVID-19 Based on Harris Hawks Optimization and Salp Swarm Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' International Journal of Online and Biomedi- cal Engineering (iJOE), 18(13), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' 113–130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='3991/ijoe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='v18i13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='33195 16 signal processing and cloud computing task scheduling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=" Furthermore, the HHO algo- rithm's searching power was used to construct a novel suggested algorithm in several aspects." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' 9 Acknowledgment The authors would like to thank the University of Technology, Baghdad, Iraq for their continuous support for this research work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' 10 References [1] Lakshmanarao, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', Raja Babu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', & Srinivasa Ravi Kiran, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' An Efficient Covid19 Epidemic Analysis and Prediction Model Using Machine Learning Algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Inter- national Journal of Online and Biomedical Engineering (iJOE), 17(11), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' 176–184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='3991/ijoe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='v17i11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='25209 [2] S.' metadata={'source': 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https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='3991/ijoe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='v17i02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='18013 11 Authors Abubakr S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Issa received his bachelor’s degree in computer science department – Artificial intelligence branch from the University of Technology (UOT) – Iraq 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Since 2014, he is working as a programmer at the Information Technology Center, at the University of Technology up till now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Meanwhile, he is an M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='Sc candidate at the University of Technology (UOT) – Iraq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Assistant Professor Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Yossra Hussain Ali.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' She received her B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='Sc, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='Sc, and Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' degrees in 1996, 2002, and 2006 respectively from Iraq, the University of Technology, Department of Computer Sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' She joined the University of Technology, Iraq in 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' During her postgraduate studies, she worked on Computer Networks, Infor- mation systems, Agent Programming and Image Processing as well as some experience in Artificial Intelligent and Computer Data Security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' She is a reviewer at many confer- ences and journals and she supervised several undergraduate and postgraduate (PhD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' and MSc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=') dissertations in Computer sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Yossra has many professional certifi- cates and she has published in well-regarded journals (e-mail: yossra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='ali@uotechnol- ogy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='iq).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Tarik A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' Rashid received his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' in Computer Science and Informatics degree from the College of Engineering, Mathematical and Physical Sciences, University College Dublin (UCD) in 2001–2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' He pursued his Post-Doctoral Fellow at the Computer Science and Informatics School, College of Engineering, Mathematical and Physical Sciences, University College Dublin (UCD) from 2006–2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' He joined the University Issa, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', Ali, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=', & Rashid , T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' An Efficient Hybrid Classification Approach for COVID-19 Based on Harris Hawks Optimization and Salp Swarm Optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' International Journal of Online and Biomedi- cal Engineering (iJOE), 18(13), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' 113–130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='3991/ijoe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='v18i13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='33195 19 of Kurdistan Hewlêr (UKH) in 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content=' He has also been included in the prestigious Stan- ford University list of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} +page_content='% of the best world researchers for the years 2020 and 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7NE4T4oBgHgl3EQf2Q0O/content/2301.05296v1.pdf'} diff --git a/89E0T4oBgHgl3EQffwAL/content/tmp_files/2301.02407v1.pdf.txt b/89E0T4oBgHgl3EQffwAL/content/tmp_files/2301.02407v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..33ba0607c9fefcb3ad05d0433a7aea1fb06e51ad --- /dev/null +++ b/89E0T4oBgHgl3EQffwAL/content/tmp_files/2301.02407v1.pdf.txt @@ -0,0 +1,2550 @@ +The detection of an extraordinarily-luminous +high-redshift optical/ultraviolet flare by Swift/UVOT +Zhi-Ping Jin1,2,3†, Hao Zhou1,2,3†, Yun Wang1,3, Jin-Jun Geng1, +Stefano Covino4, Xue-Feng Wu1,3, Xiang Li1, +Yi-Zhong Fan1,2,3∗, Da-Ming Wei1,2,3, and Jian-Yan Wei5 +1Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210023, China +2Key Laboratory of Dark Matter and Space Astronomy of Chinese Academy of Sciences, +Nanjing 210023, China +3School of Astronomy and Space Science, University of Science and Technology of China, +Hefei 230026, China +4INAF/Brera Astronomical Observatory, via Bianchi 46, I-23807 Merate (LC), Italy +5National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100049, China +∗To whom correspondence should be addressed; E-mail: yzfan@pmo.ac.cn. +†These authors contributed equally to this work. +Hyper-luminous optical/ultraviolet flares have been detected in Gamma-ray +Bursts and the record was held by naked eye event GRB 080319B. Such flares +are widely attributed to internal shock or external reverse shock radiation. +With a new dedicated method developed to derive reliable photometry from +saturated images of Swift/UVOT, here we carry out time-resolved analysis of +the initial White band 150 s exposure of GRB 220101A, a burst at the red- +shift of 4.618, and report a rapidly-evolving optical/ultraviolet flare with an +unprecedented-high absolute AB magnitude ∼ −39.4. At variance with GRB +080319B, the temporal behavior of this new flare does not trace the gamma- +ray activities. Rather than either internal shocks or reverse shock, this opti- +1 +arXiv:2301.02407v1 [astro-ph.HE] 6 Jan 2023 + +cal/ultraviolet monster is most likely from the refreshed shocks induced by the +catching-up of the late-ejected extremely-energetic material with the earlier- +launched decelerating outflow. We detect the first ultraviolet/optical flare with +an absolute AB magnitude brighter than −39 and reveal the efficient process +to power such transients. +Gamma-ray bursts are widely believed to originate from the internal energy dissipation of +a highly relativistic and narrowly collimated outflow that was launched by a nascent stellar +mass black hole or magnetized neutron star. Shortly after the onset of prompt emission of +GRBs, there could come very bright optical/ultraviolet flashes arising from either the internal +shocks in specific conditions or the external reverse shock radiation (1). An apparent ∼ 9th +mag optical radiation was detected in GRB 990123 at a redshift of z = 1.62 (2, 3). Its rapid +rise and the quick decline are consistent with the reverse shock radiation model (4–6), and +the late more-detailed afterglow modeling revealed that the reverse shock region should be +significantly more magnetized than the forward shock region (7,8). A long-holding record was +set by GRB 080319B, a burst at a redshift of z = 0.937. Its peak visual magnitude reaches 5.3 +(corresponding to an absolute AB magnitude of −38.7), which is so bright that an observer in +a dark location could have seen it with the naked eyes (9)! The correlated temporal behaviors +of the prompt gamma-ray emission and the optical radiation are in favor of the internal shock +process (10,11). In the past decade, no similar or even comparable events have been reported. +GRB 220101A was discovered simultaneously by Swift Burst Alert Telescope (BAT) (12), +the Fermi satellite (13) and the AGILE satellite (14). Before the so-called finding chart exposure +ranging from 90 to 240 seconds with the White filter (12), UVOT observed the target in V band +for 9 seconds. The estimated average magnitude in the White band for an exposure of ∼ 150 s +is ∼ 14.7th Vega mag (12,15). The redshift was measured to be z = 4.618 and in the spectrum +a broad absorption feature, which results from the Lyman alpha absorption (16, 17), is evident +2 + +centered at ∼ 6820 ˚A. The corresponding isotropic equivalent gamma-ray energy is ∼ 4 × 1054 +erg and the peak luminosity is ∼ 9 × 1053 erg s−1, both are in the rank of the brightest ones +among current GRBs (18,19). After the redshift correction, the observed optical photons were +intrinsically in the ultraviolet bands. Therefore, all the emission detected by Swift suffered from +serious absorption (in the observer’s frame, the V band absorption is about 2 mag stronger than +that in the I-band (16)) and thus the intrinsic emission would be much brighter. This is in +particular the case for the White filter because of its large effective area in the blue part (i.e., +U, UVW1, UVM2 and UVW2) and the Lyman alpha/intergalactic medium (IGM) absorption +would be very strong. We concentrate on the first ∼ 150 s exposure with White filter in the +event mode (i.e. photon counting mode) that can be efficiently divided into short bins according +to the signal-to-noise ratio (SNR). Our time-resolved analysis reveals that the measurements +in the time range of ∼ 106 − 150 s after the BAT trigger suffered from strong saturation, as +shown in Fig. 1, Table S5 and Fig. S1. The absence of clear signal of the read-out streaks in the +raw data, indicating a moderate saturation, however hampers a correction following procedures +proposed in the literature (20,21). Therefore we propose a new method to correct the saturation +effect. The basic idea is that though the pile up at the source site is so serious that can not be +reliably corrected, the surrounding but relatively “separated” pixels are possibly unaffected by +saturation and therefore the enhancement of the counts should be correlated with the intrinsic +count rate of the source. To clarify whether it is the case, we need some data with known +magnitudes as well as the count rates in external annuli. For the unsaturated data with relatively +low ring count rate, we simply take UVOT/White measurements of GRB 220101A at 150−240 +seconds after the burst trigger. For the moderate saturation that is of our great interest, we +take the UVOT measurements of GRB 130427A in the time interval of 500 − 2000 seconds. +Though the moderately saturated White band emission of GRB 130427A can not be directly +measured, we infer them with the UVOT emission in other bands since the spectrum can be +3 + +well fitted by a single power-law, see Fig. S2. With these two sets of data, we do find a tight +correlation between photon count rate in 5′′ aperture ( ˙Naper, directly measured if unsaturated, +or inferred from the “intrinsic” count rate ˙Nint measured in other ways) and in the 15′′ − 25′′ +ring ( ˙Nring, directly measured in UVOT images), which reads ˙Naper = (22.22 ± 0.84) ˙Nring +for ˙Nring ≤ 80 s−1 (see Fig. S3). The correlation efficient for such an empirical relation is +0.99 (22). The other essential correction is on the absorption of the ultraviolet photons at high +redshift. In the analysis we correct such a factor, i.e., 4.78 ± 0.10 mag in the White band, with +the wide band energy spectrum and further check it with the other two GRBs at rather similar +redshifts (see Fig. S5). +In Fig. 1 we show the lightcurves of the prompt gamma-ray emission and the very early +optical emission. The first White exposure with a duration of 150 s was in the events mode. +In our approach, a bin size of 4s is adopted. In principle, a narrower bin size is helpful in +revealing the peak or structure of the flash, but a reasonably wide bin is necessary for a high +SNR. The optical/ultraviolet flash lightcurve is relatively smooth and there is no evidence for +tracing the temporal behavior of prompt gamma-rays. This is very different from the case of +GRB 080319B, where the naked-eye optical flash shows strong variabilities and the optical +lightcurve resembles that of the gamma-rays (see the insert of Fig. 1), indicating a new origin. +We have also constructed the “prompt” optical, X-ray and gamma-ray SED. In Fig. 2 we show +three representative time intervals of the first UVOT White band exposure, including the very +beginning, the peak, and the final shallow decline phase. In the rise and the quick decline +phases, the extrapolation of the high energy radiation spectrum into the optical is well below +the White band measurements, which again points towards different physical origins of the +optical and high energy radiation. While in the t−2.3±0.3 shallow decline phase, the optical +to X-ray emission are consistent with being a single power-law, which may be dominated by +the external reverse shock radiation. In Fig. 3 we present the absolute AB magnitudes of the +4 + +very early optical emission of GRB 220101A and the other three remarkable events, including +GRB 990123 (3), GRB 050904 (23) and GRB 080319B (9), distinguished by the extremely +bright optical emission. After the proper saturation, absorption and cosmological corrections, it +turns out that GRB 220101A sets a new record. The prompt ultraviolet to X-ray spectrum at the +optical/ultraviolet emission peak time is softer than ν−1.3 (see Fig.2). If this soft spectrum could +extend to the optical band in the frame of the burst, GRB 220101A would be so far the unique +source with an absolute AB magnitude brighter than −40 in the visible band (22). Note that the +peak optical emission of GRB 220101A could be even stronger than presented here since our +current fluxes are the average of the radiation in each 4s bin. +As already mentioned before, for GRB 080319B, the internal shock model is favored by +the similar temporal behaviors of the prompt gamma-ray and optical radiation. While for GRB +990123, the external reverse shock model has been widely accepted to account for the optical +flash. The optical/ultraviolet flare detected in GRB 220101A, however, should have a differ- +ent physical origin for the following facts: (i) In contrast to GRB 080319B, the optical flash +lightcurve of GRB 220101A does not trace the variability of the prompt gamma-rays (see Fig. +1), requiring different radiation processes/sites of these two components; (ii) The t−2.3-like de- +cline of the optical/ultraviolet flare of GRB 220101A may be due to the reverse shock emission, +but the ∼ t20 increase is much quicker than that of GRB 990123 and hence strongly in tension +with the standard external reverse/forward shock emission model (5, 6). Here we present a re- +freshed shock model for the brightest optical/ultraviolet spike of GRB 220101A. Looking at the +gamma-ray lightcurve, the main burst phase consisting of two giant gamma-ray spikes appears +at ∼ 90 s after the BAT trigger, and the earlier emission was much weaker (i.e., the time- +averaged luminosity is ∼ 1052 erg s−1). As indicated by the bulk Lorentz factor−luminosity +correlation (24,25) of Γ ∝ L0.3 +γ , the weak/slow GRB outflow component launched at the early +times is expected to have a Γ ∼ 102 and the surrounding interstellar medium further decelerates +5 + +the outflow to a Lorentz factor of ΓW, while the outflow component yielding the most luminous +part of GRB 220101A likely has a Lorentz factor of ΓM ∼ 103. The first giant spike comes +from the energy release of the main outflow, either through the internal shocks or the magnetic +re-connection within it. Soon the main outflow would catch up with the decelerating weak part +at a time of ∼ Γ2 +WδtWM/Γ2 +M ∼ O(10) s, which explains the second gamma-ray spike and the +delayed onset of the optical/ultraviolet flare, where δtWM ∼ 100 s is the delay of the onset of the +main part with respect to that of the weak part (started at ∼ 60 s before the trigger, see Fig. 1). +The collision of the late/fast material shell(s) with the early/decelerating material will generate +strong refreshed shocks and then produce energetic emissions. Following the treatments pre- +sented in Sec. 2.1 of the Ref. (26), it is straightforward to show that for the internal shocks taking +place at ∼ 2Γ2 +WcδtWM/(1 + z) ∼ 1016 cm (ΓW/102)2(δtWM/102 s), the typical synchrotron +radiation frequency is indeed within the optical/ultraviolet bands. The bulk Lorentz factor of +the merged shells can be approximated to be ¯Γ ≈ +� +[MWΓW + MMΓM]/[MW/ΓW + MM/ΓM] +and the Lorentz factor of the internal shocks can be estimated as Γsh ≈ ΓM/¯Γ + ¯Γ/ΓM, where +MW and MM are the rest masses of the ejecta powering earlier weak gamma-ray emission and +the main outburst, respectively (27). Indeed, for GRB 220101A-like burst, we have the outflow +luminosity of Lm ∼ 1053 − 1054 erg s−1, with the fractions of the shock energy given to the +magnetic fields (electrons) ϵB ∼ 0.1 (ϵe ∼ 0.3), ¯Γ ∼ several × 100 and Γsh ∼ a few, it is natural +to have an optical/ultraviolet flux (26) of ∼ 1 Jy even for a redshift as high as ∼ 5 (a numerical +example is presented in (22) and Fig.S6). +Note that the very energetic prompt emission appearing at ∼ T0 +90 s, which partly overlap +with the optical/ultraviolet flash, after the BAT trigger should also effectively cool the electrons +accelerated in the collision discussed above. Such a process would produce GeV emission, +which is expected to last longer than the overlapping phase of the prompt MeV emission and +ultraviolet/optical flare. Indeed, at t ∼ 100 − 150 s after the BAT trigger, GeV emission was +6 + +detected from GRB 220101A (28). +Though the hyper-luminous very early optical/ultraviolet emission are not common, we +suggest that the bursts with prompt emission resembling GRB 220101A (i.e., the much more +energetic outflow is well separated from the early ejecta) are good candidates for hosting the +extraordinarily bright flares. The problem is how to catch such signals promptly. Small tele- +scopes with a large field of view should be very helpful and the I/R-band observation of these +telescopes can catch the monsters in a wide range of redshifts. Anyhow, such observations are +limited by the weather, the time (day or night) and the burst site. The space telescopes like +Swift/UVOT and SVOM/VT (29) may play an important role for the high redshift events. Since +the optical/ultraviolet flash of GRB 220101A was observed by Swift/UVOT, below we focus on +the upcoming 0.4m SVOM/VT with two channels, including the blue (400 − 650 nm) and the +red (650 − 1000 nm) bands. For the shortest exposure time of 1s, the saturation limit is about +9th magnitude. Given its higher sensitivity in comparison to Swift/UVOT V filter, the seriously +absorbed “ultraviolet” emission of GRB 220101A/GRB 080319B-like extra-luminous events, +even taking place at the even higher redshift (say, z ∼ 6), can still be caught by the blue channel +of SVOM/VT though the red channel might be saturated (22). Dedicated observation strategies +are needed to optimize the potential of the discoveries. +7 + + 0 + 0.1 + 0.2 + 0.3 + 0.4 + 0.5 + 0.6 + 0.7 + 0.8 + 0.9 +-50 + 0 + 50 + 100 + 150 + 200 + 250 +0 +200 +400 +600 +800 +1000 +1200 +BAT count rate (count/s/det) +UVOT count rate (count/s) +Time since trigger (s) +BAT +V +White + + 0 + 1 + 2 + 3 + 4 + 0 + 20 + 40 + 60 + 800.0E0 +5.0E4 +1.0E5 +1.5E5 +GRB 080319B V +Figure 1: +Photon count rates of the prompt gamma-ray (Swift/BAT) and optical +(Swift/UVOT V and White band) emission of GRB 220101A. The prompt gamma-ray +lightcurve is highly variable, while the prompt optical emission lightcurve is relatively smooth +and does not trace that of gamma-rays. The red filled circles are from the aperture measurement +while the open circles are obtained with the new method developed in this work. The energetic +optical/ultraviolet flash just overlaps with the late part of the giant outburst phase of the prompt +gamma-rays. The prompt gamma-ray and optical lightcurves of GRB 080319 are inserted for +comparison. +8 + +10 +3 +10 +2 +10 +1 +100 +101 +102 +103 +Energy (keV) +10 +2 +10 +1 +100 +101 +102 +103 +104 +Flux density (keV/cm2/s/keV) +Unabsorbed CPL model (91.96 - 93.62 s) +Unabsorbed CPL model (113.64 - 117.62 s) +Unabsorbed PL model (173.63 - 239.56 s) +Mape AB = 17.75 ± 0.24 (91.96 - 93.62 s) +Mring AB = 13.56 ± 0.19 (113.64 - 117.62 s) +Mape AB = 15.42 ± 0.04 (173.63 - 239.56 s) +Figure 2: The “prompt” optical to γ-ray SEDs of GRB 220101A. The data in blue (red) +are collected in the very beginning (peak) of the UVOT/White band emission. The optical +emission in both cases are well above the extrapolation of the high energy spectrum, suggesting +an origin different from the prompt X-rays and gamma-rays. While in the time interval of +t ∼ 173.6 − 239.6 seconds, the extrapolation of the X-ray and gamma-ray spectrum into the +optical is in agreement with the UVOT data. +9 + +-40 +-39 +-38 +-37 +-36 +-35 +-34 +-33 +-32 +-31 +-30 +-29 +-28 +101 +102 +103 +Absolute Magnitude (AB) +t′ (second) +GRB 220101A +GRB 080319B +GRB 050904 +GRB 990123 +-39 +-38 +-37 +-36 +1015 1016 + +ν′ (Hz) +Figure 3: The ultraviolet/optical flare of GRB 220101A (red) in comparison to that of +GRB 990123 (green) (3), GRB 050904 (pink) (23) and GRB 080319B (blue) (9) in rest +frame. The White band emission of GRB 220101A has been corrected for total extinction +of Aλ = 4.78 ± 0.1 mag, including the tiny softening of E(B − V ) = 0.0483 mag in the +Milky Way. The absolute AB magnitude of GRB 220101A exceeds that of GRB 080319B, the +so-called naked burst, rendering it the most energetic optical/ultraviolet flare recorded so far. +References +1. P. M´esz´aros, M. J. Rees, Astrophys. J. 476, 232 (1997). +2. M. I. Andersen, et al., Science 283, 2075 (1999). +3. C. Akerlof, et al., Nature 398, 400 (1999). +10 + +4. A. J. Castro-Tirado, et al., Science 283, 2069 (1999). +5. R. Sari, T. Piran, Astrophys. J. 520, 641 (1999). +6. P. M´esz´aros, M. J. Rees, Mon. Not. Roy. Astron. Soc. 306, L39 (1999). +7. Y.-Z. Fan, Z.-G. Dai, Y.-F. Huang, T. Lu, Chin. J. Astron. Astrophys. 2, 449 (2002). +8. B. Zhang, S. Kobayashi, P. M´esz´aros, Astrophys. J. 595, 950 (2003). +9. J. L. Racusin, et al., Nature 455, 183 (2008). +10. Y.-Z. Fan, B. Zhang, D.-M. Wei, Phys. Rev. D 79, 021301 (2009). +11. Z. Li, E. Waxman, Astrophys. J. Lett. 674, L65 (2008). +12. A. Tohuvavohu, et al., GRB Coordinates Network 31347, 1 (2022). +13. S. Lesage, C. Meegan, Fermi Gamma-ray Burst Monitor Team, GRB Coordinates Network +31360, 1 (2022). +14. A. Ursi, et al., GRB Coordinates Network 31354, 1 (2022). +15. N. P. M. Kuin, A. Tohuvavohu, Swift/UVOT Team, GRB Coordinates Network 31351, 1 +(2022). +16. S. Y. Fu, Z. P. Zhu, D. Xu, X. Liu, S. Q. Jiang, GRB Coordinates Network 31353, 1 (2022). +17. J. P. U. Fynbo, et al., GRB Coordinates Network 31359, 1 (2022). +18. J. L. Atteia, GRB Coordinates Network 31365, 1 (2022). +19. A. Tsvetkova, et al., GRB Coordinates Network 31433, 1 (2022). +20. M. J. Page, et al., Mon. Not. Roy. Astron. Soc. 436, 1684 (2013). +11 + +21. P. W. A. Roming, et al., Astrophys. J. Supp. 228, 13 (2017). +22. Materials and methods are available as supplementary materials. +23. M. Bo¨er, et al., Astrophys. J. Lett. 638, L71 (2006). +24. J. L¨u, et al., Astrophys. J. 751, 49 (2012). +25. Y.-Z. Fan, D.-M. Wei, F.-W. Zhang, B.-B. Zhang, Astrophys. J. Lett. 755, L6 (2012). +26. D. M. Wei, T. Yan, Y. Z. Fan, Astrophys. J. Lett. 636, L69 (2006). +27. T. Piran, Phys. Rept. 314, 575 (1999). +28. M. Arimoto, L. Scotton, F. Longo, Fermi-LAT Collaboration, GRB Coordinates Network +31350, 1 (2022). +29. S.-J. Yu, F. Gonzalez, J.-Y. Wei, S.-N. Zhang, B. Cordier, Chin. Astron. Astrophys. 44, 269 +(2020). +Acknowledgments +Funding: +This work was supported in part by NSFC under grants of No. 12225305, 11921003 +and 11933010, the China Manned Space Project (NO.CMS-CSST-2021-A13), Major Science +and Technology Project of Qinghai Province (2019-ZJ-A10), Key Research Program of Frontier +Sciences (No. QYZDJ-SSW-SYS024). SC has been supported by ASI grant I/004/11/0. +Author Contributions: +Y.Z.F and Z.P.J launched the project. Z.P.J, H.Z., Y.W, X.L, S.C and +J.Y.W carried out the data analysis. Y.Z.F, J.J.G., X.F.W, D.M.W and Z.P.J interpreted the data. +Z.P.J, H.Z. and Y.Z.F prepared the paper and all authors joined the discussion. Z.P.J and H.Z +contributed equally. +12 + +Competing Interests: +The authors declare that they have no competing financial interests. +Author Information: +Correspondence and requests for materials should be addressed to Y.Z.F +(yzfan@pmo.ac.cn). +Code availability: +The codes used in this analysis are standard in the community, as intro- +duced in the supplementary materials. +Data availability: +The Swift observation data analysed/used in this work are all publicly avail- +able. +13 + +Supplementary materials +Materials and Methods +Tables S1 to S6 +Figs. S1 to S7 +References (30-52) +Materials and Methods +1 +A new method to measure the saturated sources in Swift +UVOT images +UVOT is a photon counting detector and typical read-out rate is once every ∼ 11 ms. If the +source is bright enough (> 10 counts s−1), coincidence losses start to be significant and a +correction is necessary. When the incident photon counts rate beyond the read-out rate ∼ 86 s−1, +the source is fully saturated and proper coincidence loss correction is impractical (30). However +for extremely saturated sources with read-out streaks, a calibration method has been developed +based on the measurement of read-out streak line strength (20). Anyhow, the read-out streak +lines are only present in the extremely saturated sources or those with very long time exposure. +For the moderate saturation with relatively short exposure, it cannot be applied and our main +goal is to provide a new way. Below we focus on the White band, but our method can be applied +to other UVOT filters as well (indeed, as a validation, we also show in the end of this subsection +that a rather similar empirical correction function holds for the V band). +The saturated pattern of an UVOT image can be divided into three parts. The first is a +point source like structure at the center of saturated pattern, which represents the location of the +saturated source. The second part is a dark square structure caused by coincidence loss and the +half length of its diagonal line is about 14 arcsec. A more detailed explanation is that UVOT has +14 + +actually a 256×256 CCD which records the flash pattern produced by the incident photon after +several amplifiers and there is a centroid algorithm to calculate positions of incident photons +whose accuracy could reach 0.125 pixel. As a result, each physical pixel could be subsampled +to 8×8 virtual pixels with a resolution of 0.5 arcsec/pixel. The side length of the dark square is +about 20 arcsec, that is 40 virtual pixels, corresponding to an area of 5×5 pixels region on real +physical CCD which is the affected region of coincidence loss. The third part is the halo ring, +which is distinct for saturated sources and some unsaturated sources but with low background. +Fig. S1 shows such a saturated pattern. We attribute the halo rings to the wing of the Point +Spread Function (PSF) of UVOT detector. To test this conjecture, we will examine whether +the “intrinsic” photon counts rates of saturated sources is proportional to photon counts rates of +halo rings. +To avoid the influence of the coincidence loss, the best measurement region to get the highest +S/N ratio is the area between a circle with a radius of 25 arcsec and a square, with the same +center and with a side length of 20 arcsec, like a Chinese copper cash. However, if Swift rotated +during observations, the dark region of final stacked science image are not necessarily a square +due to that the coincidence loss square is aligned to the edge of CCD. Hence, we used an +annulus of an inner radius of 15 arcsec and an outer radius of 25 arcsec (i.e., the outer edge +of halo rings) to measure photon count rate in the ring ( ˙Nring), where the background should +be removed and the coincidence loss has been corrected. The crucial step is to reliably derive +the corresponding photon count rate of the saturated source within the standard aperture with +a radius of 5 arcsec( ˙Naper). As mentioned above, if the incident photon counts rate is beyond +the CCD readout rate, the source is saturated. Fortunately, the UVOT White band is much +wider than other 6 bands (hence, we will call them the narrow bands), which means although +a source is saturated in White band, it could be unsaturated in narrow bands. It is therefore +plausible to measure the spectrum with other filters of UVOT and then convolve it with the +15 + +White filter to get the corresponding “intrinsic” emission. This can be done for the power-law +like afterglow spectrum of GRBs and the very early time optical flash of GRB 130427A is a +nice sample. The earliest UVOT measurements of this burst were highly saturated and some of +them can be analyzed with the readout streak method (20). Moreover, as shown in Maselli et +al. (31) and the left panel of Fig. S2, when the White filter was still saturated, there were usable +measurements in other bands. In the right panel of Fig. S2, we show the ultraviolet/optical +SED of GRB 130427A with the UVOT observations. Note that these data were re-measured +in this work and they are consistent with that reported in the literature (31). We performed +the early time photometry of GRB 130424A with HEASoft and the results are summarized in +Table S1. The first exposure in B band and the first 2 exposures in U band were saturated, +hence we took the values from Maselli et al (31). Light curves of 6 narrow bands were fitted +to found their magnitudes simultaneous with White band exposures, the results are listed in +Table S2. We then carry out the power-law spectral fit to the SED and estimate the White band +magnitudes, as summarized in the last column of Table S2, which are further used in Table S3 +to yield the ˙Nint (in another word, the inferred ˙Naper). It is worth noting that in epoch 1 there +was an optical/ultraviolet flare and hence it is not suitable to evaluate the White band emission +with this method. Moreover, the White band measurement in the first, second and third epochs +were significantly saturated with readout streaks, for which the fluxes were reported before. +As show in Fig. S3, in epoch 2 our calculated flux is consistent with that reported in Maselli +et al. (31), validating the method proposed in this work. Our downloaded image of the epoch +3 mentioned in Maselli et al. (31) is distorted and we have hence focused on the subsequent +observation data with an exposure of 20 s. Our estimated flux is still well consistent with that +reported in Maselli et al. (31), which is expected because these two measurements were almost +simultaneous. Anyhow, in the plot the data point reported in Maselli et al. (31) is not shown +because we can not measure its ring count rate because of distortion. For epoch 4 to epoch +16 + +8, there were no readout streaks and the method developed by Page et al. (20) does not work +any longer. Our method mentioned above applies to these data and yield reasonable results. +As for GRB 220101A, shortly after its peak, the ultraviolet/optical flash is not saturated any +more. For these observations we can reasonably measure its White band emission. HEASoft +UVOT pipeline was used to make photometry of barely saturated images of GRB 220101A with +a circle aperture with a radius of 5 arcsec. However, a reliable measurement of the ring count +rate requires a somewhat long exposure. Therefore, we just divide the “tail” part of the flash +into two time intervals. We also notice 3 bright stars in the field and then measure them for +independent check. These five data points are summarized in Table S3. The White band fluxes +measured (indirectly and directly, respectively) in the above events and field stars are used to +clarify whether there is a tight correlation between the ring counts and the intrinsic source +emission. For such a purpose, these three data sets have been fitted with a linear function of a +model of y = ax and a least square cost function was applied, χ2 = � +i +(yi−axi)2 +y2 +err,i+(axerr,i)2, where +yi and xi represent extracted White-band photon counts rates and halo ring photon counts rates, +respectively, and yerr,i and xerr,i are the corresponding uncertainties. The Pearson correlation +coefficient is 0.99, which reveals a very strong linear correlation, and the χ2/d.o.f value is +∼ 0.90, which implies a reasonable fit, where d.o.f denotes the degree of the freedom. Hence, +we conclude that ˙Naper = 22.22 ± 0.84 ˙Nring can yield a reasonable estimation of “true” photon +counts rates of saturated sources in White band. Fig. S3 presents our best fitting result which +confirms our early speculation and suggests that the outer part of the PSFs of such sources are +nearly unmodified. +The ground-based telescopes can well measure the V-band emission of the sources, which +can thus provide an economical way to calibrate the saturated V-band observations of Swift/UVOT. +Interestingly, GRB 080319B is a nice example. For the UVOT V-band observations, in total we +have 22 sub event files, which were later converted to images with HEASoft for measurements. +17 + +The first 4 exposure duration are 30s, 40s, 50s and 55s, which are same as the time bins in Page +et al. (20). These exposures display readout streaks and have been analyzed with the method +of Page et al. (20), which are shown in the right panel of Fig. S3 (see the light green empty +squares). We measured the counts rate in the halo rings, which is defined above, with HEASoft, +but made coincidence loss correction manually. Another 18 images are unsaturated, the intrin- +sic emission were directly measured, and they are marked with dark green empty triangles in +the right panel of Fig. S3. These measurements are summarized in Table S4. In addition, the +optical emission of GRB 080319B was well measured by the ground based telescopes (32), and +the accurately measured V-band emission from RAPTOR-T can be taken as the intrinsic ones +(i.e., we have the ˙Nint, in another word, ˙Naper defined in this paper). The difference between the +V filter of UVOT and that of RAPTOR-T is small and the magnitude difference can be ignored, +as demonstrated by the overlapping data points in the left lower corner of the right panel of Fig. +S3. Since the very early UVOT/V band observations were in event mode, we can re-bin them +into the time intervals the same as that of RAPTOR-T and then get the ˙Nring. Time bins of our +measurements are listed in Table S4. Therefore, we apply the linear fit to the data sets and find +an empirical function of ˙Naper = 20.6 ± 0.4 ˙Nring with a high correlation coefficient of 0.998. +Such a correlation is nicely consistent with that for the UVOT/White band. It is worth noting +that for GRB 220101A, the photons collected in the White band are dominated by those passing +the V filter because of the serious absorption in the bluer region. Indeed we find rather sim- +ilar count rates for the (almost) simultaneous White and V-band measurements (see Fig. S4). +Therefore, the rather similar correction function for UVOT/V filter strongly suggests that our +White band analysis of GRB 220101A is robust. +18 + +2 +Data analysis +2.1 +Swift UVOT data analysis +Swift/UVOT observed GRB 220101A in V, B, U, W1, M2, W2 and White bands for several +epochs. For data in image mode, we started from the level 2 UVOT products and used standard +aperture photometry, background was measured in a nearby region without sources in stacked +images. Reliable detections were only obtained in V and White bands, and the photon count +rates were measured in 3 or 5 arcsec apertures, depending on SNR. Coincidences loss correction +and aperture correction were applied. For images without detection, upper limits were assuming +count rates would have reached the SNR of S/N = 3. Finally zeropoints including long-term +sensitivity correction were used for absolute calibrations. The results are shown in Table S5. +The first white-band exposure under event mode (incident positions and time of every pho- +ton are recorded) began at about 90 seconds after the trigger time, which lasted about 150 +seconds. Due to the fact that the luminosity of GRB 220101A changed rapidly at early epochs, +although the transient seems to be unsaturated on the image for the total 150s exposure, it could +be saturated in its peak phase. Hence, we screened the calibrated event data into slices whose +exposure time is ∼4s to check whether the situation mentioned above had happened. Follow- +ing the guidance of UVOT data process, event slices were transformed into images and image +calibrations (flat field and mod 8 corrections) were applied. Since the transient is bright and iso- +lated on reduced images, standard aperture photometry method was applied. From 90s to 100s, +the transient was brightening rapidly and then became saturated for about 50 seconds. After ∼ +150s since the trigger time, it became unsaturated, again. We found that there are halo rings +around the transient on barely saturated and saturated images, which we think are the ’wings’ +of point spread functions, hence, we analyzed the data with our calibration method described +before. The results are summarized in Table S5. +19 + +2.2 +Swift-BAT/XRT and Fermi-GBM data analysis +We processed Swift-BAT data according to standard procedures, using the software HEASoft +(ver. 6.29) and calibration database (CALDB), which are available at +https://www.swift.ac.uk/analysis/bat/setup.php. The mask weighting file used in extracting the +light curve is generated by batgrbproduct (a complete GRB processing script in HEASoft). We +extract event data at time intervals between -60 to 340 seconds related to the trigger time, the +energy range is 15-350 keV, and the time bin size is 1 second. Our BAT analysis results are +plotted with our Swift UVOT analysis results in Fig. 1. +We also present a spectral analysis in a broad gamma-ray band (0.3 - 40000 keV) from Swift- +BAT/XRT and Fermi-GBM data. The files used include the source and background spectrum +files, as well as the corresponding response functions. For BAT file extraction and correction, +we used standard procedures as in the manual (33). XRT files were created by online analysis +tools provided by Swift official website (34, 35). The Fermi-GBM data have been processed +with GBM Data Tools (36). There are different statistics used for each dataset (cstat for Swift- +XRT, χ2 for Swift-BAT and pgstat for Fermi-GBM data). We use Bilby (37) in the framework +of PyXspec for model parameter estimation. The results are shown in Fig. 2. +3 +Intrinsic optical/ultraviolet emission of GRB 220101A +To estimate a reliable un-absorbed optical/ultraviolet emission, we need an intrinsic spectrum +to evaluate the absorption in different observation bands. For such a purpose, in addition to the +UVOT V and White band observations, we adopt the g, r, i, z-band data from Liverpool tele- +scope measured at t ∼ 0.625 day after the burst (38) and the simultaneous XRT data. Such +a set of ground-based telescope observation data are chosen because they are almost simulta- +neous with one UVOT White measurement and at late times the White band emission was not +20 + +detectable any longer (see Table S5 and Fig. S4). The SED from i to g declines very rapidly, +requires a spectral index β ∼ 8 (see Fig. S5). Similar rapid declines, due to the serious Lyman +forest absorption, have been observed in GRB 000131 (39) and 100219A (40) at redshifts of +z = 4.500 and 4.667, respectively. Since the i and z observations do not suffer from strong +absorption and there is no evidence for the presence of a flare at that time, we adopt them to +construct the intrinsic optical (z band) to X-ray SED to be fν ∝ ν−0.70±0.05, with which we can +obtain the absorption correction in r, g as well as UVOT White and V bands. In the direction +of GRB 220101A, the Galactic extinction is E(B − V ) = 0.0483 (41). Basing on the intrinsic +spectrum of and assuming no extinction from the GRB host galaxy, we find an absorption in +White band as high as Aλ = 4.78 ± 0.10 mag, including Lyman absorption and the Galactic +extinction, see the right panel of Fig. S5. Note that here the central frequency of the White band +observation has been taken as the same as that of the V band because of the serious absorption +of the bluer photons, as demonstrated in Fig. S4. +In this work we adopt a cosmology with with H0 = 67.4 km s−1 Mpc −1, ΩM = 0.315 and +ΩΛ = 0.685 (42), a redshift z =4.618 leads to a distance modules DM= 48.19. The absolute +peak magnitude is calculated via Mpeak,abs = Mpeak −DM−Aλ +2.5(1−βi) log(1+z), where +the last term is the k-correction and βi is the intrinsic spectral slope. The pity is that none of the +extremely luminous flashes in GRB 990123, GRB 050904, GRB 080319B and GRB 220101A +have a measured optical/ultraviolet spectrum. For GRB 220101A, the UVOT and XRT data +suggest an “overall” optical to soft X-ray spectrum softer than ν−1.3. If this holds in the optical +band (i.e., βi ≥ 1.3) in the rest frame, then we would have Mpeak,abs ≤ −40 mag in the visible +band. It is so far the unique event to be brighter than the absolute AB magnitude of −39 mag, +see Table S6 for a comparison of the brightest flare in history. If there are spectral information +of optical flares in the future, these bursts would be able to directly compared in the same band. +21 + +4 +The numerical interpretation of the optical emission as well +as the X-ray afterglow emission +Here we call the X-ray emission after ∼ 170 s after the Swift trigger as the afterglow since the +earlier emission are most likely the low energy part of the prompt radiation arising from the +internal energy dissipation. +4.1 +Refreshed shock emission for the peak of the optical/ultraviolet flare +In the prompt γ−ray emission lightcurve, there are several weak gamma-ray spikes from earlier +outflow before the main pulse starting at ∼ T0 +65 s. The front half part (between ∼ T0 +65 to +102 s) of the giant gamma-ray pulse should come from the energy release of the main outflow, +either dissipated through internal shocks or magnetic re-connections within it. For the later part +(> T0 + 102 s) of the giant pulse, it overlaps with an energetic optical/ultraviolet flash, which +indicates the rise of an additional dissipation process. As the preceding weak outflow gets +decelerated to a bulk Lorentz factor of Γ1, a later launched but faster shell (with a bulk Lorentz +factor of Γ4) will catch up with it at a radius of R0, so that a collision between two shells would +occur. Note that Γ1 and Γ4 correspond to ΓW and ΓM mentioned in the main text, which is used +here for the convenience of the discussion below. If the fast shell is not extremely magnetized, +the collision would produce a refreshed forward shock (FS) propagating into materials of the +preceding shell, and a refreshed reverse shock (RS) propagating into the fast shell. As a result, +an optical/ultraviolet flash is expected from the radiation in the downstream of the refreshed +RS, which has been initially proposed and works well for optical flash in the early afterglow +stage (5). Here we show that this scenario could account for the prompt optical emission of +GRB 220101A with a detailed numerical approach. +Two refreshed shocks separate the system into four regions: (1) the unshocked slow shell, +(2) the shocked slow shell, (3) the shocked fast shell, and (4) the unshocked fast shell. Here- +22 + +after, Xi denotes the value of the quantity X in Region “i” in its own rest frame. Unlike the +preceding shell that exhausts the magnetic energy in the early stage (σ1 = 0), the later fast shell +may keep the magnetic fields advected from the central engine, which could be parameterized +by the magnetization of σ4 = B2 +4/4πn4mpc2, where n4 is the particle density in the comoving +frame of Region 4 and mp is the proton mass. Let’s introduce an equivalent “luminosity” of +the kinetic, internal and the magnetic energy for the two shells measured in the lab frame, Li, +the corresponding particle density is then ni = Li/4πR2βiΓ2 +i mpc3(1 + σi), where i = 1, 4, +βi = 1/ +� +(1 − 1/Γ2 +i ) and R is the radius from the central engine. Due to the highly vari- +able nature of the outflow from the central engine, the luminosity of the later fast ejecta could +be further described by L4 = Lf(R/Rpeak)qr for R ≤ Rpeak and L4 = Lf(R/Rpeak)qd for +R > Rpeak, where Rpeak is the radius that the RS reaches its peak luminosity, and qr (qd) is +the rising (decaying) index of the luminosity before(after) Rpeak. We assume that Region 2 and +Region 3 share a common bulk Lorentz factor (Γ2 = Γ3). After applying the hydrodynami- +cal/magnetohydrodynamical jump conditions (43,44) to the FS/RS respectively and the energy +conservation law to the FS-RS system (45), the evolution of Γ2 and relevant quantities within +these regions could be solved numerically given the total isotropic energy of each shell (Ef and +Es). +The kinetic particle-in-cell simulations reveal that particle acceleration is less efficient in +strongly magnetized shock than that of weakly magnetized shock (46). The shock is considered +to be moderately magnetized, and it is reasonable to assume that the distribution of electrons in- +jected downstream is Maxwellian both for the FS/RS (47), i.e., Qi(γe, t) = Ci (γe/γc,i)2 exp−γe/γc,i, +where γc,i = 1 +3ϵe,i +ei +ρic2 +mp +me is the typical Lorentz factor of the thermal distribution, ϵe,i is the frac- +tion of post-shock energy that goes into electrons for each region, ei and ρi is the energy and den- +sity of protons. The normalization constant Ci is obtained from the relevant mass conservation. +The instantaneous electron spectrum can be obtained by solving the continuity equation of elec- +23 + +trons in energy space (48). Integrating the synchrotron radiation power from the electron spec- +trum in Regions 2 and 3 and considering the effect of synchrotron self-absorption and the equal- +arrival-time surface, the radiation spectra and the light curves are then derived. With a starting +radius of R0 = 1015 cm for the collision and a set of parameters of L1 = 5.6 × 1052 erg s−1, +Lf = 4.5 × 1053 erg s−1, Γ1 = 100, Γ4 = 1000, qr = 1.3, qd = −0.5, σ4 = 0.1, ϵB,3 = 0.08, +ϵe,2 = 0.1, ϵe,3 = 0.07, Es = 5.8 × 1053 erg, Ef = 6.0 × 1054 erg. We get numerical optical +lightcurves in good agreement with the observed data. +4.2 +The external forward and reverse shock afterglow emission +In our modeling, it turns out that the shallow-declining part of the optical flare is hard to be +accounted for (see Fig. S6). A possibility is the emergence of the reverse shock, as observed in +for instance GRB 990123 (3, 5). Indeed, a reverse and forward shock scenario can reasonably +reproduce the optical and X-ray data. The magnetic field in the reverse shock region should +be stronger than that in the forward shock region by a factor of quite a few ×10 otherwise the +induced optical flash can not be brighter than the forward shock peak optical emission by a +factor of ∼ 1000 (7, 8). The following physical parameters are found to be able to reasonably +reproduce the afterglow data: the isotropic energy is Eiso = 1.0 × 1055 erg with a half open +jet angle θj = 0.025, the initial Lorentz factor is Γ = 800, the fraction of forward and reverse +shock energy given to the electrons is ϵe = 0.4, the fraction of the forward (reverse) shock +energy given to the magnetic field is ϵb,fs = 2.5 × 10−5 (ϵb,rs = 0.3), the number density of the +interstellar medium is n = 0.05 cm−3 and the power-law index for shock-accelerated electrons +is p = 2.26. Such a p is well consistent with that needed in reproducing the optical to X-ray +spectrum and lightcurves shown in Fig. S5 and Fig. S6, including Swift data analyzed in this +work and Liverpool telescope data from GCN (38,49). +24 + +5 +The prospect of detecting ultra-luminous optical/ultraviolet +flares at high redshifts with SVOM/VT +Optical/ultraviolet flares at high redshift will surfer from serious absorption. Following Moller +& Jakobsen (50), we estimate the absorption correction to be AB ∼ 5 mag (the received photons +are mainly caused by red leak of blue filter) and AR ∼ 1 mag for the sources at z ∼ 6, based +on the responses of SVOM/VT blue and red channels (i.e., B and R). For flares as luminous as +GRB 080319B or GRB 220101A, if taken place at z ∼ 6, then we would have MR ∼ 10.5 mag +and MB ∼ 15 mag. With the shortest exposure of 1s, SVOM/VT has a dynamic range of 9 − 18 +mag, which is sufficiently sensitive to catch the signals mentioned above. However, usually the +exposure time of SVOM/VT should be 10-100 seconds, for which the R filter may get saturated +but the B filter is not. We therefore conclude that SVOM/VT is a suitable equipment to detect +the extremely bright optical flares of GRBs at z ∼ 6. +Supplementary References +30. Poole, T. S. et al. Photometric calibration of the Swift ultraviolet/optical telescope. Mon. +Not. Roy. Astron. Soc. 383, 627–645 (2008). 0708.2259. +31. Maselli, A. et al. GRB 130427A: A Nearby Ordinary Monster. Science 343, 48–51 (2014). +1311.5254. +32. Wo´zniak, P. R. et al. Gamma-Ray Burst at the Extreme: “The Naked-Eye Burst” GRB +080319B. Astrophys. J. 691, 495–502 (2009). 0810.2481. +33. Markwardt, C. et al. The swift bat software guide. NASA/GSFC, Greenbelt, MD 6 (2007). +34. Evans, P. A. et al. An online repository of Swift/XRT light curves of γ-ray bursts. Astron. +Astrophys. 469, 379–385 (2007). 0704.0128. +25 + +35. Evans, P. et al. Methods and results of an automatic analysis of a complete sample of +swift-xrt observations of grbs. Monthly Notices of the Royal Astronomical Society 397, +1177–1201 (2009). +36. Goldstein, A., Cleveland, W. H. & Kocevski, D. Fermi gbm data tools: v1.1.0 (2021). URL +https://fermi.gsfc.nasa.gov/ssc/data/analysis/gbm. +37. Ashton, G. et al. Bilby: a user-friendly bayesian inference library for gravitational-wave +astronomy. The Astrophysical Journal Supplement Series 241, 27 (2019). +38. Perley, D. A. GRB 220101A: Liverpool telescope imaging of a high-redshift afterglow. +GRB Coordinates Network 31357, 1 (2022). +39. Andersen, M. I. et al. VLT identification of the optical afterglow of the gamma-ray burst +GRB 000131 at z=4.50. Astron. Astrophys. 364, L54–L61 (2000). astro-ph/0010322. +40. Th¨one, C. C. et al. GRB 100219A with X-shooter - abundances in a galaxy at z =4.7. Mon. +Not. Roy. Astron. Soc. 428, 3590–3606 (2013). 1206.2337. +41. Schlafly, E. F. & Finkbeiner, D. P. Measuring Reddening with Sloan Digital Sky Survey +Stellar Spectra and Recalibrating SFD. Astrophys. J. 737, 103 (2011). 1012.4804. +42. Planck Collaboration et al. Planck 2018 results. VI. Cosmological parameters. Astron. +Astrophys. 641, A6 (2020). 1807.06209. +43. Fan, Y. Z., Wei, D. M. & Wang, C. F. The very early afterglow powered by ultra-relativistic +mildly magnetized outflows. +Astron. Astrophys. +424, 477–484 (2004). +astro-ph/ +0405392. +26 + +44. Zhang, B. & Kobayashi, S. Gamma-Ray Burst Early Afterglows: Reverse Shock Emission +from an Arbitrarily Magnetized Ejecta. Astrophys. J. 628, 315–334 (2005). astro-ph/ +0404140. +45. Geng, J. J., Wu, X. F., Huang, Y. F., Li, L. & Dai, Z. G. Imprints of Electron-Positron +Winds on the Multiwavelength Afterglows of Gamma-ray Bursts. Astrophys. J. 825, 107 +(2016). 1605.01334. +46. Sironi, L., Keshet, U. & Lemoine, M. Relativistic Shocks: Particle Acceleration and Mag- +netization. Space Sci. Rev. 191, 519–544 (2015). 1506.02034. +47. Giannios, D. & Spitkovsky, A. Signatures of a Maxwellian component in shock-accelerated +electrons in GRBs. Mon. Not. Roy. Astron. Soc. 400, 330–336 (2009). 0905.1970. +48. Geng, J.-J., Huang, Y.-F., Wu, X.-F., Zhang, B. & Zong, H.-S. Low-energy Spectra of +Gamma-Ray Bursts from Cooling Electrons. Astrophys. J. Supp. 234, 3 (2018). 1709. +05899. +49. Perley, D. A. GRB 220101A: Additional Liverpool telescope photometry. GRB Coordi- +nates Network 31425, 1 (2022). +50. Møller, P. & Jakobsen, P. The Lyman continuum opacity at high redshifts - Through the +Lyman forest and beyond the Lyman valley. Astron. Astrophys. 228, 299–309 (1990). +51. Li, W. et al. +The Calibration of the Swift UVOT Optical Observations: A Recipe for +Photometry. Publ. Astron. Soc. Pac. 118, 37–61 (2006). astro-ph/0505504. +52. Kuin, N. P. M. & Rosen, S. R. The measurement errors in the Swift-UVOT and XMM-OM. +Mon. Not. Roy. Astron. Soc. 383, 383–386 (2008). 0709.1208. +27 + +Supplementary Tables +T-T0 +Exp +V +B +U +W1 +M2 +W2 +(s) +(s) +(AB) +(AB) +(AB) +(AB) +(AB) +(AB) +367.38 +19.46 +... +... +... +... +... +12.67±0.04 +391.76 +19.45 +12.01±0.04 +... +... +... +... +... +416.18 +19.45 +... +... +... +... +12.65±0.04 +... +440.84 +19.44 +... +... +... +12.60±0.04 +... +... +465.10 +19.44 +... +... +12.09±0.38a +... +... +... +490.09 +19.45 +... +11.28±0.40a +... +... +... +... +540.86 +19.44 +... +... +... +... +... +13.12±0.04 +565.28 +19.40 +12.41±0.04 +... +... +... +... +... +589.61 +19.46 +... +... +... +... +13.04±0.04 +... +614.82 +19.44 +... +... +... +13.06±0.04 +... +... +639.13 +19.46 +... +... +12.90±0.07a +... +... +... +663.96 +19.46 +... +12.69±0.04 +... +... +... +... +713.68 +19.45 +... +... +... +... +... +13.50±0.04 +737.97 +19.44 +12.65±0.04 +... +... +... +... +... +762.19 +19.44 +... +... +... +... +13.30±0.04 +... +786.88 +19.44 +... +... +... +13.31±0.04 +... +... +811.16 +19.44 +... +... +13.06±0.04 +... +... +... +835.91 +19.45 +... +12.98±0.04 +... +... +... +... +1136.89 +19.45 +... +... +13.50±0.04 +... +... +... +1161.73 +19.46 +... +13.39±0.04 +... +... +... +... +1213.17 +19.44 +... +... +... +... +... +14.10±0.04 +1237.51 +19.44 +13.34±0.05 +... +... +... +... +... +1261.91 +19.43 +... +... +... +... +13.92±0.05 +... +1286.75 +19.44 +... +... +... +13.90±0.04 +... +... +1311.01 +19.45 +... +... +13.73±0.04 +... +... +... +1335.68 +19.44 +... +13.60±0.04 +... +... +... +... +1385.28 +19.40 +... +... +... +... +... +14.20±0.04 +1409.66 +19.43 +13.53±0.05 +... +... +... +... +... +1433.98 +19.45 +... +... +... +... +14.09±0.05 +... +1458.64 +19.44 +... +... +... +14.08±0.04 +... +... +1482.87 +19.44 +... +... +13.82±0.04 +... +... +... +1508.07 +19.45 +... +13.77±0.04 +... +... +... +... +1557.68 +19.44 +... +... +... +... +... +14.39±0.04 +1581.95 +19.45 +13.69±0.05 +... +... +... +... +... +1606.20 +19.44 +... +... +... +... +14.25±0.05 +... +1630.88 +19.45 +... +... +... +14.24±0.04 +... +... +1655.08 +19.41 +... +... +13.98±0.04 +... +... +... +1679.93 +19.44 +... +13.87±0.04 +... +... +... +... +1729.85 +19.44 +... +... +... +... +... +14.42±0.04 +1754.32 +19.46 +13.76±0.05 +... +... +... +... +... +1779.91 +19.55 +... +... +... +... +14.28±0.05 +... +1804.55 +19.45 +... +... +... +14.30±0.04 +... +... +1828.75 +19.45 +... +... +14.08±0.04 +... +... +... +1853.48 +19.45 +... +13.90±0.04 +... +... +... +... +1903.00 +19.44 +... +... +... +... +... +14.52±0.05 +1927.21 +19.45 +13.81±0.05 +... +... +... +... +... +1951.62 +19.45 +... +... +... +... +14.38±0.05 +... +a. Taken from Maselli et al. (31). +Table S1: Early observations of GRB 130427A by Swift-UVOT. Galactic extinction AV = +0.055, AB = 0.071, AU = 0.087, AW1 = 0.118, AM2 = 0.163 and AW2 = 0.156 have been +applied. These data points have been plotted in the left panel of Fig. S2. +28 + +Epoch +T-T0 +Exp +V +B +U +W1 +M2 +W2 +Whitea +(s) +(s) +(AB) +(AB) +(AB) +(AB) +(AB) +(AB) +(AB) +1 +515.57 +19.44 +12.32±0.04 +11.51±0.35b +12.31±0.30b +12.82±0.04 +12.89±0.04 +13.05±0.04 +12.62±0.44 +2 +688.49 +19.45 +12.59±0.04 +12.79±0.04 +12.93±0.06 +13.18±0.04 +13.20±0.04 +13.45±0.04 +12.95±0.06 +3 +860.19 +19.45 +12.82±0.04 +13.00±0.04 +13.11±0.04 +13.41±0.04 +13.44±0.04 +13.75±0.04 +13.19±0.09 +4 +1187.86 +19.44 +13.28±0.05 +13.42±0.04 +13.58±0.04 +13.80±0.04 +13.85±0.04 +14.08±0.04 +13.60±0.06 +5 +1359.98 +19.45 +13.48±0.05 +13.63±0.04 +13.77±0.04 +13.98±0.04 +14.02±0.05 +14.18±0.04 +13.77±0.05 +6 +1532.32 +19.44 +13.65±0.05 +13.78±0.04 +13.86±0.04 +14.15±0.04 +14.19±0.05 +14.37±0.04 +13.93±0.05 +7 +1704.19 +19.44 +13.75±0.05 +13.88±0.04 +14.02±0.04 +14.28±0.04 +14.27±0.05 +14.41±0.04 +14.03±0.05 +8 +1877.72 +19.44 +13.80±0.05 +13.90±0.04c +14.08±0.04c +14.29±0.04c +14.32±0.05 +14.50±0.05 +14.07±0.05 +a. Interpolated White band AB magnitude of GRB 130427A. To derive the intrinsic count rate in a 5 arcsec aperture, galactic extinction +AWH = 0.0875 have been accounted. Fitting uncertainties and standard deviation of fitting residuals contribute to uncertainties have been +considered. +b. At early phase, there is an additional radiation component, hence these 2 data points are excluded from SED fitting algorithm. +c. These data points are results of extrapolation, hence they are excluded from SED fitting algorithm as well. +Table S2: White band emission interpolated by Swift-UVOT narrow bands. These data +(except for the last column) have been plotted in the right panel of Fig. S2. + +T-T0 +Exposure +˙Ntot,raw +ring +˙Nbkg,raw +ring +COItot(bkg) +LSS +˙Nring +˙Naper +(s) +(s) +(count/s) +(count/s) +(count/s) +(count/s) +GRB 220101A +165.95 +27.56 +83.08±1.86 +74.07±1.18 +1.033(1.029) +0.998 +11.23±2.75 +257.17±22.69 +209.76 +58.67 +79.73±1.25 +73.84±0.81 +1.031(1.029) +0.998 +7.28±1.85 +150.41±5.44 +GRB 130427Aa +515.57 +19.44 +139.77±3.05 +72.21±1.21 +1.056(1.028) +0.997 +80.66±4.00 +2030.98±831.79b +688.49 +19.45 +133.49±2.98 +72.09±1.21 +1.054(1.028) +0.996 +73.20±3.90 +1493.50±86.74 +860.19 +19.45 +120.21±2.83 +70.98±1.21 +1.048(1.028) +0.996 +58.19±3.69 +1204.35±101.10 +1187.86 +19.44 +102.65±2.30 +71.17±1.06 +1.041(1.028) +0.996 +36.99±3.00 +826.44±43.24 +1359.98 +19.45 +92.88±2.19 +71.45±1.06 +1.037(1.028) +0.996 +25.09±2.86 +703.53±29.41 +1532.32 +19.44 +96.50±2.23 +71.19±1.04 +1.038(1.028) +0.997 +29.68±2.90 +608.85±30.53 +1704.19 +19.44 +92.26±2.18 +71.03±1.05 +1.037(1.028) +0.997 +24.86±2.84 +553.88±26.34 +1877.72 +19.44 +89.60±2.15 +71.26±1.05 +1.036(1.028) +0.998 +21.47±2.81 +531.31±26.60 +RA +DEC +˙Ntot,raw +ring +˙Nbkg,raw +ring +COItot(bkg) +LSS +˙Nring +˙Naper +(J2000) +(J2000) +(count/s) +(count/s) +(count/s) +(count/s) +stars in GRB 220101A fieldc. +00:05:43.983 ++31:47:20.11 +80.83±0.76 +74.00±0.49 +1.032(1.029) +1.006 +8.58±1.14 +168.01±4.10 +00:05:33.844 ++31:42:10.45 +81.85±0.75 +74.00±0.48 +1.032(1.029) +1.014 +9.94±1.12 +208.39±6.07 +00:05:26.211 ++31:48:43.76 +83.42±1.06 +73.99±0.67 +1.033(1.029) +0.996 +11.73±1.56 +230.06±7.61 +a. ˙Naper is derived from SED. +b. This data is not fitted since U-band exposures were saturated around this exposure, hence it could be unreliable(see Fig. S2). +c. These data are measured with the first 150 second White band exposure in window timing mode. +Table S3: Photon count rates measured in aperture and halo ring methods in White band. +Sensitivity correction factors are 1.175 and 1.102 for GRB 220101A field and GRB 130427A +field, respectively. The factor from count rate to flux is 0.01327 mJy/(count/s) for white band. +These data points have been plotted in the left panel of Fig. S3. + +T-T0 +Exp +˙Ntot,raw +ring +˙Nbkg,raw +ring +COItot(bkg) +˙Nring +Magaper +a +˙Nb +aper +(s) +(s) +(count/s) +(count/s) +(count/s) +(AB) +(count/s) +080319B V band measurements with Wo´znika et al. as reference +180.60 +9.84 +74.86±1.19 +15.11±0.20 +1.030(1.006) +65.43±1.35 +10.06±0.02 +1349.02±24.85 +193.52 +9.84 +65.04±1.39 +14.95±0.20 +1.026(1.006) +54.63±1.56 +10.23±0.02 +1152.44±21.23 +206.25 +9.85 +54.36±1.50 +14.77±0.19 +1.021(1.006) +43.01±1.67 +10.44±0.02 +952.40±17.54 +218.97 +9.85 +52.19±1.52 +14.94±0.20 +1.020(1.006) +40.43±1.68 +10.55±0.02 +861.43±15.87 +231.70 +9.84 +47.80±1.53 +14.35±0.19 +1.019(1.006) +36.23±1.69 +10.72±0.02 +731.85±13.48 +244.43 +9.84 +42.70±1.53 +14.32±0.19 +1.017(1.006) +30.68±1.68 +10.88±0.02 +635.65±11.71 +257.45 +9.84 +39.72±1.52 +14.38±0.19 +1.015(1.006) +27.36±1.67 +11.00±0.02 +567.05±10.45 +270.38 +9.84 +38.63±1.51 +14.77±0.19 +1.015(1.006) +25.75±1.66 +11.12±0.02 +509.59±9.39 +283.30 +9.84 +36.73±1.50 +14.34±0.19 +1.014(1.006) +24.15±1.64 +11.23±0.02 +457.95±8.44 +296.23 +9.84 +35.39±1.49 +14.10±0.19 +1.014(1.005) +22.94±1.63 +11.36±0.02 +408.53±7.53 +309.15 +9.84 +31.18±1.45 +14.37±0.19 +1.012(1.006) +18.09±1.58 +11.52±0.02 +351.58±6.48 +322.08 +9.84 +28.19±1.41 +14.05±0.19 +1.011(1.005) +15.20±1.54 +11.61±0.02 +323.91±5.97 +334.80 +9.84 +28.53±1.41 +14.00±0.19 +1.011(1.005) +15.62±1.54 +11.72±0.02 +291.08±5.36 +360.05 +29.52 +24.58±0.78 +14.14±0.11 +1.010(1.005) +11.21±0.85 +11.91±0.02 +244.35±4.50 +395.50 +29.52 +23.49±0.77 +13.98±0.11 +1.009(1.005) +10.21±0.84 +12.20±0.02 +187.25±3.45 +431.04 +29.53 +22.41±0.76 +14.19±0.11 +1.009(1.005) +8.82±0.82 +12.40±0.02 +156.18±2.88 +466.29 +29.52 +21.29±0.74 +14.14±0.11 +1.008(1.005) +7.67±0.81 +12.54±0.02 +137.03±2.52 +502.44 +29.52 +20.02±0.73 +13.90±0.11 +1.008(1.005) +6.55±0.79 +12.74±0.02 +137.03±2.52 +537.68 +29.52 +19.13±0.72 +13.96±0.11 +1.007(1.005) +5.53±0.78 +12.80±0.02 +114.29±2.11 +080319B V band measurements with Page et al. as reference +189.92 +29.49 +67.15±0.79 +14.97±0.11 +1.026(1.006) +56.97±0.88 +10.07±0.26 +1335.42±319.79 +224.89 +39.36 +49.06±0.76 +14.58±0.10 +1.019(1.006) +37.38±0.84 +10.44±0.29 +949.77±253.68 +269.89 +49.21 +38.26±0.67 +14.48±0.09 +1.015(1.006) +25.67±0.74 +10.89±0.38 +627.51±219.62 +322.39 +54.13 +29.98±0.61 +14.06±0.08 +1.012(1.005) +17.13±0.67 +11.60±0.74 +326.30±222.40 +080319B V band measurements with HEASoft +322.39 +54.13 +29.98±0.61 +14.06±0.08 +1.012(1.005) +17.13±0.67 +11.75±0.02 +284.20±5.24c +357.39 +14.77 +23.47±1.09 +14.15±0.16 +1.009(1.005) +10.00±1.18 +11.91±0.04 +245.26±9.04 +372.39 +14.76 +24.23±1.10 +14.37±0.16 +1.009(1.006) +10.58±1.20 +12.10±0.04 +205.88±7.58 +387.39 +14.77 +24.79±1.11 +14.16±0.16 +1.010(1.005) +11.40±1.21 +12.14±0.04 +198.44±7.31 +402.40 +14.77 +22.70±1.08 +13.84±0.15 +1.009(1.005) +9.51±1.17 +12.21±0.04 +186.05±6.85 +417.40 +14.76 +22.44±1.07 +14.46±0.16 +1.009(1.006) +8.56±1.17 +12.35±0.04 +163.54±6.02 +432.40 +14.77 +22.51±1.07 +14.14±0.16 +1.009(1.005) +8.97±1.17 +12.44±0.04 +150.53±5.55 +447.40 +14.76 +20.96±1.05 +13.89±0.15 +1.008(1.005) +7.58±1.14 +12.49±0.04 +143.75±5.30 +462.40 +14.77 +21.64±1.06 +14.21±0.16 +1.008(1.005) +7.96±1.15 +12.51±0.04 +141.13±5.20 +477.40 +14.76 +21.09±1.05 +14.22±0.16 +1.008(1.005) +7.36±1.14 +12.60±0.04 +129.90±4.79 +492.39 +14.77 +21.44±1.06 +13.95±0.15 +1.008(1.005) +8.03±1.15 +12.62±0.04 +127.53±4.70 +507.39 +14.76 +19.91±1.03 +13.98±0.15 +1.008(1.005) +6.35±1.12 +12.77±0.04 +111.08±4.09 +522.39 +14.77 +19.06±1.01 +13.82±0.15 +1.007(1.005) +5.60±1.10 +12.79±0.04 +109.05±4.02 +537.40 +14.77 +19.94±1.03 +13.76±0.15 +1.008(1.005) +6.62±1.12 +12.88±0.04 +100.37±3.70 +552.40 +14.76 +19.08±1.01 +14.00±0.16 +1.007(1.005) +5.43±1.10 +12.89±0.04 +99.45±3.66 +567.30 +14.57 +18.62±1.01 +14.19±0.16 +1.007(1.005) +4.75±1.10 +12.94±0.04 +94.98±3.50d +719.60 +19.47 +16.60±0.84 +13.88±0.13 +1.006(1.005) +2.90±0.91 +13.35±0.04 +65.11±2.40d +1073.88 +196.67 +15.55±0.26 +14.04±0.04 +1.006(1.005) +1.61±0.28 +14.13±0.02 +31.74±0.58d +1273.77 +196.77 +15.34±0.25 +14.16±0.04 +1.006(1.005) +1.26±0.28 +14.52±0.02 +22.16±0.41d +a. Magnitudes are taken from Wo´znika et al. (32) and Page et al. (20). It is not necessary to take account for the very small(∼ 0.01) difference +between Vega magnitude and AB magnitude in V band. +b. Only values in the last sub table 080319B V band measurements with HEASoft are directly measured, others are all inferred values(i.e. +˙Nint). +c. This exposure is close to saturation, and Page et al. (20) derived a photometry result with readout streak method, which is consistent with +the aperture photometry result given by HEASoft. +d. These points are not plotted in Fig 3 and not used in fitting algorithm as well. +Table S4: Photon count rates measured in aperture and halo ring methods in v band. The +large scale structure correction factor and the sensitivity correction factor are 1.001 and 1.056, +respectively. The factor from count rate to flux is 0.25491 mJy/(count/s) for V band. These data +points used have been plotted in the right panel of Fig. 3. + +Filter +Tstart +Tend +Texp +Signala +Sky +Magb +second +second +second +count/s +count/s/pixel +(AB) +v +70.94 +80.61 +9.52 +6.137 ± 1.074 +0.0313 +16.12 ± 0.19 +white +91.96 +93.62 +1.64 +21.74 ± 4.74 +0.0145 +17.75 ± 0.24 +white +93.64 +97.62 +3.93 +50.37 ± 4.79 +0.0148 +16.83 ± 0.10 +white +97.63 +101.62 +3.94 +123.7 ± 8.9 +0.0150 +15.86 ± 0.08 +white +101.63 +105.63 +3.94 +360.4 ± 30.3 +0.0147 +14.70 ± 0.09 +white +105.64 +109.62 +3.93 +(634.4 ± 171.8)c +0.0146 +(14.08 ± 0.29) +white +109.63 +113.63 +3.94 +(765.0 ± 175.6)c +0.0147 +(13.88 ± 0.25) +white +113.64 +117.62 +3.93 +(1033 ± 184.4)c +0.0150 +(13.56 ± 0.19) +white +117.63 +121.63 +3.94 +(544.8 ± 168.7)c +0.0146 +(14.25 ± 0.34) +white +121.64 +125.62 +3.93 +(570.2 ± 170.7)c +0.0147 +(14.20 ± 0.33) +white +125.63 +129.62 +3.94 +(374.0 ± 165.0)c +0.0148 +(14.66 ± 0.48) +white +129.64 +133.62 +3.93 +(411.7 ± 167.3)c +0.0149 +(14.55 ± 0.44) +white +133.63 +137.62 +3.94 +(383.1 ± 165.9)c +0.0149 +(14.63 ± 0.47) +white +137.63 +141.62 +3.93 +(333.5 ± 163.1)c +0.0146 +(14.78 ± 0.53) +white +141.63 +145.62 +3.94 +(381.1 ± 163.3)c +0.0145 +(14.64 ± 0.47) +white +145.63 +149.63 +3.94 +(346.4 ± 163.6)c +0.0147 +(14.74 ± 0.51) +white +149.64 +153.62 +3.93 +330.2 ± 26.4 +0.0148 +14.79 ± 0.09 +white +153.63 +157.63 +3.94 +328.3 ± 26.1 +0.0147 +14.80 ± 0.09 +white +157.64 +161.62 +3.93 +303.1 ± 23.3 +0.0149 +14.89 ± 0.08 +white +161.63 +165.63 +3.94 +291.6 ± 22.0 +0.0145 +14.93 ± 0.08 +white +165.64 +169.62 +3.93 +257.8 ± 18.7 +0.0147 +15.06 ± 0.08 +white +169.63 +173.62 +3.94 +224.0 ± 15.8 +0.0149 +15.21 ± 0.08 +white +173.63 +177.62 +3.93 +215.8 ± 15.2 +0.0146 +15.26 ± 0.08 +white +177.63 +181.62 +3.94 +202.9 ± 14.2 +0.0148 +15.32 ± 0.08 +white +181.63 +185.62 +3.93 +208.6 ± 14.6 +0.0149 +15.29 ± 0.08 +white +185.63 +189.62 +3.94 +186.0 ± 12.9 +0.0147 +15.42 ± 0.08 +white +189.63 +193.63 +3.94 +192.9 ± 13.4 +0.0146 +15.38 ± 0.08 +white +193.64 +197.62 +3.93 +179.6 ± 12.5 +0.0147 +15.45 ± 0.08 +white +197.63 +201.62 +3.94 +164.6 ± 11.5 +0.0147 +15.55 ± 0.08 +white +201.64 +205.62 +3.93 +187.3 ± 13.1 +0.0149 +15.41 ± 0.08 +white +205.63 +209.62 +3.94 +146.1 ± 10.3 +0.0145 +15.68 ± 0.08 +white +209.64 +213.62 +3.93 +158.1 ± 11.1 +0.0148 +15.59 ± 0.08 +white +213.63 +217.62 +3.94 +147.3 ± 10.4 +0.0149 +15.67 ± 0.08 +white +217.63 +221.63 +3.94 +134.1 ± 9.6 +0.0149 +15.77 ± 0.08 +white +221.64 +225.62 +3.93 +135.8 ± 9.7 +0.0147 +15.76 ± 0.08 +white +225.63 +229.63 +3.94 +117.3 ± 8.5 +0.0145 +15.92 ± 0.08 +white +229.64 +233.62 +3.93 +118.3 ± 8.6 +0.0146 +15.91 ± 0.08 +white +233.63 +237.63 +3.94 +114.9 ± 8.4 +0.0143 +15.94 ± 0.08 +white +237.64 +239.56 +1.90 +102.2 ± 10.9 +0.0145 +16.07 ± 0.12 +H/Ld +u +3627.5 +3827.3 +196.6 +0.0194 ± 0.1266 +0.0384 +> 20.74 +b +3832.6 +4032.4 +196.6 +0.1181 ± 0.1624 +0.0635 +> 19.95 +white +4037.3 +4237.1 +196.6 +0.4286 ± 0.1266 +0.1187 +21.90 ± 0.32 +w2 +4242.9 +4442.6 +196.6 +−0.3402 ± 0.0613 +0.0012 +> 21.54 +v +4447.6 +4647.4 +196.6 +0.4469 ± 0.0617 +0.0151 +18.66 ± 0.15 +m2 +4652.4 +4852.1 +196.6 +0.0015 ± 0.0168 +0.0006 +> 21.25 +w1 +4857.5 +5057.3 +196.6 +0.0099 ± 0.0308 +0.0019 +> 21.33 +u +5062.2 +5201.4 +137.0 +0.0659 ± 0.0730 +0.0080 +> 20.90 +white +10266 +11051 +765 +0.3408 ± 0.0502 +0.0602 +22.15 ± 0.16 +v +21543 +22361 +798 +0.1786 ± 0.0280 +0.0141 +19.66 ± 0.17 +white +27727 +28549 +802 +0.1635 ± 0.0396 +0.0594 +22.95 ± 0.26 +v +39724 +40112 +378 +0.1419 ± 0.0340 +0.0138 +19.91 ± 0.26 +white +44845 +46039 +1137 +0.1662 ± 0.0492 +0.0601 +22.93 ± 0.36 +white +50868 +61521 +1528 +0.1369 ± 0.0530 +0.1073 +23.13 ± 0.42 +white +66833 +85013 +5485 +0.0202 ± 0.0378 +0.1000 +> 23.33 +a. Signal photon count rates have been corrected for coincidence losses (30,51) and long-term sensitivity correction. +b. Magnitudes are based on Swift/UVOT zeropoints (30), errors are adjusted by a binomial distribution (52), limiting magnitudes are 3σ upper +limits. These values have not been corrected for the Galactic extinctions of E(B − V ) = 0.0483 (41). +c. These data have been analyzed in ring apertures as introduced in supplementary materials. +d. Images taken before are in high resolution, our photometry is in 5′′ aperture, after are in low resolution, our photometry is in 3′′ aperture. +Table S5: Photometry for Swift UVOT observations of GRB 220101A. + +GRB +z +Band +Mpeak +AV,MW +β +Aλa +DMc +Mabs +Ref. +AB +fν ∝ ν−β +AB +990123 +1.600 +White to V +MV = 8.86 ± 0.02 +0.04 +0.67 +0.04 +45.42 +-36.60 +(3) +050904 +6.295 +White to 9500 ˚A +M= 12.13 ± 0.24b +0.16 +1.0 +1.25 +48.97 +-38.09 +(23) +080319B +0.937 +White to V +MV = 5.34 ± 0.04 +0.03 +0.5 +0.03 +43.99 +-38.68 +(9) +220101A +4.618 +White to V +Mwh = 13.56 ± 0.19 +0.15 +0.7 +4.78 +48.19 +-39.41 +This work +a. Aλ= Aλ,MW+Aλ,host+Aλ,IGM is derived from photometric SED fit. +b. Converted from fλ=9500 ˚A = 17 ± 4 × 10−15 erg cm−2 s−1 ˚A−1. +c. Absolute Magnitude at the peak Mpeak,abs = Mpeak − DM − Aλ. +Table S6: Properties of extremely bright GRB flares at the peak. + +Supplementary Figures +(a) GRB 130427A WH +(b) GRB 220101A WH +deep exposure +(c) GRB 220101A WH +deep exposure mask map +(d) GRB 220101A WH +T-T0=103.63s, M~14.7 +(e) GRB 220101A WH +T-T0=115.63s, M~13.56 +(f) GRB 220101A WH +T-T0=167.63s, M~15.06 +Figure S1: Swift/UVOT white (WH) band images demonstrating the halo ring photometry +method. Panel (a) is the White band image of GRB 130427A, where the solid circle represents +the standard aperture of UVOT with a radius of 5 arcsec. The dotted square region strongly +suffered from coincidence loss with a typical side length of ∼ 20 arcsec. Dashed annulus with +an inner radius of 15 arcsec and an out radius of 25 arcsec is the halo ring region defined in +this work, for which the ˙Nring is derived. Panel (b) shows the deep exposure of GRB 220101A +field in White band, which reveals 2 faint sources in the halo ring region, hence we masked the +annulus region from 95◦ to 150◦, as shown in panel (c). In addition, images of panel (b) and (c) +have a pixel scale of 1.004 arcsec/pixel instead of 0.502 arcsec/pixel for other 4 images. Panel +(d), (e) and (f) show some images around the peak time of GRB 220101A. We measured count +rates in unmasked annulus region and corrected it to the whole annulus region. + +V-0.8 +M2+0.8 +B-0.4 +W2+1.2 +U +B-0.4 Maselli +W1+0.4 +U Maselli +3×102 +5×102 +1×103 +2×103 +3×103 +10 +11 +12 +13 +14 +15 +16 +t-t0[s] +AB Magnitude +Epoch 1 -0.75 +Epoch 5 +0.25 +Epoch 2 -0.5 +Epoch 6 +0.5 +Epoch 3 -0.25 +Epoch 7 +0.75 +Epoch 4 +Epoch 8 +1. +3×1014 +5×1014 +1×1015 +2×1015 +3×1015 +10 +11 +12 +13 +14 +15 +16 +Frequency[Hz] +AB Magnitude +a +b +Figure S2: The UVOT lightcurves (left) as well as the SEDs (right) of GRB 130427A. In +the left panel, the vertical grey regions mark the observation periods of the White filter. Note +that the second U-band data is saturated, which was however a detection point in Maselli et +al. (31) if only event data in the last 6s was measured, hence the filled and the empty green +triangles coincide. The shaded colorful regions across photometry points are our interpolation +results of light curve. The right panel presents the optical to ultraviolet SEDs at the White +band observation times constructed with the extrapolated UVOT narrow band data. A single +power-law spectrum well reproduces the data, as anticipated in the fireball external forward +shock afterglow model, with which a reliable evaluation of the White band emission is yielded, +as reported in the last column of Table S2. + +0 +10 +20 +30 +40 +50 +60 +70 +80 +Nring[count/s] +0 +500 +1000 +1500 +2000 +2500 +3000 +Naper[count/s] +a +Fit Func: y=k*x +k=22.22±0.84 +2/d.o.f=0.90 +GRB130427A extracted +GRB220101A +stars in GRB 220101A field +GRB130427A Maselli +0 +10 +20 +30 +40 +50 +60 +70 +80 +Nring[count/s] +0 +500 +1000 +1500 +2000 +2500 +3000 +Naper[count/s] +b +Fit Func: y=k*x +k=20.62±0.43 +2/d.o.f=0.37 +GRB080319B Wozniak +GRB080319B Page +GRB080319B +Figure S3: Photon count rates in 5′′ aperture ˙Naper (directly measured or inferred from +the intrinsic value ˙Nint) and 15 − 25′′ ring ˙Nring (coincidence loss corrected) for some +Swift/UVOT white and V band measurements. The left panel is for the White band. The dark +green upward triangles represent the two unsaturated measurements of GRB 220101A in the +tail phase of the flash. The filled squares are for three bright stars in the filed of GRB 220101A. +The light green downward empty triangle represents inferred ˙Naper with the photometry result +of GRB 130427A derived with readout streak method (31). As for orange points, the vertical +coordinate represents the White-band emission of GRB 130427A inferred from measurements +in other UVOT bands (see Fig. S2), while the horizontal coordinate is the ˙Nring (see Fig. S1 for +definition). Black squares are 3 unsaturated stars in GRB 220101A field. The right panel is for +the V band. Empty dark green triangles are unsaturated measurements of GRB 080319B with +HEASoft and empty light green squares are photometry results of GRB 080319B derived with +the readout streak method (20). As for orange points, the vertical coordinate represents the pho- +tometry result of GRB 080319B observed with RAPTOR-T (32) when the UVOT observations +were ongoing, while the horizontal axis represents ˙Nring in the corresponding UVOT V-band +image. The linear fit is just for filled points in both panels, and the correlation coefficients of +filled points are 0.990 and 0.998 for the left and right panel, respectively. Black dashed lines +represent the saturation count rate (coincidence loss corrected, ∼ 372 count s−1) of UVOT. + +10-1 +100 +101 +102 +103 +102 +103 +104 +105 +UVOT count rate (count/s) +Time since trigger (s) +a +white +v + 0 + 10 + 20 + 30 + 40 + 50 + 60 + 70 + 80 + 90 + 2000 + 3000 + 4000 + 5000 + 6000 + 7000 + 8000 +Effective area (cm2) +Wavelength (Å) +b +Lyα +Lyman limit +V +B +U +W1 +M2 +W2 +White +Figure S4: The similarity of Swift/UVOT White and V band observations of GRB 220101A. +The left panel shows that the photon count rates in White band are almost the same as that in +V band. This is because the photons with wavelengths below the Lyman limit (in the observer +frame, it is 5124 ˚A; see the right panel) are almost fully absorbed, and the photons near the +Lyman α may also suffer from strong absorption (see Fig. S5 for this effect). Therefore the +collected photons are mainly within the V band. + + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 +1014 +1015 +1016 +1017 +1018 +10−6 +10−5 +10−4 +Magnitude (AB) +Flux density (Jy) +Frequency (Hz) +GRB 000131 −4 Mag +GRB 100219A −3 Mag +GRB 220101A + +Figure S5: Optical to X-ray SED of GRB 220101A. Swift XRT, UVOT and g, r, i, z observa- +tions of Liverpool telescope in the time interval of t ∼ 0.62−0.68 day after the burst (38). Such +a set of ground-based telescope observation data are chosen because they are almost simulta- +neous with one UVOT White exposure. Neither the X-ray nor the optical emission displays a +flare. Therefore, we construct the optical SED with the data collected at t ∼ 0.625 day. We find +that the absorption correction is AWh = 4.78 mag for intrinsic optical to X-ray spectrum with +index βoX = 0.65, it is well consistent with X-ray spectrum βX = 0.63 ± 0.09. The central fre- +quency of the White band observation has been taken as the same as that of the V band because +of the serious absorption of the bluer photons, as demonstrated in Fig. S4. The optical SEDs +of other two GRBs 000131 (39) and 100219A (40) at similar redshifts (z = 4.500 and 4.667, +respectively) are also shown for comparison. + + 10 + 15 + 20 + 25 + 30 +102 +103 +104 +105 +106 +10−9 +10−8 +10−7 +10−6 +10−5 +10−4 +10−3 +10−2 +10−1 +100 +Magnitude (AB) +Flux density (Jy) +Time since trigger (s) +z−6mag +i−4mag +r−2mag +Swift white +Swift v +g+2mag +BAT 10 keV +XRT 10 keV +Figure S6: Fit to the multi-band afterglow lightcurves of GRB 220101A. The Swift XRT +and UVOT data are analyzed in this work, and the other optical data are adopted from Liv- +erpool telescope (38, 49). The total extinction corrections, including Galactic extinction and +interstellar-medium extinction are AWh = 4.78, Av = 1.88, Ag = 3.51, Ar = 1.46, Ai = 0.24 +and Az = 0.10, respectively. The dashed and dash-dotted lines represents forward and reverse +shock emission arising from the weak/slow and main/fast outflow collision. Solid and dotted +lines are the regular external forward and reverse shock emission of the outflow. In our calcula- +tion, the main/fast outflow was launched 92 seconds after the BAT trigger. Note that the X-ray +emission at t ≤ 170 sec was attributed to the low energy part of the prompt emission and has +not been addressed in our modeling. + + 0 + 20 + 40 + 60 + 80 + 100 + 3000 + 4000 + 5000 + 6000 + 7000 + 8000 + 9000 + 10000 11000 +Response (%) +Wavelength (Å) +Lyα +Lyβ +Lyγ +Lyman limit +B +B +R +R +Figure S7: The response of the SVOM/VT and the Lyman absorption of the high redshift +(∼ 6) event. The optical/ultraviolet flash will surfer from strong absorption by intergalactic +medium. Following Moller & Jakobsen (50), we find that AB ∼ 5 mag (the received photons +are mainly caused by red leak of blue filter) and AR ∼ 1 mag for a source at z = 6, based on +the responses of SVOM/VT blue and red channels (i.e., B and R). If the initial flash is as bright +as that detected in GRB 080319B and GRB 220101A, the absorbed one would still be caught +by SVOM/VT with a dynamic range of 9 − 18 mag for the shortest exposure of 1s. Therefore +SVOM/VT is a suitable equipment to detected extremely bright optical flares of GRBs at z ∼ 6. + diff --git a/89E0T4oBgHgl3EQffwAL/content/tmp_files/load_file.txt b/89E0T4oBgHgl3EQffwAL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b29dbe0c776e2aef9545b42b7eabfc492716f69c --- /dev/null +++ b/89E0T4oBgHgl3EQffwAL/content/tmp_files/load_file.txt @@ -0,0 +1,3104 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf,len=3103 +page_content='The detection of an extraordinarily-luminous high-redshift optical/ultraviolet flare by Swift/UVOT Zhi-Ping Jin1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='3†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Hao Zhou1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='3†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Yun Wang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Jin-Jun Geng1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Stefano Covino4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Xue-Feng Wu1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Xiang Li1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Yi-Zhong Fan1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='3∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Da-Ming Wei1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' and Jian-Yan Wei5 1Purple Mountain Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Nanjing 210023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' China 2Key Laboratory of Dark Matter and Space Astronomy of Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Nanjing 210023,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' China 3School of Astronomy and Space Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' University of Science and Technology of China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Hefei 230026,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' China 4INAF/Brera Astronomical Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' via Bianchi 46,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' I-23807 Merate (LC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Italy 5National Astronomical Observatories,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Beijing 100049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' China ∗To whom correspondence should be addressed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' E-mail: yzfan@pmo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' †These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Hyper-luminous optical/ultraviolet flares have been detected in Gamma-ray Bursts and the record was held by naked eye event GRB 080319B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Such flares are widely attributed to internal shock or external reverse shock radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' With a new dedicated method developed to derive reliable photometry from saturated images of Swift/UVOT, here we carry out time-resolved analysis of the initial White band 150 s exposure of GRB 220101A, a burst at the red- shift of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='618, and report a rapidly-evolving optical/ultraviolet flare with an unprecedented-high absolute AB magnitude ∼ −39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' At variance with GRB 080319B, the temporal behavior of this new flare does not trace the gamma- ray activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Rather than either internal shocks or reverse shock, this opti- 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='02407v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='HE] 6 Jan 2023 cal/ultraviolet monster is most likely from the refreshed shocks induced by the catching-up of the late-ejected extremely-energetic material with the earlier- launched decelerating outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' We detect the first ultraviolet/optical flare with an absolute AB magnitude brighter than −39 and reveal the efficient process to power such transients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Gamma-ray bursts are widely believed to originate from the internal energy dissipation of a highly relativistic and narrowly collimated outflow that was launched by a nascent stellar mass black hole or magnetized neutron star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Shortly after the onset of prompt emission of GRBs, there could come very bright optical/ultraviolet flashes arising from either the internal shocks in specific conditions or the external reverse shock radiation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' An apparent ∼ 9th mag optical radiation was detected in GRB 990123 at a redshift of z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='62 (2, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Its rapid rise and the quick decline are consistent with the reverse shock radiation model (4–6), and the late more-detailed afterglow modeling revealed that the reverse shock region should be significantly more magnetized than the forward shock region (7,8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' A long-holding record was set by GRB 080319B, a burst at a redshift of z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='937.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Its peak visual magnitude reaches 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='3 (corresponding to an absolute AB magnitude of −38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='7), which is so bright that an observer in a dark location could have seen it with the naked eyes (9)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The correlated temporal behaviors of the prompt gamma-ray emission and the optical radiation are in favor of the internal shock process (10,11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' In the past decade, no similar or even comparable events have been reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' GRB 220101A was discovered simultaneously by Swift Burst Alert Telescope (BAT) (12), the Fermi satellite (13) and the AGILE satellite (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Before the so-called finding chart exposure ranging from 90 to 240 seconds with the White filter (12), UVOT observed the target in V band for 9 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The estimated average magnitude in the White band for an exposure of ∼ 150 s is ∼ 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='7th Vega mag (12,15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The redshift was measured to be z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='618 and in the spectrum a broad absorption feature, which results from the Lyman alpha absorption (16, 17), is evident 2 centered at ∼ 6820 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The corresponding isotropic equivalent gamma-ray energy is ∼ 4 × 1054 erg and the peak luminosity is ∼ 9 × 1053 erg s−1, both are in the rank of the brightest ones among current GRBs (18,19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' After the redshift correction, the observed optical photons were intrinsically in the ultraviolet bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Therefore, all the emission detected by Swift suffered from serious absorption (in the observer’s frame, the V band absorption is about 2 mag stronger than that in the I-band (16)) and thus the intrinsic emission would be much brighter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' This is in particular the case for the White filter because of its large effective area in the blue part (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=', U, UVW1, UVM2 and UVW2) and the Lyman alpha/intergalactic medium (IGM) absorption would be very strong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' We concentrate on the first ∼ 150 s exposure with White filter in the event mode (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' photon counting mode) that can be efficiently divided into short bins according to the signal-to-noise ratio (SNR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Our time-resolved analysis reveals that the measurements in the time range of ∼ 106 − 150 s after the BAT trigger suffered from strong saturation, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 1, Table S5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The absence of clear signal of the read-out streaks in the raw data, indicating a moderate saturation, however hampers a correction following procedures proposed in the literature (20,21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Therefore we propose a new method to correct the saturation effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The basic idea is that though the pile up at the source site is so serious that can not be reliably corrected, the surrounding but relatively “separated” pixels are possibly unaffected by saturation and therefore the enhancement of the counts should be correlated with the intrinsic count rate of the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' To clarify whether it is the case, we need some data with known magnitudes as well as the count rates in external annuli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' For the unsaturated data with relatively low ring count rate, we simply take UVOT/White measurements of GRB 220101A at 150−240 seconds after the burst trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' For the moderate saturation that is of our great interest, we take the UVOT measurements of GRB 130427A in the time interval of 500 − 2000 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Though the moderately saturated White band emission of GRB 130427A can not be directly measured, we infer them with the UVOT emission in other bands since the spectrum can be 3 well fitted by a single power-law, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' With these two sets of data, we do find a tight correlation between photon count rate in 5′′ aperture ( ˙Naper, directly measured if unsaturated, or inferred from the “intrinsic” count rate ˙Nint measured in other ways) and in the 15′′ − 25′′ ring ( ˙Nring, directly measured in UVOT images), which reads ˙Naper = (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='84) ˙Nring for ˙Nring ≤ 80 s−1 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' S3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The correlation efficient for such an empirical relation is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='99 (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The other essential correction is on the absorption of the ultraviolet photons at high redshift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' In the analysis we correct such a factor, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=', 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='10 mag in the White band, with the wide band energy spectrum and further check it with the other two GRBs at rather similar redshifts (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 1 we show the lightcurves of the prompt gamma-ray emission and the very early optical emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The first White exposure with a duration of 150 s was in the events mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' In our approach, a bin size of 4s is adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' In principle, a narrower bin size is helpful in revealing the peak or structure of the flash, but a reasonably wide bin is necessary for a high SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The optical/ultraviolet flash lightcurve is relatively smooth and there is no evidence for tracing the temporal behavior of prompt gamma-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' This is very different from the case of GRB 080319B, where the naked-eye optical flash shows strong variabilities and the optical lightcurve resembles that of the gamma-rays (see the insert of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 1), indicating a new origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' We have also constructed the “prompt” optical, X-ray and gamma-ray SED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 2 we show three representative time intervals of the first UVOT White band exposure, including the very beginning, the peak, and the final shallow decline phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' In the rise and the quick decline phases, the extrapolation of the high energy radiation spectrum into the optical is well below the White band measurements, which again points towards different physical origins of the optical and high energy radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' While in the t−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='3±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='3 shallow decline phase, the optical to X-ray emission are consistent with being a single power-law, which may be dominated by the external reverse shock radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 3 we present the absolute AB magnitudes of the 4 very early optical emission of GRB 220101A and the other three remarkable events, including GRB 990123 (3), GRB 050904 (23) and GRB 080319B (9), distinguished by the extremely bright optical emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' After the proper saturation, absorption and cosmological corrections, it turns out that GRB 220101A sets a new record.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The prompt ultraviolet to X-ray spectrum at the optical/ultraviolet emission peak time is softer than ν−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='3 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' If this soft spectrum could extend to the optical band in the frame of the burst, GRB 220101A would be so far the unique source with an absolute AB magnitude brighter than −40 in the visible band (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Note that the peak optical emission of GRB 220101A could be even stronger than presented here since our current fluxes are the average of the radiation in each 4s bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' As already mentioned before, for GRB 080319B, the internal shock model is favored by the similar temporal behaviors of the prompt gamma-ray and optical radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' While for GRB 990123, the external reverse shock model has been widely accepted to account for the optical flash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The optical/ultraviolet flare detected in GRB 220101A, however, should have a differ- ent physical origin for the following facts: (i) In contrast to GRB 080319B, the optical flash lightcurve of GRB 220101A does not trace the variability of the prompt gamma-rays (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 1), requiring different radiation processes/sites of these two components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' (ii) The t−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='3-like de- cline of the optical/ultraviolet flare of GRB 220101A may be due to the reverse shock emission, but the ∼ t20 increase is much quicker than that of GRB 990123 and hence strongly in tension with the standard external reverse/forward shock emission model (5, 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Here we present a re- freshed shock model for the brightest optical/ultraviolet spike of GRB 220101A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Looking at the gamma-ray lightcurve, the main burst phase consisting of two giant gamma-ray spikes appears at ∼ 90 s after the BAT trigger, and the earlier emission was much weaker (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=', the time- averaged luminosity is ∼ 1052 erg s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' As indicated by the bulk Lorentz factor−luminosity correlation (24,25) of Γ ∝ L0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='3 γ , the weak/slow GRB outflow component launched at the early times is expected to have a Γ ∼ 102 and the surrounding interstellar medium further decelerates 5 the outflow to a Lorentz factor of ΓW, while the outflow component yielding the most luminous part of GRB 220101A likely has a Lorentz factor of ΓM ∼ 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The first giant spike comes from the energy release of the main outflow, either through the internal shocks or the magnetic re-connection within it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Soon the main outflow would catch up with the decelerating weak part at a time of ∼ Γ2 WδtWM/Γ2 M ∼ O(10) s, which explains the second gamma-ray spike and the delayed onset of the optical/ultraviolet flare, where δtWM ∼ 100 s is the delay of the onset of the main part with respect to that of the weak part (started at ∼ 60 s before the trigger, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The collision of the late/fast material shell(s) with the early/decelerating material will generate strong refreshed shocks and then produce energetic emissions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Following the treatments pre- sented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='1 of the Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' (26), it is straightforward to show that for the internal shocks taking place at ∼ 2Γ2 WcδtWM/(1 + z) ∼ 1016 cm (ΓW/102)2(δtWM/102 s), the typical synchrotron radiation frequency is indeed within the optical/ultraviolet bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The bulk Lorentz factor of the merged shells can be approximated to be ¯Γ ≈ � [MWΓW + MMΓM]/[MW/ΓW + MM/ΓM] and the Lorentz factor of the internal shocks can be estimated as Γsh ≈ ΓM/¯Γ + ¯Γ/ΓM, where MW and MM are the rest masses of the ejecta powering earlier weak gamma-ray emission and the main outburst, respectively (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Indeed, for GRB 220101A-like burst, we have the outflow luminosity of Lm ∼ 1053 − 1054 erg s−1, with the fractions of the shock energy given to the magnetic fields (electrons) ϵB ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='1 (ϵe ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='3), ¯Γ ∼ several × 100 and Γsh ∼ a few, it is natural to have an optical/ultraviolet flux (26) of ∼ 1 Jy even for a redshift as high as ∼ 5 (a numerical example is presented in (22) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Note that the very energetic prompt emission appearing at ∼ T0 +90 s, which partly overlap with the optical/ultraviolet flash, after the BAT trigger should also effectively cool the electrons accelerated in the collision discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Such a process would produce GeV emission, which is expected to last longer than the overlapping phase of the prompt MeV emission and ultraviolet/optical flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Indeed, at t ∼ 100 − 150 s after the BAT trigger, GeV emission was 6 detected from GRB 220101A (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Though the hyper-luminous very early optical/ultraviolet emission are not common, we suggest that the bursts with prompt emission resembling GRB 220101A (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=', the much more energetic outflow is well separated from the early ejecta) are good candidates for hosting the extraordinarily bright flares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The problem is how to catch such signals promptly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Small tele- scopes with a large field of view should be very helpful and the I/R-band observation of these telescopes can catch the monsters in a wide range of redshifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Anyhow, such observations are limited by the weather, the time (day or night) and the burst site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The space telescopes like Swift/UVOT and SVOM/VT (29) may play an important role for the high redshift events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Since the optical/ultraviolet flash of GRB 220101A was observed by Swift/UVOT, below we focus on the upcoming 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='4m SVOM/VT with two channels, including the blue (400 − 650 nm) and the red (650 − 1000 nm) bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' For the shortest exposure time of 1s, the saturation limit is about 9th magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Given its higher sensitivity in comparison to Swift/UVOT V filter, the seriously absorbed “ultraviolet” emission of GRB 220101A/GRB 080319B-like extra-luminous events, even taking place at the even higher redshift (say, z ∼ 6), can still be caught by the blue channel of SVOM/VT though the red channel might be saturated (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Dedicated observation strategies are needed to optimize the potential of the discoveries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='9 50 0 50 100 150 200 250 0 200 400 600 800 1000 1200 BAT count rate (count/s/det) UVOT count rate (count/s) Time since trigger (s) BAT V White 0 1 2 3 4 0 20 40 60 800.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='0E0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='0E4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='0E5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='5E5 GRB 080319B V Figure 1: Photon count rates of the prompt gamma-ray (Swift/BAT) and optical (Swift/UVOT V and White band) emission of GRB 220101A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The prompt gamma-ray lightcurve is highly variable, while the prompt optical emission lightcurve is relatively smooth and does not trace that of gamma-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The red filled circles are from the aperture measurement while the open circles are obtained with the new method developed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The energetic optical/ultraviolet flash just overlaps with the late part of the giant outburst phase of the prompt gamma-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The prompt gamma-ray and optical lightcurves of GRB 080319 are inserted for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 8 10 3 10 2 10 1 100 101 102 103 Energy (keV) 10 2 10 1 100 101 102 103 104 Flux density (keV/cm2/s/keV) Unabsorbed CPL model (91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='96 - 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='62 s) Unabsorbed CPL model (113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='64 - 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='62 s) Unabsorbed PL model (173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='63 - 239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='56 s) Mape AB = 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='24 (91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='96 - 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='62 s) Mring AB = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='19 (113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='64 - 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='62 s) Mape AB = 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='04 (173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='63 - 239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='56 s) Figure 2: The “prompt” optical to γ-ray SEDs of GRB 220101A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The data in blue (red) are collected in the very beginning (peak) of the UVOT/White band emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The optical emission in both cases are well above the extrapolation of the high energy spectrum, suggesting an origin different from the prompt X-rays and gamma-rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' While in the time interval of t ∼ 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='6 − 239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='6 seconds, the extrapolation of the X-ray and gamma-ray spectrum into the optical is in agreement with the UVOT data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 9 40 39 38 37 36 35 34 33 32 31 30 29 28 101 102 103 Absolute Magnitude (AB) t′ (second) GRB 220101A GRB 080319B GRB 050904 GRB 990123 39 38 37 36 1015 1016 ν′ (Hz) Figure 3: The ultraviolet/optical flare of GRB 220101A (red) in comparison to that of GRB 990123 (green) (3), GRB 050904 (pink) (23) and GRB 080319B (blue) (9) in rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The White band emission of GRB 220101A has been corrected for total extinction of Aλ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='1 mag, including the tiny softening of E(B − V ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='0483 mag in the Milky Way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The absolute AB magnitude of GRB 220101A exceeds that of GRB 080319B, the so-called naked burst, rendering it the most energetic optical/ultraviolet flare recorded so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' References 1.' metadata={'source': 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Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 755, L6 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Wei, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Yan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Fan, Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 636, L69 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Piran, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 314, 575 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Arimoto, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Scotton, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Longo, Fermi-LAT Collaboration, GRB Coordinates Network 31350, 1 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Yu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Gonzalez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Wei, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='-N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Zhang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Cordier, Chin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 44, 269 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Acknowledgments Funding: This work was supported in part by NSFC under grants of No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 12225305, 11921003 and 11933010, the China Manned Space Project (NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='CMS-CSST-2021-A13), Major Science and Technology Project of Qinghai Province (2019-ZJ-A10), Key Research Program of Frontier Sciences (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' QYZDJ-SSW-SYS024).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' SC has been supported by ASI grant I/004/11/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Author Contributions: Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='F and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='J launched the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='J, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=', Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='W, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='L, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='C and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='W carried out the data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='F, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=', X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='W, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='W and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='J interpreted the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='J, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='F prepared the paper and all authors joined the discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='J and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='Z contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 12 Competing Interests: The authors declare that they have no competing financial interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Author Information: Correspondence and requests for materials should be addressed to Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='F (yzfan@pmo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Code availability: The codes used in this analysis are standard in the community, as intro- duced in the supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Data availability: The Swift observation data analysed/used in this work are all publicly avail- able.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 13 Supplementary materials Materials and Methods Tables S1 to S6 Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' S1 to S7 References (30-52) Materials and Methods 1 A new method to measure the saturated sources in Swift UVOT images UVOT is a photon counting detector and typical read-out rate is once every ∼ 11 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' If the source is bright enough (> 10 counts s−1), coincidence losses start to be significant and a correction is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' When the incident photon counts rate beyond the read-out rate ∼ 86 s−1, the source is fully saturated and proper coincidence loss correction is impractical (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' However for extremely saturated sources with read-out streaks, a calibration method has been developed based on the measurement of read-out streak line strength (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Anyhow, the read-out streak lines are only present in the extremely saturated sources or those with very long time exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' For the moderate saturation with relatively short exposure, it cannot be applied and our main goal is to provide a new way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Below we focus on the White band, but our method can be applied to other UVOT filters as well (indeed, as a validation, we also show in the end of this subsection that a rather similar empirical correction function holds for the V band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The saturated pattern of an UVOT image can be divided into three parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The first is a point source like structure at the center of saturated pattern, which represents the location of the saturated source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The second part is a dark square structure caused by coincidence loss and the half length of its diagonal line is about 14 arcsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' A more detailed explanation is that UVOT has 14 actually a 256×256 CCD which records the flash pattern produced by the incident photon after several amplifiers and there is a centroid algorithm to calculate positions of incident photons whose accuracy could reach 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='125 pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' As a result, each physical pixel could be subsampled to 8×8 virtual pixels with a resolution of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='5 arcsec/pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The side length of the dark square is about 20 arcsec, that is 40 virtual pixels, corresponding to an area of 5×5 pixels region on real physical CCD which is the affected region of coincidence loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The third part is the halo ring, which is distinct for saturated sources and some unsaturated sources but with low background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' S1 shows such a saturated pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' We attribute the halo rings to the wing of the Point Spread Function (PSF) of UVOT detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' To test this conjecture, we will examine whether the “intrinsic” photon counts rates of saturated sources is proportional to photon counts rates of halo rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' To avoid the influence of the coincidence loss, the best measurement region to get the highest S/N ratio is the area between a circle with a radius of 25 arcsec and a square, with the same center and with a side length of 20 arcsec, like a Chinese copper cash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' However, if Swift rotated during observations, the dark region of final stacked science image are not necessarily a square due to that the coincidence loss square is aligned to the edge of CCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Hence, we used an annulus of an inner radius of 15 arcsec and an outer radius of 25 arcsec (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=', the outer edge of halo rings) to measure photon count rate in the ring ( ˙Nring), where the background should be removed and the coincidence loss has been corrected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The crucial step is to reliably derive the corresponding photon count rate of the saturated source within the standard aperture with a radius of 5 arcsec( ˙Naper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' As mentioned above, if the incident photon counts rate is beyond the CCD readout rate, the source is saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Fortunately, the UVOT White band is much wider than other 6 bands (hence, we will call them the narrow bands), which means although a source is saturated in White band, it could be unsaturated in narrow bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' It is therefore plausible to measure the spectrum with other filters of UVOT and then convolve it with the 15 White filter to get the corresponding “intrinsic” emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' This can be done for the power-law like afterglow spectrum of GRBs and the very early time optical flash of GRB 130427A is a nice sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The earliest UVOT measurements of this burst were highly saturated and some of them can be analyzed with the readout streak method (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Moreover, as shown in Maselli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' (31) and the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' S2, when the White filter was still saturated, there were usable measurements in other bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' In the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' S2, we show the ultraviolet/optical SED of GRB 130427A with the UVOT observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Note that these data were re-measured in this work and they are consistent with that reported in the literature (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' We performed the early time photometry of GRB 130424A with HEASoft and the results are summarized in Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The first exposure in B band and the first 2 exposures in U band were saturated, hence we took the values from Maselli et al (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Light curves of 6 narrow bands were fitted to found their magnitudes simultaneous with White band exposures, the results are listed in Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' We then carry out the power-law spectral fit to the SED and estimate the White band magnitudes, as summarized in the last column of Table S2, which are further used in Table S3 to yield the ˙Nint (in another word, the inferred ˙Naper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' It is worth noting that in epoch 1 there was an optical/ultraviolet flare and hence it is not suitable to evaluate the White band emission with this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Moreover, the White band measurement in the first, second and third epochs were significantly saturated with readout streaks, for which the fluxes were reported before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' As show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' S3, in epoch 2 our calculated flux is consistent with that reported in Maselli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' (31), validating the method proposed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Our downloaded image of the epoch 3 mentioned in Maselli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' (31) is distorted and we have hence focused on the subsequent observation data with an exposure of 20 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Our estimated flux is still well consistent with that reported in Maselli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' (31), which is expected because these two measurements were almost simultaneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Anyhow, in the plot the data point reported in Maselli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' (31) is not shown because we can not measure its ring count rate because of distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' For epoch 4 to epoch 16 8, there were no readout streaks and the method developed by Page et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' (20) does not work any longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Our method mentioned above applies to these data and yield reasonable results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' As for GRB 220101A, shortly after its peak, the ultraviolet/optical flash is not saturated any more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' For these observations we can reasonably measure its White band emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' HEASoft UVOT pipeline was used to make photometry of barely saturated images of GRB 220101A with a circle aperture with a radius of 5 arcsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' However, a reliable measurement of the ring count rate requires a somewhat long exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Therefore, we just divide the “tail” part of the flash into two time intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' We also notice 3 bright stars in the field and then measure them for independent check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' These five data points are summarized in Table S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The White band fluxes measured (indirectly and directly, respectively) in the above events and field stars are used to clarify whether there is a tight correlation between the ring counts and the intrinsic source emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' For such a purpose, these three data sets have been fitted with a linear function of a model of y = ax and a least square cost function was applied, χ2 = � i (yi−axi)2 y2 err,i+(axerr,i)2, where yi and xi represent extracted White-band photon counts rates and halo ring photon counts rates, respectively, and yerr,i and xerr,i are the corresponding uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The Pearson correlation coefficient is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='99, which reveals a very strong linear correlation, and the χ2/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='f value is ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='90, which implies a reasonable fit, where d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='f denotes the degree of the freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Hence, we conclude that ˙Naper = 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='84 ˙Nring can yield a reasonable estimation of “true” photon counts rates of saturated sources in White band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' S3 presents our best fitting result which confirms our early speculation and suggests that the outer part of the PSFs of such sources are nearly unmodified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The ground-based telescopes can well measure the V-band emission of the sources, which can thus provide an economical way to calibrate the saturated V-band observations of Swift/UVOT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Interestingly, GRB 080319B is a nice example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' For the UVOT V-band observations, in total we have 22 sub event files, which were later converted to images with HEASoft for measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 17 The first 4 exposure duration are 30s, 40s, 50s and 55s, which are same as the time bins in Page et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' These exposures display readout streaks and have been analyzed with the method of Page et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' (20), which are shown in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' S3 (see the light green empty squares).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' We measured the counts rate in the halo rings, which is defined above, with HEASoft, but made coincidence loss correction manually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Another 18 images are unsaturated, the intrin- sic emission were directly measured, and they are marked with dark green empty triangles in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' These measurements are summarized in Table S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' In addition, the optical emission of GRB 080319B was well measured by the ground based telescopes (32), and the accurately measured V-band emission from RAPTOR-T can be taken as the intrinsic ones (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=', we have the ˙Nint, in another word, ˙Naper defined in this paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The difference between the V filter of UVOT and that of RAPTOR-T is small and the magnitude difference can be ignored, as demonstrated by the overlapping data points in the left lower corner of the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Since the very early UVOT/V band observations were in event mode, we can re-bin them into the time intervals the same as that of RAPTOR-T and then get the ˙Nring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Time bins of our measurements are listed in Table S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Therefore, we apply the linear fit to the data sets and find an empirical function of ˙Naper = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='4 ˙Nring with a high correlation coefficient of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Such a correlation is nicely consistent with that for the UVOT/White band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' It is worth noting that for GRB 220101A, the photons collected in the White band are dominated by those passing the V filter because of the serious absorption in the bluer region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Indeed we find rather sim- ilar count rates for the (almost) simultaneous White and V-band measurements (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Therefore, the rather similar correction function for UVOT/V filter strongly suggests that our White band analysis of GRB 220101A is robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 18 2 Data analysis 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='1 Swift UVOT data analysis Swift/UVOT observed GRB 220101A in V, B, U, W1, M2, W2 and White bands for several epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' For data in image mode, we started from the level 2 UVOT products and used standard aperture photometry, background was measured in a nearby region without sources in stacked images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Reliable detections were only obtained in V and White bands, and the photon count rates were measured in 3 or 5 arcsec apertures, depending on SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Coincidences loss correction and aperture correction were applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' For images without detection, upper limits were assuming count rates would have reached the SNR of S/N = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Finally zeropoints including long-term sensitivity correction were used for absolute calibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The results are shown in Table S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The first white-band exposure under event mode (incident positions and time of every pho- ton are recorded) began at about 90 seconds after the trigger time, which lasted about 150 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Due to the fact that the luminosity of GRB 220101A changed rapidly at early epochs, although the transient seems to be unsaturated on the image for the total 150s exposure, it could be saturated in its peak phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Hence, we screened the calibrated event data into slices whose exposure time is ∼4s to check whether the situation mentioned above had happened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Follow- ing the guidance of UVOT data process, event slices were transformed into images and image calibrations (flat field and mod 8 corrections) were applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Since the transient is bright and iso- lated on reduced images, standard aperture photometry method was applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' From 90s to 100s, the transient was brightening rapidly and then became saturated for about 50 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' After ∼ 150s since the trigger time, it became unsaturated, again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' We found that there are halo rings around the transient on barely saturated and saturated images, which we think are the ’wings’ of point spread functions, hence, we analyzed the data with our calibration method described before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The results are summarized in Table S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 19 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='2 Swift-BAT/XRT and Fermi-GBM data analysis We processed Swift-BAT data according to standard procedures, using the software HEASoft (ver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='29) and calibration database (CALDB), which are available at https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='swift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='uk/analysis/bat/setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='php.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The mask weighting file used in extracting the light curve is generated by batgrbproduct (a complete GRB processing script in HEASoft).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' We extract event data at time intervals between -60 to 340 seconds related to the trigger time, the energy range is 15-350 keV, and the time bin size is 1 second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Our BAT analysis results are plotted with our Swift UVOT analysis results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' We also present a spectral analysis in a broad gamma-ray band (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='3 - 40000 keV) from Swift- BAT/XRT and Fermi-GBM data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The files used include the source and background spectrum files, as well as the corresponding response functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' For BAT file extraction and correction, we used standard procedures as in the manual (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' XRT files were created by online analysis tools provided by Swift official website (34, 35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The Fermi-GBM data have been processed with GBM Data Tools (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' There are different statistics used for each dataset (cstat for Swift- XRT, χ2 for Swift-BAT and pgstat for Fermi-GBM data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' We use Bilby (37) in the framework of PyXspec for model parameter estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 3 Intrinsic optical/ultraviolet emission of GRB 220101A To estimate a reliable un-absorbed optical/ultraviolet emission, we need an intrinsic spectrum to evaluate the absorption in different observation bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' For such a purpose, in addition to the UVOT V and White band observations, we adopt the g, r, i, z-band data from Liverpool tele- scope measured at t ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='625 day after the burst (38) and the simultaneous XRT data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Such a set of ground-based telescope observation data are chosen because they are almost simulta- neous with one UVOT White measurement and at late times the White band emission was not 20 detectable any longer (see Table S5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The SED from i to g declines very rapidly, requires a spectral index β ∼ 8 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Similar rapid declines, due to the serious Lyman forest absorption, have been observed in GRB 000131 (39) and 100219A (40) at redshifts of z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='500 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='667, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Since the i and z observations do not suffer from strong absorption and there is no evidence for the presence of a flare at that time, we adopt them to construct the intrinsic optical (z band) to X-ray SED to be fν ∝ ν−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='70±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='05, with which we can obtain the absorption correction in r, g as well as UVOT White and V bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' In the direction of GRB 220101A, the Galactic extinction is E(B − V ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='0483 (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Basing on the intrinsic spectrum of and assuming no extinction from the GRB host galaxy, we find an absorption in White band as high as Aλ = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='78 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='10 mag, including Lyman absorption and the Galactic extinction, see the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Note that here the central frequency of the White band observation has been taken as the same as that of the V band because of the serious absorption of the bluer photons, as demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' In this work we adopt a cosmology with with H0 = 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='4 km s−1 Mpc −1, ΩM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='315 and ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='685 (42), a redshift z =4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='618 leads to a distance modules DM= 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The absolute peak magnitude is calculated via Mpeak,abs = Mpeak −DM−Aλ +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='5(1−βi) log(1+z), where the last term is the k-correction and βi is the intrinsic spectral slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The pity is that none of the extremely luminous flashes in GRB 990123, GRB 050904, GRB 080319B and GRB 220101A have a measured optical/ultraviolet spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' For GRB 220101A, the UVOT and XRT data suggest an “overall” optical to soft X-ray spectrum softer than ν−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' If this holds in the optical band (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=', βi ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='3) in the rest frame, then we would have Mpeak,abs ≤ −40 mag in the visible band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' It is so far the unique event to be brighter than the absolute AB magnitude of −39 mag, see Table S6 for a comparison of the brightest flare in history.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' If there are spectral information of optical flares in the future, these bursts would be able to directly compared in the same band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 21 4 The numerical interpretation of the optical emission as well as the X-ray afterglow emission Here we call the X-ray emission after ∼ 170 s after the Swift trigger as the afterglow since the earlier emission are most likely the low energy part of the prompt radiation arising from the internal energy dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='1 Refreshed shock emission for the peak of the optical/ultraviolet flare In the prompt γ−ray emission lightcurve, there are several weak gamma-ray spikes from earlier outflow before the main pulse starting at ∼ T0 +65 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The front half part (between ∼ T0 +65 to 102 s) of the giant gamma-ray pulse should come from the energy release of the main outflow, either dissipated through internal shocks or magnetic re-connections within it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' For the later part (> T0 + 102 s) of the giant pulse, it overlaps with an energetic optical/ultraviolet flash, which indicates the rise of an additional dissipation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' As the preceding weak outflow gets decelerated to a bulk Lorentz factor of Γ1, a later launched but faster shell (with a bulk Lorentz factor of Γ4) will catch up with it at a radius of R0, so that a collision between two shells would occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Note that Γ1 and Γ4 correspond to ΓW and ΓM mentioned in the main text, which is used here for the convenience of the discussion below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' If the fast shell is not extremely magnetized, the collision would produce a refreshed forward shock (FS) propagating into materials of the preceding shell, and a refreshed reverse shock (RS) propagating into the fast shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' As a result, an optical/ultraviolet flash is expected from the radiation in the downstream of the refreshed RS, which has been initially proposed and works well for optical flash in the early afterglow stage (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Here we show that this scenario could account for the prompt optical emission of GRB 220101A with a detailed numerical approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Two refreshed shocks separate the system into four regions: (1) the unshocked slow shell, (2) the shocked slow shell, (3) the shocked fast shell, and (4) the unshocked fast shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Here- 22 after, Xi denotes the value of the quantity X in Region “i” in its own rest frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Unlike the preceding shell that exhausts the magnetic energy in the early stage (σ1 = 0), the later fast shell may keep the magnetic fields advected from the central engine, which could be parameterized by the magnetization of σ4 = B2 4/4πn4mpc2, where n4 is the particle density in the comoving frame of Region 4 and mp is the proton mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Let’s introduce an equivalent “luminosity” of the kinetic, internal and the magnetic energy for the two shells measured in the lab frame, Li, the corresponding particle density is then ni = Li/4πR2βiΓ2 i mpc3(1 + σi), where i = 1, 4, βi = 1/ � (1 − 1/Γ2 i ) and R is the radius from the central engine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Due to the highly vari- able nature of the outflow from the central engine, the luminosity of the later fast ejecta could be further described by L4 = Lf(R/Rpeak)qr for R ≤ Rpeak and L4 = Lf(R/Rpeak)qd for R > Rpeak, where Rpeak is the radius that the RS reaches its peak luminosity, and qr (qd) is the rising (decaying) index of the luminosity before(after) Rpeak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' We assume that Region 2 and Region 3 share a common bulk Lorentz factor (Γ2 = Γ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' After applying the hydrodynami- cal/magnetohydrodynamical jump conditions (43,44) to the FS/RS respectively and the energy conservation law to the FS-RS system (45), the evolution of Γ2 and relevant quantities within these regions could be solved numerically given the total isotropic energy of each shell (Ef and Es).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The kinetic particle-in-cell simulations reveal that particle acceleration is less efficient in strongly magnetized shock than that of weakly magnetized shock (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The shock is considered to be moderately magnetized, and it is reasonable to assume that the distribution of electrons in- jected downstream is Maxwellian both for the FS/RS (47), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=', Qi(γe, t) = Ci (γe/γc,i)2 exp−γe/γc,i, where γc,i = 1 3ϵe,i ei ρic2 mp me is the typical Lorentz factor of the thermal distribution, ϵe,i is the frac- tion of post-shock energy that goes into electrons for each region, ei and ρi is the energy and den- sity of protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The normalization constant Ci is obtained from the relevant mass conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The instantaneous electron spectrum can be obtained by solving the continuity equation of elec- 23 trons in energy space (48).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Integrating the synchrotron radiation power from the electron spec- trum in Regions 2 and 3 and considering the effect of synchrotron self-absorption and the equal- arrival-time surface, the radiation spectra and the light curves are then derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' With a starting radius of R0 = 1015 cm for the collision and a set of parameters of L1 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='6 × 1052 erg s−1, Lf = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='5 × 1053 erg s−1, Γ1 = 100, Γ4 = 1000, qr = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='3, qd = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='5, σ4 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='1, ϵB,3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='08, ϵe,2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='1, ϵe,3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='07, Es = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='8 × 1053 erg, Ef = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='0 × 1054 erg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' We get numerical optical lightcurves in good agreement with the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='2 The external forward and reverse shock afterglow emission In our modeling, it turns out that the shallow-declining part of the optical flare is hard to be accounted for (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' A possibility is the emergence of the reverse shock, as observed in for instance GRB 990123 (3, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Indeed, a reverse and forward shock scenario can reasonably reproduce the optical and X-ray data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The magnetic field in the reverse shock region should be stronger than that in the forward shock region by a factor of quite a few ×10 otherwise the induced optical flash can not be brighter than the forward shock peak optical emission by a factor of ∼ 1000 (7, 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The following physical parameters are found to be able to reasonably reproduce the afterglow data: the isotropic energy is Eiso = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='0 × 1055 erg with a half open jet angle θj = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='025, the initial Lorentz factor is Γ = 800, the fraction of forward and reverse shock energy given to the electrons is ϵe = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='4, the fraction of the forward (reverse) shock energy given to the magnetic field is ϵb,fs = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='5 × 10−5 (ϵb,rs = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='3), the number density of the interstellar medium is n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='05 cm−3 and the power-law index for shock-accelerated electrons is p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Such a p is well consistent with that needed in reproducing the optical to X-ray spectrum and lightcurves shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' S5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' S6, including Swift data analyzed in this work and Liverpool telescope data from GCN (38,49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 24 5 The prospect of detecting ultra-luminous optical/ultraviolet flares at high redshifts with SVOM/VT Optical/ultraviolet flares at high redshift will surfer from serious absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Following Moller & Jakobsen (50), we estimate the absorption correction to be AB ∼ 5 mag (the received photons are mainly caused by red leak of blue filter) and AR ∼ 1 mag for the sources at z ∼ 6, based on the responses of SVOM/VT blue and red channels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=', B and R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' For flares as luminous as GRB 080319B or GRB 220101A, if taken place at z ∼ 6, then we would have MR ∼ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='5 mag and MB ∼ 15 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' With the shortest exposure of 1s, SVOM/VT has a dynamic range of 9 − 18 mag, which is sufficiently sensitive to catch the signals mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' However, usually the exposure time of SVOM/VT should be 10-100 seconds, for which the R filter may get saturated but the B filter is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' We therefore conclude that SVOM/VT is a suitable equipment to detect the extremely bright optical flares of GRBs at z ∼ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Supplementary References 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Poole, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Photometric calibration of the Swift ultraviolet/optical telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 383, 627–645 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 0708.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='2259.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Maselli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' GRB 130427A: A Nearby Ordinary Monster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Science 343, 48–51 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 1311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='5254.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Wo´zniak, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Gamma-Ray Burst at the Extreme: “The Naked-Eye Burst” GRB 080319B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Astrophys.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=', Cleveland, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' & Kocevski, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Fermi gbm data tools: v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='0 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} 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high-redshift afterglow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' GRB Coordinates Network 31357, 1 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Andersen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' VLT identification of the optical afterglow of the gamma-ray burst GRB 000131 at z=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 364, L54–L61 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' astro-ph/0010322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Th¨one, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' GRB 100219A with X-shooter - abundances in a galaxy at z =4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 428, 3590–3606 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 1206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='2337.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 41.' metadata={'source': 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Lyman valley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Astrophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 228, 299–309 (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The Calibration of the Swift UVOT Optical Observations: A Recipe for Photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Publ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Pac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 118, 37–61 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' astro-ph/0505504.' metadata={'source': 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XMM-OM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 383, 383–386 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='05 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Taken from Maselli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Table S1: Early observations of GRB 130427A by Swift-UVOT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Galactic extinction AV = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='055, AB = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='071, AU = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='087, AW1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='118, AM2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='163 and AW2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='156 have been applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' These data points have been plotted in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 28 Epoch T-T0 Exp V B U W1 M2 W2 Whitea (s) (s) (AB) (AB) (AB) (AB) (AB) (AB) (AB) 1 515.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='57 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='44 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='32±0.' metadata={'source': 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130427A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' To derive the intrinsic count rate in a 5 arcsec aperture, galactic extinction AWH = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='0875 have been accounted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Fitting uncertainties and standard deviation of fitting residuals contribute to uncertainties have been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' At early phase, there is an additional radiation component, hence these 2 data points are excluded from SED fitting algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' These data points are results of extrapolation, hence they are excluded from SED fitting algorithm as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Table S2: White band emission interpolated by Swift-UVOT narrow bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' These data (except for the last column) have been plotted in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' S2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='72 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='44 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='60±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='15 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='26±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='036(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='028) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='998 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='47±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='81 531.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='31±26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='60 RA DEC ˙Ntot,raw ring ˙Nbkg,raw ring COItot(bkg) LSS ˙Nring ˙Naper (J2000) (J2000) (count/s) (count/s) (count/s) (count/s) stars in GRB 220101A fieldc.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='76 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='42±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='06 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='99±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='67 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='033(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='029) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='996 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='73±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='56 230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='06±7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='61 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' ˙Naper is derived from SED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' This data is not fitted since U-band exposures were saturated around this exposure, hence it could be unreliable(see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' These data are measured with the first 150 second White band exposure in window timing mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Table S3: Photon count rates measured in aperture and halo ring methods in White band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Sensitivity correction factors are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='175 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='102 for GRB 220101A field and GRB 130427A field, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The factor from count rate to flux is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='01327 mJy/(count/s) for white band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' These data points have been plotted in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' T-T0 Exp ˙Ntot,raw ring ˙Nbkg,raw ring COItot(bkg) ˙Nring Magaper a ˙Nb aper (s) (s) (count/s) (count/s) (count/s) (AB) (count/s) 080319B V band measurements with Wo´znika et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' as reference 180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='60 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='84 74.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='26±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='28 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='52±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='02 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='16±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='41d a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Magnitudes are taken from Wo´znika et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' (32) and Page et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' It is not necessary to take account for the very small(∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='01) difference between Vega magnitude and AB magnitude in V band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Only values in the last sub table 080319B V band measurements with HEASoft are directly measured, others are all inferred values(i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' ˙Nint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' This exposure is close to saturation, and Page et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' (20) derived a photometry result with readout streak method, which is consistent with the aperture photometry result given by HEASoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' These points are not plotted in Fig 3 and not used in fitting algorithm as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Table S4: Photon count rates measured in aperture and halo ring methods in v band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The large scale structure correction factor and the sensitivity correction factor are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='001 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='056, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The factor from count rate to flux is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='25491 mJy/(count/s) for V band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' These data points used have been plotted in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Filter Tstart Tend Texp Signala Sky Magb second second second count/s count/s/pixel (AB) v 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='94 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='61 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='52 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='137 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='074 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='0313 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='08 white 177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='63 181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='62 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='94 202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='9 ± 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='0148 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='32 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='08 white 181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='63 185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='62 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='93 208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='6 ± 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='0149 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='08 white 185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='63 189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='62 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='94 186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='0 ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='0147 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='08 white 189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='63 193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='63 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='94 192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='9 ± 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='0146 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='08 white 193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='64 197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='62 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='93 179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='6 ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='0147 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='08 white 197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='63 201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='62 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='94 164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='6 ± 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='0147 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='55 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='08 white 201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='64 205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='62 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='93 187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='3 ± 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='0149 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='41 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='08 white 205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='63 209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='62 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='94 146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='1 ± 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='0145 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='08 white 209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='64 213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='62 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='93 158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='1 ± 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='0148 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='59 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='08 white 213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='63 217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='62 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='94 147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='3 ± 10.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='0145 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='12 H/Ld u 3627.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='5 3827.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='3 196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='0194 ± 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='17 white 27727 28549 802 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='1635 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='0396 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='0594 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='26 v 39724 40112 378 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='1419 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='0340 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='0138 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='26 white 44845 46039 1137 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='1662 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='0492 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='0601 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='36 white 50868 61521 1528 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='1369 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='0530 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='1073 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='42 white 66833 85013 5485 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='0202 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='0378 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='1000 > 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='33 a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Signal photon count rates have been corrected for coincidence losses (30,51) and long-term sensitivity correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Magnitudes are based on Swift/UVOT zeropoints (30), errors are adjusted by a binomial distribution (52), limiting magnitudes are 3σ upper limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' These values have not been corrected for the Galactic extinctions of E(B − V ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='0483 (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' These data have been analyzed in ring apertures as introduced in supplementary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Images taken before are in high resolution, our photometry is in 5′′ aperture, after are in low resolution, our photometry is in 3′′ aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Table S5: Photometry for Swift UVOT observations of GRB 220101A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' GRB z Band Mpeak AV,MW β Aλa DMc Mabs Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' AB fν ∝ ν−β AB 990123 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='600 White to V MV = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='04 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='42 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='60 (3) 050904 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='295 White to 9500 ˚A M= 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='24b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='25 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='97 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='09 (23) 080319B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='937 White to V MV = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='03 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='99 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='68 (9) 220101A 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='618 White to V Mwh = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='78 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='19 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='41 This work a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Aλ= Aλ,MW+Aλ,host+Aλ,IGM is derived from photometric SED fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Converted from fλ=9500 ˚A = 17 ± 4 × 10−15 erg cm−2 s−1 ˚A−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Absolute Magnitude at the peak Mpeak,abs = Mpeak − DM − Aλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Table S6: Properties of extremely bright GRB flares at the peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Supplementary Figures (a) GRB 130427A WH (b) GRB 220101A WH deep exposure (c) GRB 220101A WH deep exposure mask map (d) GRB 220101A WH T-T0=103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='63s, M~14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='7 (e) GRB 220101A WH T-T0=115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='63s, M~13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='56 (f) GRB 220101A WH T-T0=167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='63s, M~15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='06 Figure S1: Swift/UVOT white (WH) band images demonstrating the halo ring photometry method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Panel (a) is the White band image of GRB 130427A, where the solid circle represents the standard aperture of UVOT with a radius of 5 arcsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The dotted square region strongly suffered from coincidence loss with a typical side length of ∼ 20 arcsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Dashed annulus with an inner radius of 15 arcsec and an out radius of 25 arcsec is the halo ring region defined in this work, for which the ˙Nring is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Panel (b) shows the deep exposure of GRB 220101A field in White band, which reveals 2 faint sources in the halo ring region, hence we masked the annulus region from 95◦ to 150◦, as shown in panel (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' In addition, images of panel (b) and (c) have a pixel scale of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='004 arcsec/pixel instead of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='502 arcsec/pixel for other 4 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Panel (d), (e) and (f) show some images around the peak time of GRB 220101A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' We measured count rates in unmasked annulus region and corrected it to the whole annulus region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' V-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='8 M2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='8 B-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='4 W2+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='2 U B-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='4 Maselli W1+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='4 U Maselli 3×102 5×102 1×103 2×103 3×103 10 11 12 13 14 15 16 t-t0[s] AB Magnitude Epoch 1 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='75 Epoch 5 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='25 Epoch 2 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='5 Epoch 6 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='5 Epoch 3 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='25 Epoch 7 +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='75 Epoch 4 Epoch 8 +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 3×1014 5×1014 1×1015 2×1015 3×1015 10 11 12 13 14 15 16 Frequency[Hz] AB Magnitude a b Figure S2: The UVOT lightcurves (left) as well as the SEDs (right) of GRB 130427A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' In the left panel, the vertical grey regions mark the observation periods of the White filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Note that the second U-band data is saturated, which was however a detection point in Maselli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' (31) if only event data in the last 6s was measured, hence the filled and the empty green triangles coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The shaded colorful regions across photometry points are our interpolation results of light curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The right panel presents the optical to ultraviolet SEDs at the White band observation times constructed with the extrapolated UVOT narrow band data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' A single power-law spectrum well reproduces the data, as anticipated in the fireball external forward shock afterglow model, with which a reliable evaluation of the White band emission is yielded, as reported in the last column of Table S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 0 10 20 30 40 50 60 70 80 Nring[count/s] 0 500 1000 1500 2000 2500 3000 Naper[count/s] a Fit Func: y=k*x k=22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='22±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='84 2/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='f=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='90 GRB130427A extracted GRB220101A stars in GRB 220101A field GRB130427A Maselli 0 10 20 30 40 50 60 70 80 Nring[count/s] 0 500 1000 1500 2000 2500 3000 Naper[count/s] b Fit Func: y=k*x k=20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='62±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='43 2/d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='f=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='37 GRB080319B Wozniak GRB080319B Page GRB080319B Figure S3: Photon count rates in 5′′ aperture ˙Naper (directly measured or inferred from the intrinsic value ˙Nint) and 15 − 25′′ ring ˙Nring (coincidence loss corrected) for some Swift/UVOT white and V band measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The left panel is for the White band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The dark green upward triangles represent the two unsaturated measurements of GRB 220101A in the tail phase of the flash.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The filled squares are for three bright stars in the filed of GRB 220101A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The light green downward empty triangle represents inferred ˙Naper with the photometry result of GRB 130427A derived with readout streak method (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' As for orange points, the vertical coordinate represents the White-band emission of GRB 130427A inferred from measurements in other UVOT bands (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' S2), while the horizontal coordinate is the ˙Nring (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' S1 for definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Black squares are 3 unsaturated stars in GRB 220101A field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The right panel is for the V band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Empty dark green triangles are unsaturated measurements of GRB 080319B with HEASoft and empty light green squares are photometry results of GRB 080319B derived with the readout streak method (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' As for orange points, the vertical coordinate represents the pho- tometry result of GRB 080319B observed with RAPTOR-T (32) when the UVOT observations were ongoing, while the horizontal axis represents ˙Nring in the corresponding UVOT V-band image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The linear fit is just for filled points in both panels, and the correlation coefficients of filled points are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='990 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='998 for the left and right panel, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Black dashed lines represent the saturation count rate (coincidence loss corrected, ∼ 372 count s−1) of UVOT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 10-1 100 101 102 103 102 103 104 105 UVOT count rate (count/s) Time since trigger (s) a white v 0 10 20 30 40 50 60 70 80 90 2000 3000 4000 5000 6000 7000 8000 Effective area (cm2) Wavelength (Å) b Lyα Lyman limit V B U W1 M2 W2 White Figure S4: The similarity of Swift/UVOT White and V band observations of GRB 220101A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The left panel shows that the photon count rates in White band are almost the same as that in V band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' This is because the photons with wavelengths below the Lyman limit (in the observer frame, it is 5124 ˚A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' see the right panel) are almost fully absorbed, and the photons near the Lyman α may also suffer from strong absorption (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' S5 for this effect).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Therefore the collected photons are mainly within the V band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 17 18 19 20 21 22 23 24 25 1014 1015 1016 1017 1018 10−6 10−5 10−4 Magnitude (AB) Flux density (Jy) Frequency (Hz) GRB 000131 −4 Mag GRB 100219A −3 Mag GRB 220101A Figure S5: Optical to X-ray SED of GRB 220101A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Swift XRT, UVOT and g, r, i, z observa- tions of Liverpool telescope in the time interval of t ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='62−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='68 day after the burst (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Such a set of ground-based telescope observation data are chosen because they are almost simulta- neous with one UVOT White exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Neither the X-ray nor the optical emission displays a flare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Therefore, we construct the optical SED with the data collected at t ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='625 day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' We find that the absorption correction is AWh = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='78 mag for intrinsic optical to X-ray spectrum with index βoX = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='65, it is well consistent with X-ray spectrum βX = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The central fre- quency of the White band observation has been taken as the same as that of the V band because of the serious absorption of the bluer photons, as demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The optical SEDs of other two GRBs 000131 (39) and 100219A (40) at similar redshifts (z = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='500 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='667, respectively) are also shown for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 10 15 20 25 30 102 103 104 105 106 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 Magnitude (AB) Flux density (Jy) Time since trigger (s) z−6mag i−4mag r−2mag Swift white Swift v g+2mag BAT 10 keV XRT 10 keV Figure S6: Fit to the multi-band afterglow lightcurves of GRB 220101A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The Swift XRT and UVOT data are analyzed in this work, and the other optical data are adopted from Liv- erpool telescope (38, 49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The total extinction corrections, including Galactic extinction and interstellar-medium extinction are AWh = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='78, Av = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='88, Ag = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='51, Ar = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='46, Ai = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='24 and Az = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='10, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The dashed and dash-dotted lines represents forward and reverse shock emission arising from the weak/slow and main/fast outflow collision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Solid and dotted lines are the regular external forward and reverse shock emission of the outflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' In our calcula- tion, the main/fast outflow was launched 92 seconds after the BAT trigger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Note that the X-ray emission at t ≤ 170 sec was attributed to the low energy part of the prompt emission and has not been addressed in our modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' 0 20 40 60 80 100 3000 4000 5000 6000 7000 8000 9000 10000 11000 Response (%) Wavelength (Å) Lyα Lyβ Lyγ Lyman limit B B R R Figure S7: The response of the SVOM/VT and the Lyman absorption of the high redshift (∼ 6) event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' The optical/ultraviolet flash will surfer from strong absorption by intergalactic medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Following Moller & Jakobsen (50), we find that AB ∼ 5 mag (the received photons are mainly caused by red leak of blue filter) and AR ∼ 1 mag for a source at z = 6, based on the responses of SVOM/VT blue and red channels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=', B and R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' If the initial flash is as bright as that detected in GRB 080319B and GRB 220101A, the absorbed one would still be caught by SVOM/VT with a dynamic range of 9 − 18 mag for the shortest exposure of 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} +page_content=' Therefore SVOM/VT is a suitable equipment to detected extremely bright optical flares of GRBs at z ∼ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/89E0T4oBgHgl3EQffwAL/content/2301.02407v1.pdf'} diff --git a/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf b/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..25de8f4b60a72be89a30fd972e94790cfbdd125b --- /dev/null +++ b/8tAzT4oBgHgl3EQfE_rp/content/2301.01005v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:357a92735e5f6c6df0882b1a187b42da295660d7041f670d65ea47ec31088dc3 +size 298106 diff --git a/8tAzT4oBgHgl3EQfE_rp/vector_store/index.faiss b/8tAzT4oBgHgl3EQfE_rp/vector_store/index.faiss new file mode 100644 index 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0000000000000000000000000000000000000000..de3aec834ebc4a92de8b45d3a3cb469f4bb7578c --- /dev/null +++ b/D9AyT4oBgHgl3EQfSPfV/content/tmp_files/2301.00084v1.pdf.txt @@ -0,0 +1,966 @@ +Selection of Centrality Measures Using Self-Consistency +and Bridge Axioms +Pavel Chebotarev∗ +Moscow Institute of Physics and Technology +9 Inststitutskii per., Dolgoprudny, Moscow Region, 141700 Russia +January 3, 2023 +Abstract +We consider several families of network centrality measures induced by graph ker- +nels. The Self-consistency and Bridge axioms that appeared earlier in the literature +turn out to be closely related to two of these families. We obtain a necessary and suffi- +cient condition of Self-consistency, a sufficient condition of the Bridge axiom, indicate +specific measures that satisfy these axioms and show that under some additional con- +ditions they are incompatible. It is also shown that PageRank centrality violates most +conditions under consideration, and has a property that, according to some authors, +is hardly imaginable for a centrality measure. Adopting such conditions as the Self- +consistency or Bridge axioms allows one to dramatically reduce the length of a survey +for selecting the most appropriate centrality measures in the culling method proposed +in [1]. +Keywords: network | centrality measure | axiomatic approach | self-consistency | bridge +axiom | PageRank +1 +Introduction +The number of network centrality measures studied in the literature exceeds 400 [2] and many +new measures appear every year. This diversity needs to be structured. The main means +of structuring it is to establish a correspondence between the measures and their properties, +some of which can be considered as normative conditions or axioms. The purpose of this +paper is to advance this work by studying two natural axiomatic conditions, namely, the Self- +consistency and Bridge axioms, which are closely related to special classes of kernel-based +centrality measures. We establish a sufficient condition of the Bridge axiom, a necessary +and sufficient condition of Self-consistency, and indicate centralities, some of which are well +known and others are new, that satisfy these axioms. +Very often, centrality is identified with structural importance [3–7]. However, there are +concepts of importance that are not reducible to centrality. Say, in a chain of moving people +modeled by a path graph, the most important actors may be the leader and the trailer, i.e., +∗pavel4e@gmail.com +1 +arXiv:2301.00084v1 [physics.soc-ph] 31 Dec 2022 + +the least central end elements of the chain. Moreover, the central elements of such a chain +may not be of particular importance. Thus, the importance of nodes in networks is not +necessarily manifested through centrality. +Anyway, each point centrality measures some structural capital of the nodes. It turns out +that the types of capital accounted for by the centralities that satisfy the Bridge axiom on the +one hand and by centralities satisfying the conjunction of Self-consistency and Monotonicity +on the other hand are different, and therefore these conditions are incompatible, provided +that Equivalence is assumed. +Similarly, the Bridge axiom is incompatible with Transit +monotonicity. +PageRank is a centrality measure that attracts a lot of attention. +In this paper, we +show that it does not satisfy the most of the conditions under consideration and give an +explanation of this phenomenon. +The paper is organized as follows. After introducing the basic notation in Section 2, +in Section 3 we consider several families of centralities associated with graph kernels. In +Section 4, the Bridge and Self-consistency axioms are introduced. +Section 5 presents a +sufficient condition of the Bridge axiom as well as a number of measures that satisfy it. +In Section 6, we prove a necessary and sufficient condition of Self-consistency and present +centralities that satisfy it. In Section 7, simple general properties of centrality measures are +discussed. Axioms of Monotonicity and Transit monotonicity are considered in Section 8 and +we prove that the addition of these axioms is sufficient to ensure the properties of Section 7 +and to form conditions incompatible with the Bridge axiom. In the concluding Section 9, +we propose some interpretations of the results obtained. +2 +Notation +Let G = (V, E) be an undirected graph with node set V = V (G) and edge set E = E(G). +The order of G is |V | = n. Graph nodes will be denoted by letters u, v, w, ui, vi, etc., numbers +0, 1, 2, . . . , or names: Medici, Pazzi, etc. We consider graphs with n > 1, without loops and +multiple edges. Since some centrality measures under study are applicable only to connected +graphs, we confine ourselves to them. +Nodes u and v of G are neighbors iff {u, v} ∈ E(G). Let Nu denote the set of neighbors +of node u. +The adjacency matrix of G is denoted by A = A(G) = (auv)n×n: auv = 1 when u and v +are neighbors and auv = 0, otherwise. Let ρ(A) be the spectral radius of A. +The degree du of a node u is the number of neighbors of u: du = |Nu|. The vector of node +degrees is d = (d1, . . . , dn)T = A1, where 1 = (1, . . . , 1)T. A leaf is a node that has exactly +one neighbor. Nodes u and v are equivalent in G if there exists an automorphism of G that +takes u to v; in this case we write u ∼ v. +The Laplacian matrix of G is +L = diag(A1) − A, +where diag(x) is the diagonal matrix with vector x on the diagonal. +2 + +The union of graphs G = G1 ∪ G2 (not necessarily disjoint) is defined by: V (G) = +V (G1) ∪ V (G2) and E(G) = E(G1) ∪ E(G2). +Given a graph G, a centrality measure (or centrality; sometimes, point centrality) f +attaches a real number f(v) to each node v ∈ V (G). Thus, f depends on G, however, for +simplicity we do not reflect this dependence in the notation. In most cases G is fixed, and +when it is not, we explicitly specify the graph to which centrality applies. Formally, for a +fixed graph G, a centrality on G is a function f : V (G) → R+ ∪ {0}. It associates a non- +negative real number f(v) with every node v ∈ V (G) based only on the graph structure [4]. +Various conceptions of centrality are quite diverse. +In this regard, there is no generally +accepted definition of centrality that would semantically distinguish it from other types of +point structural measures. On some attempts to make such a distinction, see Section 7. +When a centrality measure f(·) on G is fixed, we will write u ≻ v, u ⪰ v, and u ∼= v as +short versions of f(u) > f(v), f(u) ≥ f(v), and f(u) = f(v), respectively. Moreover, if, for +instance, V = {1, . . . , 7}, then ({1, 6}, {2, 3, 4}, 5, 7) is an example of centrality ranking of +nodes 1 to 7 in which f(1) = f(6) > f(2) = f(3) = f(4) > f(5) > f(7). +3 +Centrality measures induced by graph kernels +In this section, we consider several families of centrality measures. +Let d(u, v) be the shortest path distance [8] between nodes u and v in a graph, i.e., the +length of a shortest path between u and v. Two popular1 distance based centrality measures +are the [Shortest path] Closeness [10,11] +f(u) = +� � +v∈V +d(u, v) +�−1 +(1) +and [Shortest path] Eccentricity [10,12] +f(u) = (max +v∈V d(u, v))−1. +(2) +General classes of Closeness and Eccentricity centralities are defined by (1) and (2) with +d(u, v) being arbitrary distances for graph nodes. In the literature, several classes of such +distances and, more generally, dissimilarity measures have been proposed (see, e.g., [13,14]). +Substituting them in (1) and (2) provides centralities whose properties may vary. Most of +the alternative distances and dissimilarity measures are defined via graph kernels. Let us +consider several of them. +1. The parametric Katz [15] kernels (also referred to as Walk [16] or Neumann diffusion +[17] kernels) are defined as +P Walk(t) = +∞ +� +k=0 +(tA)k = (I − tA)−1 +(3) +1For example, in the recent study [9], the authors come to the conclusion that in the infection source +identification problem “a combination of eccentricity and closeness... generally performs better than several +state-of-the-art source identification techniques, with higher accuracy and lower average hop error.” +3 + +with 0 < t < (ρ(A))−1. +2. The Communicability kernels [18,19] are +P Comm(t) = +∞ +� +k=0 +(tA)k +k! += exp(tA), +t > 0. +Two other classes of kernels are defined similarly via the Laplacian matrix L = L(G). +3. The Forest kernels or regularized Laplacian kernels [20,21] are +P For(t) = (I + tL)−1, where t > 0. +(4) +4. The Heat kernels are the Laplacian exponential diffusion kernels [22] +P Heat(t) = +∞ +� +k=0 +(−tL)k +k! += exp(−tL), +t > 0. +By Schoenberg’s theorem [23,24], if matrix P = (puv) is a kernel (i.e., is positive semidef- +inite), then it produces a Euclidean distance d(u, v) by means of the transformation +d(u, v) = +� 1 +2(puu + pvv − puv − pvu) +� 1 +2, +u, v ∈ V, +(5) +where factor 1 +2 determines the scale. +Thus, all Walk, Communicability, Forest, and Heat kernels with appropriate parameters t +provide distances that can be substituted in (1) and (2) to obtain Closeness and Eccentricity +centralities. We will denote them by Closeness(Kernel) and Eccentricity(Kernel) with the +corresponding kernels substituted. +Furthermore, if Pn×n = (puv) determines a proximity measure (which means that for any +x, y, z ∈ V, pxy + pxz − pyz ≤ pxx, and the inequality is strict whenever z = y and y ̸= x), +then [25] transformation +d(u, v) = 1 +2(puu + pvv − puv − pvu), +u, v ∈ V +(6) +provides a distance function that satisfies the axioms of a metric. The Forest kernel with +any t > 0 produces a proximity measure, while kernels in the remaining three families do +so when t is sufficiently small [14]. The centralities obtained from a Proximity measure by +transformation (6) and substitution of the resulting distance into (1) and (2) will be denoted +by Closeness∗(Proximity) and Eccentricity∗(Proximity), respectively. +Moreover, if P represents a strictly positive transitional measure on G (i.e., pxy pyz ≤ +pxz pyy for all nodes x, y, and z, with pxy pyz = pxz pyy whenever every path in G from x to z +visits y), then transformation +ˆpuv = ln puv, +u, v ∈ V +produces [13,26] a proximity measure. In this case, (6) applied to ˆP = (ˆpuv) reduces to +d(u, v) = 1 +2(ln puu + ln pvv − ln puv − ln pvu) +(7) +4 + +and generates [13] a cutpoint additive distance d(u, v), viz., such a distance that d(u, v) + +d(v, w) = d(u, w) whenever v is a cutpoint between u and w in G (or, equivalently, whenever +all paths connecting u and w visit v). The centralities obtained from anyTransitional Measure +by transformation (7) and substitution of the resulting distance into (1) and (2) will be +denoted by Closeness∗(logTransitionalMeasure) and Eccentricity∗(logTransitionalMeasure), +respectively. +Since the Walk and Forest kernels determine [26] strictly positive transitional measures, +transformation (7) applied to them generates cutpoint additive distances. Substituting them +into (1) and (2) produces Closeness∗(logForest), Closeness∗(logWalk) and the corresponding +Eccentricity∗(·) centrality measures. +Thus, based on the above results, we define Closeness and Eccentricity centrality mea- +sures obtained by substituting the: +• Forest kernel; +• Heat kernel; +• logarithmic Forest kernel; +• logarithmic Walk kernel; +• logarithmic Heat kernel, and +• logarithmic Communicability kernel +transformed by (5) or (6) into (1) and (2). +These centralities were used in the survey +proposed in [1] with parameter t = 1 for the Forest, Heat, and Communicability kernels and +t = (ρ(A) + 1)−1 for the Walk kernel. +While the above measures are promising kernel-based centralities, they do not exhaust all +kernels and transformations [14,17] that can be used to obtain such measures. To mention +some alternative constructions, note that every distance on graph nodes can be integrated +in the p-Means framework [27] or in the framework developed in [28]. +The Closeness(Forest) centrality was examined in [29] with the conclusion that “forest +distance centrality has a better discriminating power than alternate metrics such as be- +tweenness, harmonic centrality, eigenvector centrality, and PageRank.” Along with this, the +authors note that the order of node importance induced by forest distances on some simple +graphs is consistent with their intuition. +In addition to the above approaches, kernels and similarity/proximity measures can be +used to obtain centralities directly, without transformations into distances. +An example +of such measures is the Estrada subgraph centrality [18]. +This index of a graph node u +is equal to the diagonal entry pComm +uu +of the Communicability kernel, so we denote it by +Communicability(Kii). Similarly, Walk(Kii) is the measure f(u) = pWalk +uu +, u ∈ V determined +by the diagonal entries of the Walk kernel. +One more type of centrality measures is constructed by summing the non-diagonal entries +of the rows of a kernel matrix. We consider the measures of this kind Communicability(Kij) +and Walk(Kij) defined by f(u) = � +v̸=u pComm +uv +and f(u) = � +v̸=u pWalk +uv +, u ∈ V, respectively. +Finally, Total communicability [30] is obtained by summing all row entries of the Commu- +nicability kernel: f(u) = � +v∈V pComm +uv +; it can be described [31] in terms of “potential gain,” +as well as the corresponding Walk measure. +5 + +The existence of hundreds of types and subtypes of centralities compounded by the +existence of infinite families of them highlights the need for powerful tools for comparing +centrality measures and choosing the most appropriate ones. The axiomatic approach is +indispensable in this regard. +4 +Axioms of Bridge and Self-consistency +The axioms considered in this section determine the relation between the centrality values of +two nodes in a graph of a special structure. As mentioned above, the measures under study +assign centrality to nodes based solely on the graph structure. The Equivalence axiom is a +partial embodiment of this idea (cf. [32, axiom A3]). +Axiom E (Equivalence). +If u, v ∈ V (G) and u ∼ v, then f(u) = f(v). +All measures under consideration satisfy axiom E; it will be assumed by default. +Among the most appealing axioms characterizing various classes of “reasonable” centrality +measures are those of an ordinal nature. Such axioms allow one to compare the centrality of +some nodes, but they do not determine specific computational algorithms. In other words, +they are not fingerprints of particular centrality measures. +Positive responsiveness is a type of axiom, which is of primary importance in many +axiomatic constructions. The template of these axioms is as follows: “an increase in input +(making a node more central from some point of view) leads to an increase in output (i.e., +raises its centrality).” Now we present two axioms of this kind. In the next two sections, we +will find centrality measures that satisfy them. +Recall that a bridge in a graph is an edge whose deletion increases the number of graph’s +connected components. The following axiom [33] relates the centrality of the endpoints of +any bridge. +Axiom B (Bridge). +If edge {u, v} is a bridge in G, i.e., the removal of {u, v} from E(G) +separates G into two connected components (with node sets Vu ∋ u and Vv ∋ v), then +|Vu| < |Vv| ⇔ f(u) < f(v). +A strengthening of this axiom is the Ratio property [34], which holds when under the +same premise, f(w) > 0 for all w ∈ V and f(u)/f(v) = |Vu|/|Vv|. +The idea of the second axiom is quite different. We assume that the vector of centrality +values of the neighbors of any node u carries a lot of information about the centrality of +u itself (cf. Consistency in [35]). A more specific form of this idea is that “the higher the +centrality values of a node’s neighbors, the higher the centrality of the node itself.” +This is in line with the justification given by Bonacich and Lloyd [36] to the Eigenvector +centrality, a measure satisfying (Section 6) the axiom we are going to introduce: “The eigen- +vector is an appropriate measure when one believes that actors’ status is determined by those +with whom they are in contact. This conception of importance or centrality makes sense in +a variety of circumstances. Social status rubs off on one’s associates. Receiving information +6 + +from knowledgeable sources adds more to one’s own knowledge. However, eigenvectors can +give weird and misleading results when misapplied.” +The final step in refining this concept leads to the axiom of Self-consistency. In the case +of directed graphs that express paired comparisons, it appeared in [37–39]; for undirected +graphs, in [40, 41] under the name of Structural consistency. It strengthens Preservation +of neighborhood-inclusion [42], whose directed version goes back to Preservation of cover +relation [43]. +Axiom S (Self-consistency). +If for u, v ∈ V, there is a bijection between Nu to Nv such +that every element of Nu is, according to f(·), no more central than the corresponding element +of Nv, then f(u) ≤ f(v). If “no more” is actually “less” at least once, then f(u) < f(v). +Both the Bridge and Self-consistency axioms belong to the class of positive responsive- +ness axioms, however, the positivity requirement in the premise of Self-consistency is not +objective: it reduces to positivity in terms of f(·). This implies that when f(·) satisfies ax- +iom S and the values of ¯f(·) are ordered oppositely to those of f(·), then ¯f(·) also satisfies S. +Consequently, the sole axiom S allows in some cases to conclude that f(u) = f(v), but never +that f(u) > f(v). In particular, if f(u) = f(v) for all u, v ∈ V, then f(·) satisfies S for any +graph. Therefore, Self-consistency is usually combined with other axioms indicating how +centrality is related to the graph structure itself rather than to the neighbors’ centrality. +In the following two sections, we present several results on the centrality measures that +satisfy the Bridge or Self-consistency axioms. +5 +Centrality measures satisfying the Bridge axiom +In the statements of this section, the notion of a cutpoint additive distance and the Close- +ness∗(logForest) and Closeness∗(logWalk) measures are those introduced in Section 3. +The Connectivity centrality [34] of vertex u is equal to the number of permutations +π = (π1, . . . , π|V |) of V (G) such that π1 = u and for every j ∈ {2, . . . , |V | − 1}, the induced +subgraph of G with node set {π1, . . . , πj} is connected. +Lemma 1. Any Closeness centrality of the form (1) such that the corresponding distance +d(·, ·) is cutpoint additive satisfies axiom B. +Proof. +For any connected G, consider a Closeness centrality f(u) = +�� +v∈V d(u, v) +�−1, +where d(·, ·) is a cutpoint additive distance. +Let {u, v} be a bridge in G. Since v is a +cutpoint between u and any node w ∈ Vv∖{v}, it holds that +(f(u))−1 += +� +w∈V (G) +d(u, w) = +� +w∈Vu +d(u, w) + +� +w∈Vv +d(u, w) += +� +w∈Vu +d(u, w) + |Vv| d(u, v) + +� +w∈Vv +d(v, w). +7 + +Figure 1: A tree on which Betweenness violates axiom B. +Similarly, (f(v))−1 = � +w∈Vv d(v, w) + |Vu| d(v, u) + � +w∈Vu d(u, w). Hence +(f(u))−1 − (f(v))−1 = (|Vv| − |Vu|) d(u, v), +consequently, f(u) < f(v) ⇔ (f(v))−1 < (f(u))−1 ⇔ |Vu| < |Vv|. Therefore, f(·) satisfies the +Bridge axiom. +Proposition 1. The Shortest path Closeness, Connectivity, Closeness∗(logWalk), and +Closeness∗(logForest) centralities satisfy the Bridge axiom. +Proof. The fulfilment of the Bridge axiom for the Shortest path Closeness is due to Skibski +and Sosnowska [33]. Alternatively, it follows from Lemma 1. +The Bridge axiom holds for Connectivity since this centrality measure satisfies the +stronger Ratio property [34]. +The Walk (3) and Forest (4) kernels represent [26] strictly positive transitional measures +on any connected graph. Therefore, definition (7) transforms [13] them into cutpoint additive +distances d(u, v). By Lemma 1 this implies that the Closeness centralities corresponding +to these distances, namely, the Closeness∗(logWalk) and Closeness∗(logForest) centralities, +satisfy the Bridge axiom. +Similarly, other strictly positive transitional measures [26] and cutpoint additive distances +also produce centralities that satisfy the Bridge axiom. +Remark 1. It is worth noting that the Betweenness centrality [44] satisfies the Bridge axiom +for many graphs, however, generally this is not the case. The simplest graph on which Be- +tweenness violates this axiom is shown in Fig. 1. Here, axiom B requires that the centralities +of nodes 0 and 5 are equal since |V0| = |V5|. However, the Betweenness centrality of node 0 +is higher than that of node 5, as 0 lies on the shortest path from 1 to 2. +6 +Centrality measures satisfying Self-consistency +To formulate a necessary and sufficient condition of Self-consistency, we introduce two defi- +nitions. +8 + +Definition 1. A function ϕ : Mk → R, where Mk = {M : 0 < |M| < k}, M being a +multiset2 of real numbers, will be called a scoring function if ϕ(M) is strictly increasing in +any element of M, while the remaining elements, including those equal to the varying one, +are fixed. +Definition 2. A centrality vector x = (x1, . . . , xn)T assigned to a connected graph G with +V (G) = {1, . . . , n} (xu = f(u), u ∈ V (G), where f is the corresponding centrality measure) +has a monotonic neighborhood representation if there exists a scoring function ϕ such that +x satisfies the system of equations +xu = ϕ({xw : w ∈ Nu}), +u = 1, . . . , n. +(8) +In Definition 2, {xw : w ∈ Nu} is the multiset of the components of x that correspond to +the neighbors of node u in G. If a centrality vector has a monotonic neighborhood represen- +tation, then finding this vector reduces to solving the system (8). +Lemma 2. A centrality measure on G satisfies Self-consistency if and only if the centrality +vector this measure attaches to G has a monotonic neighborhood representation. +Proof. +Suppose that the centrality vector x = (x1, . . . , xn)T associated with G has a +monotonic neighborhood representation (8). Let the premise of Self-consistency be true for +nodes u and v. Consider the equations (8) corresponding to u and v: +xu += +ϕ({xw : w ∈ Nu}), +(9) +xv += +ϕ({xw : w ∈ Nv}). +(10) +Since there is a bijection that maps each element of Nu to an element of Nv with a greater +or equal centrality, step by step replacing in (9) the xw value of each element of Nu by the +x component of the corresponding element of Nv and using the definition of monotonic +neighborhood representation, we get a growth or preservation of the value of ϕ(·) at each +step, yielding the value xv in the last step. This implies that xu ≤ xv, or, stronger, xu < xv +whenever xw has been strictly increased at least once. Therefore, Self-consistency is satisfied. +Conversely, suppose that a centrality measure on G is Self-consistent. Let us construct a +scoring function ϕ(·) that provides a monotonic neighborhood representation of the centrality +vector x = (x1, . . . , xn)T associated with G. First, we set ϕ({xw : w ∈ Nu}) +def += xu for all +u ∈ {1, . . . , n}. Whenever {xw : w ∈ Nu} = {xw : w ∈ Nv} for some u, v ∈ V, Self-consistency +implies xu = xv, i.e., the above definition of ϕ(·) on the set of multisets P = {{xw : w ∈ Nu}, +1 ≤ u ≤ n} ⊂ Mk is not contradictory. Thus, we defined the function ϕP(·) on P. Now, +to obtain a monotonic neighborhood representation of x, it suffices to extend ϕP(·) to the +entire set Mk (k = max{|Nu|, 1 ≤ u ≤ n}) of multisets of real numbers in such a way that +the resulting ϕ(·) is strictly increasing on Mk. +2A finite multiset is an equivalence class of vectors such that two vectors z and z′ are equivalent whenever +z′ can be obtained from z by permuting its components. As distinct from a set, a multiset may contain +several copies of the same element, as the components of a vector may be equal. +9 + +By the definition of a scoring function, the strict increase of ϕ(·) is required with respect +to the following preorder ≽ on Mk: for X, Y ∈ Mk, X ≽ Y ⇔ [there is a bijection between +X to Y such that every element of Y does not exceed the corresponding element of X]. The +condition of strict increase reduces to the implication [X ≽ Y and Y ̸≽ X] ⇒ ϕ(X) > ϕ(Y ), +since the second necessary implication [X ≽ Y and Y ≽ X] ⇒ ϕ(X) = ϕ(Y ) is trivial as its +premise implies X = Y. +Observe that the preorder ≽ has a numerical [utility] representation. This means that +there exists a function u: Mk → R such that for all X, Y ∈ Mk, X ≻ Y ⇒ u(X) > u(Y ), +where, by definition, X ≻ Y ⇔ [X ≽ Y and Y ̸≽ X]. Indeed, u(X) can be defined, say, +as the sum of the elements of multiset X. Then X ≻ Y ⇒ u(X) > u(Y ) and so u(·) is a +numerical representation of ≽. +By Self-consistency, ϕP(·) strictly increases on P, i.e., ϕP(·) is a numerical representation +of ≽P, the restriction of ≽ to P. Since ≽ has a numerical representation, it follows from [45, +Theorem 1] that ϕP(·) has a strictly increasing extension to Mk if and only if ϕP(·) is gap- +safe increasing, i.e., is weakly increasing and for any X, Y ∈ Mk ∪ {−∞, +∞}, Y ≻ X +implies +inf{ϕP(Z) : Z ≽ Y, Z ∈ P} > sup{ϕP(Z) : X ≽ Z, Z ∈ P}, +(11) +where, by convention, sup ∅ = −∞ and inf ∅ = +∞. +To prove that ϕP(·) is gap-safe increasing, first observe that since P is finite, sup and inf +in (11) can be replaced by max and min, respectively, under the convention that max ∅ = −∞ +and min ∅ = +∞. Then, if the [multi]sets on the left-hand and right-hand sides of (11) are +both nonempty, then for any Z′′ and Z′ minimizing ϕP(Z) on the left and maximizing ϕP(Z) +on the right, respectively, Z′′ ≽ Y ≻ X ≽ Z′ holds, and by the “mixed” strict transitivity3 of +≽, Z′′ ≻ Z′. By Self-consistency this implies ϕP(Z′′) > ϕP(Z′) and (11) is valid. Otherwise, +if some multiset in (11) is empty, then we have +∞ on the left or/and −∞ on the right, in a +possible combination with a finite number on one of the sides. In all these cases, (11) is valid, +hence ϕP(·) is gap-safe increasing. Therefore, by [45, Theorem 1], ϕP(·) can be extended +to Mk so that its extension ϕ(·) is a strictly increasing function and therefore, provides +a monotonic neighborhood representation of the centrality vector x = (x1, . . . , xn)T. This +completes the proof. The extension of ϕP(·) to Mk can be made, in particular, using the +approach proposed in [45]. +The following propositions involve five centrality measures; we now recall their definitions +using the notation introduced in Section 2. +For a connected graph G of order n, vector x = (x1, . . . , xn)T presents: +• the Walk centrality [15] if +x = +∞ +� +k=1 +(tA)k1 = ((I − tA)−1 − I)1, +(12) +where t ∈ R is a parameter such that 0 < t < (ρ(A))−1; +3This means that for any X, Y, Z ∈ Mk, Z ≽ Y ≻ X ⇒ Z ≻ X and Y ≻ X ≽ Z ⇒ Y ≻ Z. +10 + +• the Bonacich centrality [46] with real parameters α and β > 0 if x satisfies the system +of equations +xu = +� +w∈Nu +(α + βxw), +u = 1, . . . , n; +(13) +• the Generalized Degree centrality [47] if x satisfies the system of equations +(I + εL)x = d, +(14) +where ε > 0 is a real parameter; +• the Eigenvector centrality [48,49] if x is positive and satisfies the equation +Ax = ρ(A)x; +(15) +• the PageRank centrality [50] if x is positive and satisfies the equation4 +x = +� +αAT(diag(A1))−1 + (1 − α)J +� +x, +(16) +where J = 1 +n11T, while α ∈ R is the “teleportation” parameter such that 0 < α < 1. +Proposition 2. The Generalized Degree, Walk, Eigenvector, and Bonacich centralities sat- +isfy Self-consistency. +Proof. 1. +Since for any u, du = |Nu|, Eq. (14) can be written in component form as +xu(1 + ε|Nu|) − ε +� +w∈Nu +xw = |Nu|, +u = 1, . . . , n, +which is equivalent to +xu = (1 + ε|Nu|)−1 � +w∈Nu +(1 + εxw), +u = 1, . . . , n. +(17) +Eq. (17) is a monotonic neighborhood representation of vector x, therefore, by Lemma 2, +the Generalized Degree centrality satisfies Self-consistency. +2. It follows from (12) that +(I − tA)x = td, +from which +xu = t +� +w∈Nu +(1 + xw), +u = 1, . . . , n. +(18) +Since for any t > 0, (18) is a monotonic neighborhood representation of x, Lemma 2 +implies that the Walk centrality satisfies Self-consistency. +3. +A component form of (15) is +xu = (ρ(A))−1 � +w∈Nu +xw, +u = 1, . . . , n, +(19) +4In the case of simple graphs considered in this paper, AT = A. +11 + +which is a monotonic neighborhood representation of x. Hence, by Lemma 2, the Eigenvector +centrality satisfies Self-consistency. +4. +The equations (13) of the Bonacich centrality provide a monotonic neighborhood +representation of x. By Lemma 2, these centralities satisfy Self-consistency. It follows from +the comparison of (18) and (13) that the Walk centralities are the Bonacich centralities with +α = β = t. +To prove that a centrality measure satisfies Self-consistency, it suffices to find its mono- +tonic neighborhood representation, as we did, e.g., for the Walk centrality. Disproving the +hypothesis of the Self-consistency of some measure reduces to giving a refuting example, i.e., +an appropriate pair of nodes in some network. Here, among others, the famous network of +Florentine ruling families (Fig. 2) can be of help, as we show in Lemma 3 and Proposition 3. +Figure 2: Marriage network of the Florentine ruling families of the 15th century (without +the isolated Pucci family). +Let f(·) be a centrality measure on a graph G. We say that two arrays (u1, . . . , uk) +and (v1, . . . , vk) of the nodes of G are f(·) order equivalent iff for any i, j ∈ {1, . . . , k}, +sign(f(ui) − f(uj)) = sign(f(vi) − f(vj)). +Lemma 3. If a centrality measure f(·) satisfies axiom S, then for the Florentine families +graph of Fig. 2, the following arrays of nodes are f(·) order equivalent: +(a) (Tornabuoni, Albizzi) and (Ridolfi, Ginori); +(b) (Bischeri, Peruzzi) and (Guadagni, Castellani); +(c) (Bischeri, Castellani) and (Guadagni, Barbadori); +12 + +Lamberteschi +Ginori +Guadagni +Albizzi +Bischeri +Tornabuoni +Acciaiuoli +Medici +Ridolfi +Strozzi +Peruzzi +Salviati +Barbadori +Castellahi +Pazzi(d) (Peruzzi, Castellani) and (Bischeri, Barbadori); +(e) (Tornabuoni, Ridolfi) and (Guadagni, Strozzi); +(f) (Barbadori, Salviati) and (Castellani, Pazzi); +(g) (Ginori, Aciaiuoli, Pazzi, Lamberteschi) and (Albizzi, Medici, Salviati, Guadagni). +Proof. (a) Observe that Tornabuoni and Albizzi have three neighbors each, and they share +two neighbors. Therefore, by S, the relation between them is the same as the relation between +the remaining neighbors, Ridolfi and Ginori. (b) Bischeri and Peruzzi are adjacent and have +a common neighbor Strozzi; in addition, Bischeri has a neighbor Guadagni, while Peruzzi has +a neighbor Castellani. Due to S, the relation between Bischeri and Peruzzi coincides with +that between Guadagni and Castellani. Indeed, it is easy to see that the edge {Bischeri, +Peruzzi} cannot correct the violation of Self-consistency that may occur in the absence of +this edge. This completes the proof of (b). The remaining parts are proved similarly. +The following proposition demonstrates that Lemma 3 can be quite useful in proving +that certain measures violate Self-consistency. +Proposition 3. Walk(Kii), Communicability(Kii), Closeness(Forest), Closeness(Heat), +Closeness∗(logWalk), Closeness∗(logCommunicability), Closeness∗(logForest), and Close- +ness∗(logHeat) centralities violate axiom S. +Proof. +For the graph in Fig. 2, Walk(Kii) and Communicability(Kii) provide a central- +ity ranking in which Peruzzi ≻ Bischeri, but Guadagni ≻ Castellani. Thus, by part (b) +of Lemma 3, these measures violate Self-consistency. Measures Closeness(Forest), Close- +ness∗(logWalk), Closeness∗(logCommunicability), and Closeness∗(logHeat) provide rankings +in which Ridolfi ≻ Tornabuoni, but Guadagni ≻ Strozzi. Thus, by part (e) of Lemma 3, these +measures violate Self-consistency. Measures Closeness(Heat), and Closeness∗(logForest) pro- +vide rankings in which Castellani ≻ Peruzzi, but Bischeri ≻ Barbadori. Thus, by part (d) +of Lemma 3, these measures violate Self-consistency. +7 +On core intuition behind centrality +The best example of a “central” node is the center of a star of order more than 2. +A star of order n is a graph with one node (the center) having degree n − 1 and n − 1 +nodes of degree 1. The edges of a star are sometimes called rays. +As Freeman [51] noted, “one general intuitive theme seems to have run through all the +earlier thinking about point centrality in social networks: the point at the center of a star +[...] is the most central possible position.” +Definition 3. We say that a centrality measure on a star G with n ≥ 3 nodes satisfies the +star condition if it assigns maximum centrality to the center of this star. +For an example of a centrality measure that violates the star condition, see [1, Section 1]. +13 + +Self-consistency is a strong axiom, however, as was noted, it is not comprehensive. One +of its features is that it only applies to nodes of the same degree. Therefore, it does not +imply the star condition. As distinct from it, the Bridge axiom implies this condition. +Proposition 4. On a star with two or more rays, any centrality measure that satisfies +axiom B also satisfies the star condition. +Proof. This is true as each ray of a star is a bridge, and among the components formed after +its removal, the component containing a leaf is smaller than that containing the center. +However, axiom B does not imply that the centrality of all leaves of a star is the same, +which is immediate from Self-consistency (or from axiom E, as the leaves are equivalent). +Roy and Tredan [6], trying to capture the intuition underlying the concept of centrality +claim that for a path graph with nodes 1, . . . , n, where each node u such that 1 < u < n is +linked to u − 1 and u + 1, it is (converting to our notation) “hard to imagine a centrality f +such that, given a node u (u ̸= n+1 +2 ), we have f(u) ̸∈ [f(u − 1), f(u + 1)].” +Definition 4. Let G be a path graph where each node u such that 1 < u < n is linked to +u − 1 and u + 1. A centrality measure f on G is said to satisfy the +• Roy-Tredan (RT) condition if for any node u, u ̸= n+1 +2 +⇒ f(u) ∈ [f(u − 1), f(u + 1)]; +• path centripetal condition if the centrality of a node strictly increases with increasing +shortest path distance from the nearest leaf. +Obviously, the path centripetal condition is generally stronger than the RT condition. +Proposition 5 states that the path centripetal condition is fulfilled for all centralities that +satisfy axioms B and E. +Proposition 5. For a path graph, any centrality measure that satisfies axioms B and E also +satisfies the path centripetal condition. +Proof. Let f(·) satisfy axioms B and E. Consider the path graph 1—2—· · · —n, where “—” +denotes an edge. Let 1 ≤ u = v − 1 < n. Suppose that v ≤ n+1 +2 . Then {u, v} ∈ E is a +bridge and by axiom B, f(u) < f(v), since |Vu| < |Vv|. Hence for such u and v, the centrality +strictly increases with increasing distance from the nearest leaf 1. The case of u ≥ n+1 +2 +is +considered similarly. Finally, if u − 1 = n − v, i.e., u and v have the same distance from the +nearest leaf, then u ∼ v and by axiom E, f(u) = f(v). +It is all the more remarkable that PageRank, one of the most popular5 centrality measures, +according to Roy and Tredan, is “hard to imagine” as it violates the RT condition. +Proposition 6. For the path graph 1—2—3—4—5, the PageRank centrality f PR +α (·) with any +parameter α violates the RT condition. Namely, f PR +α (2) > f PR +α (1) and f PR +α (2) > f PR +α (3). +5According to [52], “PageRank centrality is probably the most well-known and frequently used measure.” +14 + +Proof. +For the path graph 1—2—3—4—5, let us search the solution of (16) in the form +x = (x1, x2, x3, x2, x1)T, where x1 = 1. Then the first two equations of (16) have the form +α +2 x2 + 1 − α +5 +(2 + 2x2 + x3) += +1; +α + α +2 x3 + 1 − α +5 +(2 + 2x2 + x3) += +x2. +The solution of this system is: +x2 += +2(−α2 + 2α + 4)−1(3α + 2); +x3 += +2(−α2 + 2α + 4)−1(α2 + 2α + 2). +Since both differences x2 − x1 = α(α + 4)(−α2 + 2α + 4)−1 and x2 − x3 = 2α(1 − α)(−α2 + +2α + 4)−1 are strictly positive for all α ∈ (0, 1), PageRank centralities with all appropriate +parameter values violate the RT condition. Namely, x2 > x1 and x2 > x3. +Node 3 of the 1—2—3—4—5 path can be considered as its center. It follows from the +proof of Proposition 6 that PageRank never assigns maximum centrality to this center. It +can be shown that PageRank centrality also violates the RT condition on paths with n > 5. +It is worth noting that if the user considers the Self-consistency or Bridge axiom as an +indispensable property of a centrality measure, then this leads to a dramatic reduction of +the set of candidate measures (see [1], where the corresponding reduced surveys for choosing +the most appropriate centrality measure are shown in Figures 7 and 8). +8 +Combinations with monotonicity axioms +In this section, we focus on edge-monotonicity conditions, which, as well as the Self- +consistency and Bridge axioms, belongs to the class of positive responsiveness axioms. It is +proved that together with axiom E (and once S) they imply the star and path centripetal +conditions and contradict axiom B, while PageRank violates axioms B, S, and Transit mono- +tonicity. +The edge-monotonicity axioms of this section involve two graphs: an original graph G0 +and a graph G obtained from G0 by adding an extra edge (extra edges). These axioms +restrict a universal centrality measure fG(·) operating on any connected graph G. The word +“universal” in the formulations of Propositions 7 to 10 is implied, not explicit. +Axiom M (Monotonicity). Suppose that u, v ∈ V (G0), fG0(u) ≤ fG0(v), u ̸= v, and G = +G0 ∪ G′ ̸= G0, where V (G′) = {v, w}, E(G′) = {{v, w}}, and w ̸= u. Then fG(u) < fG(v). +According to Monotonicity, if u is no more central than v and a new edge not adjacent +to u is attached to v, then v becomes or remains more central than u. +Similar axioms called Adding rank monotonicity and Strict rank-monotonicity have been +proposed in [47] and [53] (for directed graphs), respectively. Item 1.2 of Dynamic monotonic- +ity in [54] is the corresponding condition for directed graphs representing paired comparisons. +Monotonicity together with axiom E imply the star condition. +15 + +Proposition 7. For a star with two or more rays, any centrality measure that satisfies +axioms E and M also satisfies the star condition and assigns the same centrality to all leaves. +Proof. +By E, the centrality of the two nodes of a 1-ray star is the same. By M, adding one +more node adjacent to the “center” of the 1-ray star makes the centrality of the center greater +than the same centrality of the leaves, and attaching additional leaves preserves this. +Transit monotonicity is a natural strengthening of M. +Axiom T (Transit monotonicity). +If u, v ∈ V (G0), fG0(u) ≤ fG0(v), u ̸= v, G = +G0 ∪ G′ ̸= G0, and any path in G from a node of G′ to u contains v, then fG(u) < fG(v). +According to Transit monotonicity, if u is no more central than v and v is a cutpoint +between the new edges and u, then v becomes or remains more central than u. +Together with E, Transit monotonicity implies the path centripetal condition. +Proposition 8. For a path graph, any centrality measure that satisfies axioms E and T also +satisfies the path centripetal condition. +Proof. +By E, the conclusion holds for the 1—2 path graph on two nodes. Assume that it +holds for the path graph 1—· · · —2k. Then for all i ∈ {1, . . . , k}, f(i) ≤ f(i + 1). Attaching +a new node 2k +1 and the edge {2k, 2k +1} provides the path graph 1—· · · —(2k +1). Since +any path in the new graph from 2k + 1 to i contains i + 1, axiom T implies f(i) < f(i + 1). +Therefore, the centrality of the nodes i ∈ {1, . . . , k + 1} of the new graph strictly increases +with the increase of the shortest path distance from the nearest leaf. This is also the case +for the remaining nodes i ∈ {k + 2, . . . , 2k + 1} by axiom E, since for them i ∼ (2k + 2 − i). +Thus, the conclusion of Proposition 8 is true for the 1—· · · —(2k + 1) graph. Adding node +2k + 2 and edge {2k + 1, 2k + 2} to it, we similarly derive that this conclusion also holds for +the resulting 1—· · · —(2k + 2) graph. This completes the proof by induction. +As a corollary of Proposition 8 we obtain that PageRank centrality violates axiom T. +Moreover, it does not satisfy axioms B and S. +Proposition 9. The PageRank centrality with any parameter α violates axioms T, B, and S. +Proof. PageRank centrality violates axiom T since otherwise, by Proposition 8, PageRank, +obeying axiom E, satisfies the path centripetal condition and therefore the RT condition, +which is not true by Proposition 6. +By Proposition 5, axioms B and E imply the path centripetal condition. Thus, PageRank +centrality similarly violates axiom B. +In the path graph 1—2—3—4—5, node 2 has neighbors 1 and 3, 3 has neighbors 2 and 4; +by Proposition 6, for any α ∈ (0, 1), f PR +α (2) > f PR +α (1) and f PR +α (4) = f PR +α (2) > f PR +α (3), i.e., +the neighbors of 3 have higher centrality values than the corresponding neighbors of 2. In +this case, axiom S requires f PR +α (3) > f PR +α (2), which is not the case. Therefore, axiom S is +violated. +On some other peculiarities of the PageRank centrality, see [1, Section 1]. +We conclude by showing that under Equivalence, the conjunction of M and S is incom- +patible with axiom B, and so is T. +16 + +Proposition 10. If a centrality measure satisfies axioms E, M, and S or axioms E and T, +then it violates axiom B. +Proof. +Let a universal centrality measure satisfy axioms E and B. For the graph G in +Fig. 3a, fG(4) = fG(3) by B. For the graph G0 in Fig. 3b, fG0(4) = fG0(3) by E. Observe +that G = G0 ∪ G′, where V (G′) = {0, 1} and E(G′) = {{0, 1}}. +(a) +(b) +Figure 3: (a) A graph G on which axiom B is incompatible with E&M&S as well as with +E&T; (b) G0, a subgraph of G used in the proof of Proposition 10. +Assume that this universal centrality measure satisfies axioms M and S. By E and M, +fG(0) = fG(1) > fG(2) = fG(5). Therefore, by S, fG(4) > fG(3), a contradiction. +Now assume that, instead of M&S, this centrality measure satisfies axiom T. Since all +paths in G from 0 or 1 to 3 contain 4, by T, fG(4) > fG(3) holds, a contradiction. +Axioms B and T are incompatible under Equivalence for the following reason. Suppose +that {u, v} is a bridge in G and |Vu| = |Vv|. Then B implies f(u) = f(v). However, if +the restriction of E(G) to Vu is sparse, while its restriction to Vv is dense, then T requires +f(u) < f(v). The logic of axiom S is similar to that of T in terms of transferring influences, +however, S is not “grounded” as it does not require any direct effect of density on centrality. +In the conjunction M&S, axiom M provides this “grounding.” +9 +Discussion +Each point centrality measures some structural capital of the nodes. +According to the +Bridge axiom, one end-node of a bridge is more central than the other if and only if the +removal of the bridge leaves the first one in a greater (in terms of the number of nodes) +component. In this sense, the corresponding capital is node-based: it does not depend on +the density of the components. Self-consistency states that a node’s capital increases with +the capital of its neighbors. By the Monotonicity axiom, edges incident to a node contribute +to its capital, i.e., the corresponding capital is locally edge-based. The conjunction of the +Self-consistency and Monotonicity makes this impact of edges global. +As a result, this +conjunction turns out to be incompatible (under Equivalence) with the node-based Bridge +axiom (Proposition 10). Similarly, by the same proposition, the Bridge axiom is incompatible +with the Transit monotonicity axiom, which postulates the edge nature of the capital globally. +17 + +5 +4 +3 +2 +05 +4 +3 +2 +0 +GoAn additional subject of this paper is the properties of the PageRank centrality measure +related to the main topic. It turns out that this measure violates most of the conditions +we consider and even has a property that, according to some authors, “is hard to imag- +ine” for a measure of centrality. The reason for this is the stochastic normalization used +in PageRank. In the path graph 1—2—3—4—5 used in Proposition 6, nodes 2 and 4 have +maximum PageRank centrality as they are linked to the leaves: these links receive a maxi- +mum weight of 1, since normalization does not change them. This maximum weight can be +interpreted as the specific importance of these links for the leaves, and not for the nodes 2 +and 4, which profit from this weight. It is this counterintuitive normalization that violates +the RT condition. +The axioms of Self-consistency and Bridge are quite strong, so the adoption of either +of them dramatically reduces the set of centrality measures under consideration. This fact +is used in [1], where the “culling” method for determining the most appropriate centrality +measure is proposed. +This method consists in compiling and completing a survey that +allows the user to find a measure that matches their underlying concept of centrality. In the +framework of this method, adopting a certain axiom results in compiling a shorter survey +on the set of measures that satisfy this axiom. In [1], the surveys reduced to the measures +obeying the Self-consistency or Bridge axioms are shown in Figures 7 and 8, respectively. +Acknowledgement +The author thanks Anna Khmelnitskaya and Dmitry Gubanov for helpful discussions. +References +[1] P. Chebotarev, D. Gubanov, How to choose the most appropriate centrality measure?, Preprint +[physics.soc-ph] 2003.01052, arXiv, https://doi.org/10.48550/arXiv.2003.01052 (2020). +[2] Centiserver: the most comprehensive centrality resource and web application for centrality +measures calculation, https://www.centiserver.org/?q1=centrality (2022). +[3] R. E. Preston, Two centrality models, Yearbook of the Association of Pacific Coast Geographers +32 (1) (1970) 59–78. +[4] S. Wasserman, K. Faust, Social Network Analysis: Methods and Applications, Cambridge +University Press, 1994. +[5] R. 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Chebotarev, Aggregation of preferences by the generalized row sum method, Mathemat- +ical Social Sciences 27 (1994) 293–320. +21 + diff --git a/D9AyT4oBgHgl3EQfSPfV/content/tmp_files/load_file.txt b/D9AyT4oBgHgl3EQfSPfV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b0642488093766223097b883d9b442343440631d --- /dev/null +++ b/D9AyT4oBgHgl3EQfSPfV/content/tmp_files/load_file.txt @@ -0,0 +1,782 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf,len=781 +page_content='Selection of Centrality Measures Using Self-Consistency and Bridge Axioms Pavel Chebotarev∗ Moscow Institute of Physics and Technology 9 Inststitutskii per.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', Dolgoprudny, Moscow Region, 141700 Russia January 3, 2023 Abstract We consider several families of network centrality measures induced by graph ker- nels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The Self-consistency and Bridge axioms that appeared earlier in the literature turn out to be closely related to two of these families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' We obtain a necessary and suffi- cient condition of Self-consistency, a sufficient condition of the Bridge axiom, indicate specific measures that satisfy these axioms and show that under some additional con- ditions they are incompatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' It is also shown that PageRank centrality violates most conditions under consideration, and has a property that, according to some authors, is hardly imaginable for a centrality measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Adopting such conditions as the Self- consistency or Bridge axioms allows one to dramatically reduce the length of a survey for selecting the most appropriate centrality measures in the culling method proposed in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Keywords: network | centrality measure | axiomatic approach | self-consistency | bridge axiom | PageRank 1 Introduction The number of network centrality measures studied in the literature exceeds 400 [2] and many new measures appear every year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' This diversity needs to be structured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The main means of structuring it is to establish a correspondence between the measures and their properties, some of which can be considered as normative conditions or axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The purpose of this paper is to advance this work by studying two natural axiomatic conditions, namely, the Self- consistency and Bridge axioms, which are closely related to special classes of kernel-based centrality measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' We establish a sufficient condition of the Bridge axiom, a necessary and sufficient condition of Self-consistency, and indicate centralities, some of which are well known and others are new, that satisfy these axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Very often, centrality is identified with structural importance [3–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' However, there are concepts of importance that are not reducible to centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Say, in a chain of moving people modeled by a path graph, the most important actors may be the leader and the trailer, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', ∗pavel4e@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='com 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='00084v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='soc-ph] 31 Dec 2022 the least central end elements of the chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Moreover, the central elements of such a chain may not be of particular importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Thus, the importance of nodes in networks is not necessarily manifested through centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Anyway, each point centrality measures some structural capital of the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' It turns out that the types of capital accounted for by the centralities that satisfy the Bridge axiom on the one hand and by centralities satisfying the conjunction of Self-consistency and Monotonicity on the other hand are different, and therefore these conditions are incompatible, provided that Equivalence is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Similarly, the Bridge axiom is incompatible with Transit monotonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' PageRank is a centrality measure that attracts a lot of attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In this paper, we show that it does not satisfy the most of the conditions under consideration and give an explanation of this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' After introducing the basic notation in Section 2, in Section 3 we consider several families of centralities associated with graph kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In Section 4, the Bridge and Self-consistency axioms are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Section 5 presents a sufficient condition of the Bridge axiom as well as a number of measures that satisfy it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In Section 6, we prove a necessary and sufficient condition of Self-consistency and present centralities that satisfy it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In Section 7, simple general properties of centrality measures are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Axioms of Monotonicity and Transit monotonicity are considered in Section 8 and we prove that the addition of these axioms is sufficient to ensure the properties of Section 7 and to form conditions incompatible with the Bridge axiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In the concluding Section 9, we propose some interpretations of the results obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 2 Notation Let G = (V, E) be an undirected graph with node set V = V (G) and edge set E = E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The order of G is |V | = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Graph nodes will be denoted by letters u, v, w, ui, vi, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', numbers 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , or names: Medici, Pazzi, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' We consider graphs with n > 1, without loops and multiple edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Since some centrality measures under study are applicable only to connected graphs, we confine ourselves to them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Nodes u and v of G are neighbors iff {u, v} ∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Let Nu denote the set of neighbors of node u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The adjacency matrix of G is denoted by A = A(G) = (auv)n×n: auv = 1 when u and v are neighbors and auv = 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Let ρ(A) be the spectral radius of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The degree du of a node u is the number of neighbors of u: du = |Nu|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The vector of node degrees is d = (d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , dn)T = A1, where 1 = (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , 1)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' A leaf is a node that has exactly one neighbor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Nodes u and v are equivalent in G if there exists an automorphism of G that takes u to v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' in this case we write u ∼ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The Laplacian matrix of G is L = diag(A1) − A, where diag(x) is the diagonal matrix with vector x on the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 2 The union of graphs G = G1 ∪ G2 (not necessarily disjoint) is defined by: V (G) = V (G1) ∪ V (G2) and E(G) = E(G1) ∪ E(G2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Given a graph G, a centrality measure (or centrality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' sometimes, point centrality) f attaches a real number f(v) to each node v ∈ V (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Thus, f depends on G, however, for simplicity we do not reflect this dependence in the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In most cases G is fixed, and when it is not, we explicitly specify the graph to which centrality applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Formally, for a fixed graph G, a centrality on G is a function f : V (G) → R+ ∪ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' It associates a non- negative real number f(v) with every node v ∈ V (G) based only on the graph structure [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Various conceptions of centrality are quite diverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In this regard, there is no generally accepted definition of centrality that would semantically distinguish it from other types of point structural measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' On some attempts to make such a distinction, see Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' When a centrality measure f(·) on G is fixed, we will write u ≻ v, u ⪰ v, and u ∼= v as short versions of f(u) > f(v), f(u) ≥ f(v), and f(u) = f(v), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Moreover, if, for instance, V = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , 7}, then ({1, 6}, {2, 3, 4}, 5, 7) is an example of centrality ranking of nodes 1 to 7 in which f(1) = f(6) > f(2) = f(3) = f(4) > f(5) > f(7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 3 Centrality measures induced by graph kernels In this section, we consider several families of centrality measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Let d(u, v) be the shortest path distance [8] between nodes u and v in a graph, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', the length of a shortest path between u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Two popular1 distance based centrality measures are the [Shortest path] Closeness [10,11] f(u) = � � v∈V d(u, v) �−1 (1) and [Shortest path] Eccentricity [10,12] f(u) = (max v∈V d(u, v))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (2) General classes of Closeness and Eccentricity centralities are defined by (1) and (2) with d(u, v) being arbitrary distances for graph nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In the literature, several classes of such distances and, more generally, dissimilarity measures have been proposed (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', [13,14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Substituting them in (1) and (2) provides centralities whose properties may vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Most of the alternative distances and dissimilarity measures are defined via graph kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Let us consider several of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The parametric Katz [15] kernels (also referred to as Walk [16] or Neumann diffusion [17] kernels) are defined as P Walk(t) = ∞ � k=0 (tA)k = (I − tA)−1 (3) 1For example, in the recent study [9], the authors come to the conclusion that in the infection source identification problem “a combination of eccentricity and closeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' generally performs better than several state-of-the-art source identification techniques, with higher accuracy and lower average hop error.” 3 with 0 < t < (ρ(A))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The Communicability kernels [18,19] are P Comm(t) = ∞ � k=0 (tA)k k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' = exp(tA), t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Two other classes of kernels are defined similarly via the Laplacian matrix L = L(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The Forest kernels or regularized Laplacian kernels [20,21] are P For(t) = (I + tL)−1, where t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (4) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The Heat kernels are the Laplacian exponential diffusion kernels [22] P Heat(t) = ∞ � k=0 (−tL)k k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' = exp(−tL), t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' By Schoenberg’s theorem [23,24], if matrix P = (puv) is a kernel (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', is positive semidef- inite), then it produces a Euclidean distance d(u, v) by means of the transformation d(u, v) = � 1 2(puu + pvv − puv − pvu) � 1 2, u, v ∈ V, (5) where factor 1 2 determines the scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Thus, all Walk, Communicability, Forest, and Heat kernels with appropriate parameters t provide distances that can be substituted in (1) and (2) to obtain Closeness and Eccentricity centralities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' We will denote them by Closeness(Kernel) and Eccentricity(Kernel) with the corresponding kernels substituted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Furthermore, if Pn×n = (puv) determines a proximity measure (which means that for any x, y, z ∈ V, pxy + pxz − pyz ≤ pxx, and the inequality is strict whenever z = y and y ̸= x), then [25] transformation d(u, v) = 1 2(puu + pvv − puv − pvu), u, v ∈ V (6) provides a distance function that satisfies the axioms of a metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The Forest kernel with any t > 0 produces a proximity measure, while kernels in the remaining three families do so when t is sufficiently small [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The centralities obtained from a Proximity measure by transformation (6) and substitution of the resulting distance into (1) and (2) will be denoted by Closeness∗(Proximity) and Eccentricity∗(Proximity), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Moreover, if P represents a strictly positive transitional measure on G (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', pxy pyz ≤ pxz pyy for all nodes x, y, and z, with pxy pyz = pxz pyy whenever every path in G from x to z visits y), then transformation ˆpuv = ln puv, u, v ∈ V produces [13,26] a proximity measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In this case, (6) applied to ˆP = (ˆpuv) reduces to d(u, v) = 1 2(ln puu + ln pvv − ln puv − ln pvu) (7) 4 and generates [13] a cutpoint additive distance d(u, v), viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', such a distance that d(u, v) + d(v, w) = d(u, w) whenever v is a cutpoint between u and w in G (or, equivalently, whenever all paths connecting u and w visit v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The centralities obtained from anyTransitional Measure by transformation (7) and substitution of the resulting distance into (1) and (2) will be denoted by Closeness∗(logTransitionalMeasure) and Eccentricity∗(logTransitionalMeasure), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Since the Walk and Forest kernels determine [26] strictly positive transitional measures, transformation (7) applied to them generates cutpoint additive distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Substituting them into (1) and (2) produces Closeness∗(logForest), Closeness∗(logWalk) and the corresponding Eccentricity∗(·) centrality measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Thus, based on the above results, we define Closeness and Eccentricity centrality mea- sures obtained by substituting the: Forest kernel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Heat kernel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' logarithmic Forest kernel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' logarithmic Walk kernel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' logarithmic Heat kernel, and logarithmic Communicability kernel transformed by (5) or (6) into (1) and (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' These centralities were used in the survey proposed in [1] with parameter t = 1 for the Forest, Heat, and Communicability kernels and t = (ρ(A) + 1)−1 for the Walk kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' While the above measures are promising kernel-based centralities, they do not exhaust all kernels and transformations [14,17] that can be used to obtain such measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' To mention some alternative constructions, note that every distance on graph nodes can be integrated in the p-Means framework [27] or in the framework developed in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The Closeness(Forest) centrality was examined in [29] with the conclusion that “forest distance centrality has a better discriminating power than alternate metrics such as be- tweenness, harmonic centrality, eigenvector centrality, and PageRank.” Along with this, the authors note that the order of node importance induced by forest distances on some simple graphs is consistent with their intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In addition to the above approaches, kernels and similarity/proximity measures can be used to obtain centralities directly, without transformations into distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' An example of such measures is the Estrada subgraph centrality [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' This index of a graph node u is equal to the diagonal entry pComm uu of the Communicability kernel, so we denote it by Communicability(Kii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Similarly, Walk(Kii) is the measure f(u) = pWalk uu , u ∈ V determined by the diagonal entries of the Walk kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' One more type of centrality measures is constructed by summing the non-diagonal entries of the rows of a kernel matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' We consider the measures of this kind Communicability(Kij) and Walk(Kij) defined by f(u) = � v̸=u pComm uv and f(u) = � v̸=u pWalk uv , u ∈ V, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Finally, Total communicability [30] is obtained by summing all row entries of the Commu- nicability kernel: f(u) = � v∈V pComm uv ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' it can be described [31] in terms of “potential gain,” as well as the corresponding Walk measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 5 The existence of hundreds of types and subtypes of centralities compounded by the existence of infinite families of them highlights the need for powerful tools for comparing centrality measures and choosing the most appropriate ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The axiomatic approach is indispensable in this regard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 4 Axioms of Bridge and Self-consistency The axioms considered in this section determine the relation between the centrality values of two nodes in a graph of a special structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' As mentioned above, the measures under study assign centrality to nodes based solely on the graph structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The Equivalence axiom is a partial embodiment of this idea (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' [32, axiom A3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Axiom E (Equivalence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' If u, v ∈ V (G) and u ∼ v, then f(u) = f(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' All measures under consideration satisfy axiom E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' it will be assumed by default.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Among the most appealing axioms characterizing various classes of “reasonable” centrality measures are those of an ordinal nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Such axioms allow one to compare the centrality of some nodes, but they do not determine specific computational algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In other words, they are not fingerprints of particular centrality measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Positive responsiveness is a type of axiom, which is of primary importance in many axiomatic constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The template of these axioms is as follows: “an increase in input (making a node more central from some point of view) leads to an increase in output (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', raises its centrality).” Now we present two axioms of this kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In the next two sections, we will find centrality measures that satisfy them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Recall that a bridge in a graph is an edge whose deletion increases the number of graph’s connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The following axiom [33] relates the centrality of the endpoints of any bridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Axiom B (Bridge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' If edge {u, v} is a bridge in G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', the removal of {u, v} from E(G) separates G into two connected components (with node sets Vu ∋ u and Vv ∋ v), then |Vu| < |Vv| ⇔ f(u) < f(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' A strengthening of this axiom is the Ratio property [34], which holds when under the same premise, f(w) > 0 for all w ∈ V and f(u)/f(v) = |Vu|/|Vv|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The idea of the second axiom is quite different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' We assume that the vector of centrality values of the neighbors of any node u carries a lot of information about the centrality of u itself (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Consistency in [35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' A more specific form of this idea is that “the higher the centrality values of a node’s neighbors, the higher the centrality of the node itself.” This is in line with the justification given by Bonacich and Lloyd [36] to the Eigenvector centrality, a measure satisfying (Section 6) the axiom we are going to introduce: “The eigen- vector is an appropriate measure when one believes that actors’ status is determined by those with whom they are in contact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' This conception of importance or centrality makes sense in a variety of circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Social status rubs off on one’s associates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Receiving information 6 from knowledgeable sources adds more to one’s own knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' However, eigenvectors can give weird and misleading results when misapplied.” The final step in refining this concept leads to the axiom of Self-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In the case of directed graphs that express paired comparisons, it appeared in [37–39];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' for undirected graphs, in [40, 41] under the name of Structural consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' It strengthens Preservation of neighborhood-inclusion [42], whose directed version goes back to Preservation of cover relation [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Axiom S (Self-consistency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' If for u, v ∈ V, there is a bijection between Nu to Nv such that every element of Nu is, according to f(·), no more central than the corresponding element of Nv, then f(u) ≤ f(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' If “no more” is actually “less” at least once, then f(u) < f(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Both the Bridge and Self-consistency axioms belong to the class of positive responsive- ness axioms, however, the positivity requirement in the premise of Self-consistency is not objective: it reduces to positivity in terms of f(·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' This implies that when f(·) satisfies ax- iom S and the values of ¯f(·) are ordered oppositely to those of f(·), then ¯f(·) also satisfies S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Consequently, the sole axiom S allows in some cases to conclude that f(u) = f(v), but never that f(u) > f(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In particular, if f(u) = f(v) for all u, v ∈ V, then f(·) satisfies S for any graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Therefore, Self-consistency is usually combined with other axioms indicating how centrality is related to the graph structure itself rather than to the neighbors’ centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In the following two sections, we present several results on the centrality measures that satisfy the Bridge or Self-consistency axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 5 Centrality measures satisfying the Bridge axiom In the statements of this section, the notion of a cutpoint additive distance and the Close- ness∗(logForest) and Closeness∗(logWalk) measures are those introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The Connectivity centrality [34] of vertex u is equal to the number of permutations π = (π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , π|V |) of V (G) such that π1 = u and for every j ∈ {2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , |V | − 1}, the induced subgraph of G with node set {π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , πj} is connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Any Closeness centrality of the form (1) such that the corresponding distance d(·, ·) is cutpoint additive satisfies axiom B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' For any connected G, consider a Closeness centrality f(u) = �� v∈V d(u, v) �−1, where d(·, ·) is a cutpoint additive distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Let {u, v} be a bridge in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Since v is a cutpoint between u and any node w ∈ Vv∖{v}, it holds that (f(u))−1 = � w∈V (G) d(u, w) = � w∈Vu d(u, w) + � w∈Vv d(u, w) = � w∈Vu d(u, w) + |Vv| d(u, v) + � w∈Vv d(v, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 7 Figure 1: A tree on which Betweenness violates axiom B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Similarly, (f(v))−1 = � w∈Vv d(v, w) + |Vu| d(v, u) + � w∈Vu d(u, w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Hence (f(u))−1 − (f(v))−1 = (|Vv| − |Vu|) d(u, v), consequently, f(u) < f(v) ⇔ (f(v))−1 < (f(u))−1 ⇔ |Vu| < |Vv|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Therefore, f(·) satisfies the Bridge axiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The Shortest path Closeness, Connectivity, Closeness∗(logWalk), and Closeness∗(logForest) centralities satisfy the Bridge axiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The fulfilment of the Bridge axiom for the Shortest path Closeness is due to Skibski and Sosnowska [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Alternatively, it follows from Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The Bridge axiom holds for Connectivity since this centrality measure satisfies the stronger Ratio property [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The Walk (3) and Forest (4) kernels represent [26] strictly positive transitional measures on any connected graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Therefore, definition (7) transforms [13] them into cutpoint additive distances d(u, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' By Lemma 1 this implies that the Closeness centralities corresponding to these distances, namely, the Closeness∗(logWalk) and Closeness∗(logForest) centralities, satisfy the Bridge axiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Similarly, other strictly positive transitional measures [26] and cutpoint additive distances also produce centralities that satisfy the Bridge axiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' It is worth noting that the Betweenness centrality [44] satisfies the Bridge axiom for many graphs, however, generally this is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The simplest graph on which Be- tweenness violates this axiom is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Here, axiom B requires that the centralities of nodes 0 and 5 are equal since |V0| = |V5|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' However, the Betweenness centrality of node 0 is higher than that of node 5, as 0 lies on the shortest path from 1 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 6 Centrality measures satisfying Self-consistency To formulate a necessary and sufficient condition of Self-consistency, we introduce two defi- nitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 8 Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' A function ϕ : Mk → R, where Mk = {M : 0 < |M| < k}, M being a multiset2 of real numbers, will be called a scoring function if ϕ(M) is strictly increasing in any element of M, while the remaining elements, including those equal to the varying one, are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' A centrality vector x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , xn)T assigned to a connected graph G with V (G) = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , n} (xu = f(u), u ∈ V (G), where f is the corresponding centrality measure) has a monotonic neighborhood representation if there exists a scoring function ϕ such that x satisfies the system of equations xu = ϕ({xw : w ∈ Nu}), u = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (8) In Definition 2, {xw : w ∈ Nu} is the multiset of the components of x that correspond to the neighbors of node u in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' If a centrality vector has a monotonic neighborhood represen- tation, then finding this vector reduces to solving the system (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' A centrality measure on G satisfies Self-consistency if and only if the centrality vector this measure attaches to G has a monotonic neighborhood representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Suppose that the centrality vector x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , xn)T associated with G has a monotonic neighborhood representation (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Let the premise of Self-consistency be true for nodes u and v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Consider the equations (8) corresponding to u and v: xu = ϕ({xw : w ∈ Nu}), (9) xv = ϕ({xw : w ∈ Nv}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (10) Since there is a bijection that maps each element of Nu to an element of Nv with a greater or equal centrality, step by step replacing in (9) the xw value of each element of Nu by the x component of the corresponding element of Nv and using the definition of monotonic neighborhood representation, we get a growth or preservation of the value of ϕ(·) at each step, yielding the value xv in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' This implies that xu ≤ xv, or, stronger, xu < xv whenever xw has been strictly increased at least once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Therefore, Self-consistency is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Conversely, suppose that a centrality measure on G is Self-consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Let us construct a scoring function ϕ(·) that provides a monotonic neighborhood representation of the centrality vector x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , xn)T associated with G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' First, we set ϕ({xw : w ∈ Nu}) def = xu for all u ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Whenever {xw : w ∈ Nu} = {xw : w ∈ Nv} for some u, v ∈ V, Self-consistency implies xu = xv, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', the above definition of ϕ(·) on the set of multisets P = {{xw : w ∈ Nu}, 1 ≤ u ≤ n} ⊂ Mk is not contradictory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Thus, we defined the function ϕP(·) on P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Now, to obtain a monotonic neighborhood representation of x, it suffices to extend ϕP(·) to the entire set Mk (k = max{|Nu|, 1 ≤ u ≤ n}) of multisets of real numbers in such a way that the resulting ϕ(·) is strictly increasing on Mk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 2A finite multiset is an equivalence class of vectors such that two vectors z and z′ are equivalent whenever z′ can be obtained from z by permuting its components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' As distinct from a set, a multiset may contain several copies of the same element, as the components of a vector may be equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 9 By the definition of a scoring function, the strict increase of ϕ(·) is required with respect to the following preorder ≽ on Mk: for X, Y ∈ Mk, X ≽ Y ⇔ [there is a bijection between X to Y such that every element of Y does not exceed the corresponding element of X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The condition of strict increase reduces to the implication [X ≽ Y and Y ̸≽ X] ⇒ ϕ(X) > ϕ(Y ), since the second necessary implication [X ≽ Y and Y ≽ X] ⇒ ϕ(X) = ϕ(Y ) is trivial as its premise implies X = Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Observe that the preorder ≽ has a numerical [utility] representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' This means that there exists a function u: Mk → R such that for all X, Y ∈ Mk, X ≻ Y ⇒ u(X) > u(Y ), where, by definition, X ≻ Y ⇔ [X ≽ Y and Y ̸≽ X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Indeed, u(X) can be defined, say, as the sum of the elements of multiset X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Then X ≻ Y ⇒ u(X) > u(Y ) and so u(·) is a numerical representation of ≽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' By Self-consistency, ϕP(·) strictly increases on P, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', ϕP(·) is a numerical representation of ≽P, the restriction of ≽ to P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Since ≽ has a numerical representation, it follows from [45, Theorem 1] that ϕP(·) has a strictly increasing extension to Mk if and only if ϕP(·) is gap- safe increasing, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', is weakly increasing and for any X, Y ∈ Mk ∪ {−∞, +∞}, Y ≻ X implies inf{ϕP(Z) : Z ≽ Y, Z ∈ P} > sup{ϕP(Z) : X ≽ Z, Z ∈ P}, (11) where, by convention, sup ∅ = −∞ and inf ∅ = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' To prove that ϕP(·) is gap-safe increasing, first observe that since P is finite, sup and inf in (11) can be replaced by max and min, respectively, under the convention that max ∅ = −∞ and min ∅ = +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Then, if the [multi]sets on the left-hand and right-hand sides of (11) are both nonempty, then for any Z′′ and Z′ minimizing ϕP(Z) on the left and maximizing ϕP(Z) on the right, respectively, Z′′ ≽ Y ≻ X ≽ Z′ holds, and by the “mixed” strict transitivity3 of ≽, Z′′ ≻ Z′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' By Self-consistency this implies ϕP(Z′′) > ϕP(Z′) and (11) is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Otherwise, if some multiset in (11) is empty, then we have +∞ on the left or/and −∞ on the right, in a possible combination with a finite number on one of the sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In all these cases, (11) is valid, hence ϕP(·) is gap-safe increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Therefore, by [45, Theorem 1], ϕP(·) can be extended to Mk so that its extension ϕ(·) is a strictly increasing function and therefore, provides a monotonic neighborhood representation of the centrality vector x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , xn)T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The extension of ϕP(·) to Mk can be made, in particular, using the approach proposed in [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The following propositions involve five centrality measures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' we now recall their definitions using the notation introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' For a connected graph G of order n, vector x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , xn)T presents: the Walk centrality [15] if x = ∞ � k=1 (tA)k1 = ((I − tA)−1 − I)1, (12) where t ∈ R is a parameter such that 0 < t < (ρ(A))−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 3This means that for any X, Y, Z ∈ Mk, Z ≽ Y ≻ X ⇒ Z ≻ X and Y ≻ X ≽ Z ⇒ Y ≻ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 10 the Bonacich centrality [46] with real parameters α and β > 0 if x satisfies the system of equations xu = � w∈Nu (α + βxw), u = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (13) the Generalized Degree centrality [47] if x satisfies the system of equations (I + εL)x = d, (14) where ε > 0 is a real parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' the Eigenvector centrality [48,49] if x is positive and satisfies the equation Ax = ρ(A)x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (15) the PageRank centrality [50] if x is positive and satisfies the equation4 x = � αAT(diag(A1))−1 + (1 − α)J � x, (16) where J = 1 n11T, while α ∈ R is the “teleportation” parameter such that 0 < α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The Generalized Degree, Walk, Eigenvector, and Bonacich centralities sat- isfy Self-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Since for any u, du = |Nu|, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (14) can be written in component form as xu(1 + ε|Nu|) − ε � w∈Nu xw = |Nu|, u = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , n, which is equivalent to xu = (1 + ε|Nu|)−1 � w∈Nu (1 + εxw), u = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (17) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (17) is a monotonic neighborhood representation of vector x, therefore, by Lemma 2, the Generalized Degree centrality satisfies Self-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' It follows from (12) that (I − tA)x = td, from which xu = t � w∈Nu (1 + xw), u = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (18) Since for any t > 0, (18) is a monotonic neighborhood representation of x, Lemma 2 implies that the Walk centrality satisfies Self-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' A component form of (15) is xu = (ρ(A))−1 � w∈Nu xw, u = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , n, (19) 4In the case of simple graphs considered in this paper, AT = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 11 which is a monotonic neighborhood representation of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Hence, by Lemma 2, the Eigenvector centrality satisfies Self-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The equations (13) of the Bonacich centrality provide a monotonic neighborhood representation of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' By Lemma 2, these centralities satisfy Self-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' It follows from the comparison of (18) and (13) that the Walk centralities are the Bonacich centralities with α = β = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' To prove that a centrality measure satisfies Self-consistency, it suffices to find its mono- tonic neighborhood representation, as we did, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', for the Walk centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Disproving the hypothesis of the Self-consistency of some measure reduces to giving a refuting example, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', an appropriate pair of nodes in some network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Here, among others, the famous network of Florentine ruling families (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 2) can be of help, as we show in Lemma 3 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Figure 2: Marriage network of the Florentine ruling families of the 15th century (without the isolated Pucci family).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Let f(·) be a centrality measure on a graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' We say that two arrays (u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , uk) and (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , vk) of the nodes of G are f(·) order equivalent iff for any i, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , k}, sign(f(ui) − f(uj)) = sign(f(vi) − f(vj)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' If a centrality measure f(·) satisfies axiom S, then for the Florentine families graph of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 2, the following arrays of nodes are f(·) order equivalent: (a) (Tornabuoni, Albizzi) and (Ridolfi, Ginori);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (b) (Bischeri, Peruzzi) and (Guadagni, Castellani);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (c) (Bischeri, Castellani) and (Guadagni, Barbadori);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 12 Lamberteschi Ginori Guadagni Albizzi Bischeri Tornabuoni Acciaiuoli Medici Ridolfi Strozzi Peruzzi Salviati Barbadori Castellahi Pazzi(d) (Peruzzi, Castellani) and (Bischeri, Barbadori);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (e) (Tornabuoni, Ridolfi) and (Guadagni, Strozzi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (f) (Barbadori, Salviati) and (Castellani, Pazzi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (g) (Ginori, Aciaiuoli, Pazzi, Lamberteschi) and (Albizzi, Medici, Salviati, Guadagni).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (a) Observe that Tornabuoni and Albizzi have three neighbors each, and they share two neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Therefore, by S, the relation between them is the same as the relation between the remaining neighbors, Ridolfi and Ginori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (b) Bischeri and Peruzzi are adjacent and have a common neighbor Strozzi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' in addition, Bischeri has a neighbor Guadagni, while Peruzzi has a neighbor Castellani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Due to S, the relation between Bischeri and Peruzzi coincides with that between Guadagni and Castellani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Indeed, it is easy to see that the edge {Bischeri, Peruzzi} cannot correct the violation of Self-consistency that may occur in the absence of this edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' This completes the proof of (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The remaining parts are proved similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The following proposition demonstrates that Lemma 3 can be quite useful in proving that certain measures violate Self-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Walk(Kii), Communicability(Kii), Closeness(Forest), Closeness(Heat), Closeness∗(logWalk), Closeness∗(logCommunicability), Closeness∗(logForest), and Close- ness∗(logHeat) centralities violate axiom S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' For the graph in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 2, Walk(Kii) and Communicability(Kii) provide a central- ity ranking in which Peruzzi ≻ Bischeri, but Guadagni ≻ Castellani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Thus, by part (b) of Lemma 3, these measures violate Self-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Measures Closeness(Forest), Close- ness∗(logWalk), Closeness∗(logCommunicability), and Closeness∗(logHeat) provide rankings in which Ridolfi ≻ Tornabuoni, but Guadagni ≻ Strozzi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Thus, by part (e) of Lemma 3, these measures violate Self-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Measures Closeness(Heat), and Closeness∗(logForest) pro- vide rankings in which Castellani ≻ Peruzzi, but Bischeri ≻ Barbadori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Thus, by part (d) of Lemma 3, these measures violate Self-consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 7 On core intuition behind centrality The best example of a “central” node is the center of a star of order more than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' A star of order n is a graph with one node (the center) having degree n − 1 and n − 1 nodes of degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The edges of a star are sometimes called rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' As Freeman [51] noted, “one general intuitive theme seems to have run through all the earlier thinking about point centrality in social networks: the point at the center of a star [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='] is the most central possible position.” Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' We say that a centrality measure on a star G with n ≥ 3 nodes satisfies the star condition if it assigns maximum centrality to the center of this star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' For an example of a centrality measure that violates the star condition, see [1, Section 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 13 Self-consistency is a strong axiom, however, as was noted, it is not comprehensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' One of its features is that it only applies to nodes of the same degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Therefore, it does not imply the star condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' As distinct from it, the Bridge axiom implies this condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' On a star with two or more rays, any centrality measure that satisfies axiom B also satisfies the star condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' This is true as each ray of a star is a bridge, and among the components formed after its removal, the component containing a leaf is smaller than that containing the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' However, axiom B does not imply that the centrality of all leaves of a star is the same, which is immediate from Self-consistency (or from axiom E, as the leaves are equivalent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Roy and Tredan [6], trying to capture the intuition underlying the concept of centrality claim that for a path graph with nodes 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , n, where each node u such that 1 < u < n is linked to u − 1 and u + 1, it is (converting to our notation) “hard to imagine a centrality f such that, given a node u (u ̸= n+1 2 ), we have f(u) ̸∈ [f(u − 1), f(u + 1)].” Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Let G be a path graph where each node u such that 1 < u < n is linked to u − 1 and u + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' A centrality measure f on G is said to satisfy the Roy-Tredan (RT) condition if for any node u, u ̸= n+1 2 ⇒ f(u) ∈ [f(u − 1), f(u + 1)];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' path centripetal condition if the centrality of a node strictly increases with increasing shortest path distance from the nearest leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Obviously, the path centripetal condition is generally stronger than the RT condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proposition 5 states that the path centripetal condition is fulfilled for all centralities that satisfy axioms B and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' For a path graph, any centrality measure that satisfies axioms B and E also satisfies the path centripetal condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Let f(·) satisfy axioms B and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Consider the path graph 1—2—· · · —n, where “—” denotes an edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Let 1 ≤ u = v − 1 < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Suppose that v ≤ n+1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Then {u, v} ∈ E is a bridge and by axiom B, f(u) < f(v), since |Vu| < |Vv|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Hence for such u and v, the centrality strictly increases with increasing distance from the nearest leaf 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The case of u ≥ n+1 2 is considered similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Finally, if u − 1 = n − v, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', u and v have the same distance from the nearest leaf, then u ∼ v and by axiom E, f(u) = f(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' It is all the more remarkable that PageRank, one of the most popular5 centrality measures, according to Roy and Tredan, is “hard to imagine” as it violates the RT condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' For the path graph 1—2—3—4—5, the PageRank centrality f PR α (·) with any parameter α violates the RT condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Namely, f PR α (2) > f PR α (1) and f PR α (2) > f PR α (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 5According to [52], “PageRank centrality is probably the most well-known and frequently used measure.” 14 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' For the path graph 1—2—3—4—5, let us search the solution of (16) in the form x = (x1, x2, x3, x2, x1)T, where x1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Then the first two equations of (16) have the form α 2 x2 + 1 − α 5 (2 + 2x2 + x3) = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' α + α 2 x3 + 1 − α 5 (2 + 2x2 + x3) = x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The solution of this system is: x2 = 2(−α2 + 2α + 4)−1(3α + 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' x3 = 2(−α2 + 2α + 4)−1(α2 + 2α + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Since both differences x2 − x1 = α(α + 4)(−α2 + 2α + 4)−1 and x2 − x3 = 2α(1 − α)(−α2 + 2α + 4)−1 are strictly positive for all α ∈ (0, 1), PageRank centralities with all appropriate parameter values violate the RT condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Namely, x2 > x1 and x2 > x3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Node 3 of the 1—2—3—4—5 path can be considered as its center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' It follows from the proof of Proposition 6 that PageRank never assigns maximum centrality to this center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' It can be shown that PageRank centrality also violates the RT condition on paths with n > 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' It is worth noting that if the user considers the Self-consistency or Bridge axiom as an indispensable property of a centrality measure, then this leads to a dramatic reduction of the set of candidate measures (see [1], where the corresponding reduced surveys for choosing the most appropriate centrality measure are shown in Figures 7 and 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 8 Combinations with monotonicity axioms In this section, we focus on edge-monotonicity conditions, which, as well as the Self- consistency and Bridge axioms, belongs to the class of positive responsiveness axioms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' It is proved that together with axiom E (and once S) they imply the star and path centripetal conditions and contradict axiom B, while PageRank violates axioms B, S, and Transit mono- tonicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The edge-monotonicity axioms of this section involve two graphs: an original graph G0 and a graph G obtained from G0 by adding an extra edge (extra edges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' These axioms restrict a universal centrality measure fG(·) operating on any connected graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The word “universal” in the formulations of Propositions 7 to 10 is implied, not explicit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Axiom M (Monotonicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Suppose that u, v ∈ V (G0), fG0(u) ≤ fG0(v), u ̸= v, and G = G0 ∪ G′ ̸= G0, where V (G′) = {v, w}, E(G′) = {{v, w}}, and w ̸= u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Then fG(u) < fG(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' According to Monotonicity, if u is no more central than v and a new edge not adjacent to u is attached to v, then v becomes or remains more central than u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Similar axioms called Adding rank monotonicity and Strict rank-monotonicity have been proposed in [47] and [53] (for directed graphs), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Item 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='2 of Dynamic monotonic- ity in [54] is the corresponding condition for directed graphs representing paired comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Monotonicity together with axiom E imply the star condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 15 Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' For a star with two or more rays, any centrality measure that satisfies axioms E and M also satisfies the star condition and assigns the same centrality to all leaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' By E, the centrality of the two nodes of a 1-ray star is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' By M, adding one more node adjacent to the “center” of the 1-ray star makes the centrality of the center greater than the same centrality of the leaves, and attaching additional leaves preserves this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Transit monotonicity is a natural strengthening of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Axiom T (Transit monotonicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' If u, v ∈ V (G0), fG0(u) ≤ fG0(v), u ̸= v, G = G0 ∪ G′ ̸= G0, and any path in G from a node of G′ to u contains v, then fG(u) < fG(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' According to Transit monotonicity, if u is no more central than v and v is a cutpoint between the new edges and u, then v becomes or remains more central than u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Together with E, Transit monotonicity implies the path centripetal condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' For a path graph, any centrality measure that satisfies axioms E and T also satisfies the path centripetal condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' By E, the conclusion holds for the 1—2 path graph on two nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Assume that it holds for the path graph 1—· · · —2k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Then for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , k}, f(i) ≤ f(i + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Attaching a new node 2k +1 and the edge {2k, 2k +1} provides the path graph 1—· · · —(2k +1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Since any path in the new graph from 2k + 1 to i contains i + 1, axiom T implies f(i) < f(i + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Therefore, the centrality of the nodes i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , k + 1} of the new graph strictly increases with the increase of the shortest path distance from the nearest leaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' This is also the case for the remaining nodes i ∈ {k + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' , 2k + 1} by axiom E, since for them i ∼ (2k + 2 − i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Thus, the conclusion of Proposition 8 is true for the 1—· · · —(2k + 1) graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Adding node 2k + 2 and edge {2k + 1, 2k + 2} to it, we similarly derive that this conclusion also holds for the resulting 1—· · · —(2k + 2) graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' This completes the proof by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' As a corollary of Proposition 8 we obtain that PageRank centrality violates axiom T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Moreover, it does not satisfy axioms B and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The PageRank centrality with any parameter α violates axioms T, B, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' PageRank centrality violates axiom T since otherwise, by Proposition 8, PageRank, obeying axiom E, satisfies the path centripetal condition and therefore the RT condition, which is not true by Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' By Proposition 5, axioms B and E imply the path centripetal condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Thus, PageRank centrality similarly violates axiom B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In the path graph 1—2—3—4—5, node 2 has neighbors 1 and 3, 3 has neighbors 2 and 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' by Proposition 6, for any α ∈ (0, 1), f PR α (2) > f PR α (1) and f PR α (4) = f PR α (2) > f PR α (3), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', the neighbors of 3 have higher centrality values than the corresponding neighbors of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In this case, axiom S requires f PR α (3) > f PR α (2), which is not the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Therefore, axiom S is violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' On some other peculiarities of the PageRank centrality, see [1, Section 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' We conclude by showing that under Equivalence, the conjunction of M and S is incom- patible with axiom B, and so is T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 16 Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' If a centrality measure satisfies axioms E, M, and S or axioms E and T, then it violates axiom B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Let a universal centrality measure satisfy axioms E and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' For the graph G in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 3a, fG(4) = fG(3) by B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' For the graph G0 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 3b, fG0(4) = fG0(3) by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Observe that G = G0 ∪ G′, where V (G′) = {0, 1} and E(G′) = {{0, 1}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (a) (b) Figure 3: (a) A graph G on which axiom B is incompatible with E&M&S as well as with E&T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' (b) G0, a subgraph of G used in the proof of Proposition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Assume that this universal centrality measure satisfies axioms M and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' By E and M, fG(0) = fG(1) > fG(2) = fG(5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Therefore, by S, fG(4) > fG(3), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Now assume that, instead of M&S, this centrality measure satisfies axiom T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Since all paths in G from 0 or 1 to 3 contain 4, by T, fG(4) > fG(3) holds, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Axioms B and T are incompatible under Equivalence for the following reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Suppose that {u, v} is a bridge in G and |Vu| = |Vv|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Then B implies f(u) = f(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' However, if the restriction of E(G) to Vu is sparse, while its restriction to Vv is dense, then T requires f(u) < f(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The logic of axiom S is similar to that of T in terms of transferring influences, however, S is not “grounded” as it does not require any direct effect of density on centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In the conjunction M&S, axiom M provides this “grounding.” 9 Discussion Each point centrality measures some structural capital of the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' According to the Bridge axiom, one end-node of a bridge is more central than the other if and only if the removal of the bridge leaves the first one in a greater (in terms of the number of nodes) component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In this sense, the corresponding capital is node-based: it does not depend on the density of the components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Self-consistency states that a node’s capital increases with the capital of its neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' By the Monotonicity axiom, edges incident to a node contribute to its capital, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=', the corresponding capital is locally edge-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The conjunction of the Self-consistency and Monotonicity makes this impact of edges global.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' As a result, this conjunction turns out to be incompatible (under Equivalence) with the node-based Bridge axiom (Proposition 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Similarly, by the same proposition, the Bridge axiom is incompatible with the Transit monotonicity axiom, which postulates the edge nature of the capital globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 17 5 4 3 2 05 4 3 2 0 GoAn additional subject of this paper is the properties of the PageRank centrality measure related to the main topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' It turns out that this measure violates most of the conditions we consider and even has a property that, according to some authors, “is hard to imag- ine” for a measure of centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The reason for this is the stochastic normalization used in PageRank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In the path graph 1—2—3—4—5 used in Proposition 6, nodes 2 and 4 have maximum PageRank centrality as they are linked to the leaves: these links receive a maxi- mum weight of 1, since normalization does not change them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' This maximum weight can be interpreted as the specific importance of these links for the leaves, and not for the nodes 2 and 4, which profit from this weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' It is this counterintuitive normalization that violates the RT condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' The axioms of Self-consistency and Bridge are quite strong, so the adoption of either of them dramatically reduces the set of centrality measures under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' This fact is used in [1], where the “culling” method for determining the most appropriate centrality measure is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' This method consists in compiling and completing a survey that allows the user to find a measure that matches their underlying concept of centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In the framework of this method, adopting a certain axiom results in compiling a shorter survey on the set of measures that satisfy this axiom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' In [1], the surveys reduced to the measures obeying the Self-consistency or Bridge axioms are shown in Figures 7 and 8, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Acknowledgement The author thanks Anna Khmelnitskaya and Dmitry Gubanov for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' References [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Chebotarev, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Gubanov, How to choose the most appropriate centrality measure?' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' Chebotarev, Aggregation of preferences by the generalized row sum method, Mathemat- ical Social Sciences 27 (1994) 293–320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} +page_content=' 21' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/D9AyT4oBgHgl3EQfSPfV/content/2301.00084v1.pdf'} diff --git a/DNAyT4oBgHgl3EQfefhw/content/tmp_files/2301.00322v1.pdf.txt b/DNAyT4oBgHgl3EQfefhw/content/tmp_files/2301.00322v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a52ec76b974f77ff51178e5a52742527d2d16287 --- /dev/null +++ b/DNAyT4oBgHgl3EQfefhw/content/tmp_files/2301.00322v1.pdf.txt @@ -0,0 +1,871 @@ +Encrypted Data-driven Predictive Cloud Control with Disturbance +Observer +Qiwen Li, Runze Gao and Yuanqing Xia∗ +Abstract— In data-driven predictive cloud control tasks, the +privacy of data stored and used in cloud services could be leaked +to malicious attackers or curious eavesdroppers. Homomorphic +encryption technique could be used to protect data privacy +while allowing computation. However, extra errors are intro- +duced by the homomorphic encryption extension to ensure the +privacy-preserving properties, and the real number truncation +also brings uncertainty. Also, process and measure noise existed +in system input and output may bring disturbance. In this work, +a data-driven predictive cloud controller is developed based +on homomorphic encryption to protect the cloud data privacy. +Besides, a disturbance observer is introduced to estimate and +compensate the encrypted control signal sequence computed in +the cloud. The privacy of data is guaranteed by encryption and +experiment results show the effect of our cloud-edge cooperative +design. +Index Terms— Cloud Control Systems, Data-Driven Predic- +tive Control, Disturbance Observer, Homomorphic Encryption. +I. INTRODUCTION +Cloud computing provides enormous computing and stor- +age resources for the implementation of control applications, +which brings the concept of cloud control systems (CCSs) +[1]–[3]. In CCSs, control algorithms are outsourced and +executed on cloud platforms to offer control services for +local plants. With the development of CCSs, there is an +emerging requirement of cloud control for complex systems. +However, the complexity and scale of control systems bring +new difficulty in designing model-based cloud control laws, +since system models are difficult to obtain. As a kind of +model-free control approach, data-driven predictive control +(DPC) [4] directly computes control sequences based on the +input-output data of the system, which avoids the process +of system modeling. Therefore, the combination of CCSs +and DPC, i.e., data-driven predictive cloud control (DPCC) +[5]–[7], takes advantage of data storage and computation in +the cloud, as well as the model-free manner in control of +complex systems, becoming a potential candidate in CCSs. +However, in DPCC scenarios, the input-output data and +control law of systems are stored and computed in the +cloud with no data privacy protection, leading to the risk +of privacy leakage. To be specific, an eavesdropper could +get access to the private system data through communication +channel, cloud storage and memory. The eavesdropper could +consequently infer the state and model of the system for +malicious purposes, such as advanced persistent threat (APT) +Q. Li, R. Gao and Y. Xia are with School of Automation, Beijing +Institute of Technology, Beijing 100081, P. R. China. (Corresponding +author: Yuanqing Xia). E-mail address: penguinlee@bit.edu.cn (Q. Li), +runze gao@bit.edu.cn (R. Gao), xia yuanqing@bit.edu.cn (Y. Xia). +design and system state tracking. Thus, the privacy issues in +DPCC should be seriously considered. +As a solution, we use homomorphic encryption (HE) +approaches to protect data privacy while computing the +DPCC control law, since HE schemes allow computations +on encrypted data. Specifically, we use CKKS scheme [8], +which is a RLWE-based HE protocol that ensures the privacy +of the scheme through introducing errors to satisfy the +hardness of the RLWE problem. In CKKS scheme, complex- +number vectors are mapped to integer-coefficient polyno- +mials through interpolation, amplification and truncation. +Consequently, the addition and multiplication of ciphertext +in polynomials are homomorphically equivalent to element- +wise addition and multiplication of plaintext in vectors. In +this work, we design a privacy-preserving DPCC controller +based on CKKS scheme to compute control sequences while +keeping system information invisible to potential attackers. +When performing the privacy-preserving DPCC tasks +described above, we should consider the effects on the +control quality induced by system noise and uncertainty. +Firstly, errors are introduced to the privacy-preserving DPCC +procedure through HE scheme. To be specific, errors are +introduced to public keys in CKKS scheme to protect the +semantic security properties. Moreover, the amplification and +truncation procedure bring noises into ciphertexts. Besides, +measurement noise, process noise and system uncertainty are +ubiquitous in control systems, which consequently influence +the control effect of data-driven approaches. +Hence, disturbance observer (DOB) [5], [9], [10] is used +to guarantee the control accuracy under the uncertainty, +including system noise and errors induced by HE scheme. +The function of DOB is to estimate the effects performed +on a system based on an auxiliary system. If estimated, the +system uncertainty could be properly compensated with a +suitable magnitude. +Motivated by the above reasons, the main contributions of +the privacy-preserving DPCC based on HE scheme are listed +as follows: +• We design a private DPCC protocol based on CKKS +scheme, which preserves the privacy of sensitive system +input-output data. +• We apply the DOB technique to estimate and com- +pensate for the uncertainty induced by the HE scheme +and system noise under the privacy-preserving DPCC +scenario. +• A numerical example shows the effectiveness of +privacy-preserving DPCC with DOB, compared to un- +encrypted non-DOB and encrypted non-DOB condi- +arXiv:2301.00322v1 [eess.SY] 1 Jan 2023 + +tions. +The remainder of this work is shown as follows. DPCC +approaches and their privacy issues are briefly surveyed in +Section II, based on which we develop a privacy-preserving +data-driven control protocol in Section III. In Section IV, +a disturbance observer is proposed to compensate for the +error induced by encryption and data noise. In Section V +a numerical example of our proposed method is shown to +demonstrate its effectiveness. Section VI concludes the paper. +II. RELATED WORKS +Showing potential in model-free control scenarios, DPC +approaches compute the control input directly from the input- +output data of the system, and have been widely used in +extended situations. [11] propose a model-free approach for +linear parameter-varying systems. A data-driven error model +is learned with precollected data in [12] to achieve accurate +position tracking with a robot arm. +DPC approaches may require extensive data to estimate +system models or generate control inputs, in which cases the +computation time of system input may become the bottleneck +of implementation. Thus cloud computing and distributed +computing are gathering more and more attention in DPC +tasks for the possibility of computation acceleration by prop- +erly utilizing elastic resources in the cloud. [6], [7] develop a +cloud-edge-endpoint DPC prototype, showing the feasibility +of cloud-based control systems. To optimize the effort of +subspace identification task, which is the basis of data-driven +control, [13] decomposes the identification algorithm to inter- +connected containerized tasks through parallel computing. +A further implementation of cloud-edge cooperative DPCC +[5] uses workflow-based parallel cloud control and edge +compensation. +The privacy of data and models could be leaked through +outsourced tasks, since the communication channel and +execution environment could be eavesdropped by untrusted +third-parties. Therefore, encrypted control approaches have +been widely studied since it could simultaneously allow +the computation of control signals and preservation of data +privacy. Encrypted linear feedback controllers are realized +in [14]. Moreover, the encrypted realization of more ef- +ficient and complex control schemes are proposed to fit +integrated cloud scenarios. In [15], a privacy-preserving sub- +space identification approach based on partially HE scheme +is proposed. Alexandru et al. [16] offer offline and online +encrypted cloud control designs, both based on HE, to +protect the input-output data of DPC based on a single cloud +server. Subsequently a privacy-preserving distributed alter- +nating direction method of multipliers approach is designed +to perform the system estimation process in ciphertexts [17]. +III. PRELIMINARIES +In this section, we sketch the preliminaries of DPC and +RLWE-based HE. +A. Implementation of data-driven predictive control +We consider a state-space expression of discrete linear +time-invariant (LTI) system: +x(k + 1) =Ax(k) + Bu(k) + ϵp, +y(k) =Cx(k) + ϵs, +(1) +where x(k) ∈ Rn, u(k) ∈ Rm, y(k) ∈ Rp are the state, +input and output vector of the system, ϵp, ϵs are process +noise and measure noise of suitable shapes, respectively. In +the following statements, vectors are all viewed as column +vectors, except for additional specifications. +In DPC, we cannot access the specific parameter A, B +and C mentioned in (1). Therefore, data-driven approaches +are used to infer the system information and perform control +task. Specifically, we have the input-output data series of the +system through time: +{u(n), y(n), n = 1, 2, ..., T}. +At every time step k, we use some slices of the input- +output data series as prior information of the system for +identification, which are denoted as: +uf(k) = +� +���� +u(k) +u(k + 1) +... +u(k + N − 1) +� +���� , yf(k) = +� +���� +y(k) +y(k + 1) +... +y(k + N − 1) +� +���� , +up(k) = +� +���� +u(k − N) +u(k − N + 1) +... +u(k − 1) +� +���� , yp(k) = +� +���� +y(k − N) +y(k − N + 1) +... +y(k − 1) +� +���� , +(2) +and +vp(k) = +� yp(k) +up(k) +� +, +(3) +where the subscript ”p” and ”f” indicate ”past” and ”future”, +respectively. +Based on the slices shown above, we can fit the implicit +system expression with linear regression: +yf(k) = Lvvp(k) + Luuf(k) + e(k), +(4) +where Lv and Lu are coefficient matrices to be fit with +appropriate shapes that contain system information, e(k) is +a noise vector. +Aiming at sufficiently utilizing prior information, we con- +catenate the slices of data into the form of Hankel matrix: +Uf(k) = [uf(N) uf(N + 1) · · · uf(N + j − 1)], +(5) +Yf(k) = [yf(N) yf(N + 1) · · · yf(N + j − 1)], +(6) +Vp(k) = [vp(N) vp(N + 1) · · · vp(N + j − 1)]. +(7) +Thus the linear regression problem (4) can be viewed as: +Yf(k) = LvVp(k) + LuUf(k) + E(k). +(8) + +After solving this linear regression problem, i.e. Lv, Lu +being obtained, we consider an optimal control problem with +the loss function +J = (rf(k) − yf(k))⊤Q(rf(k) − yf(k)) + uf(k)⊤Ruf(k), +(9) +where Q and R are positive-definite matrices of appropriate +shapes, rf is the reference signal. Problem (9) could be +solved by taking derivative of J with respect to uf after +substituting (4) to (9): +uf(k) = (R + L⊤ +u QLu)−1L⊤ +u Q(rf − Lvvp(k)), +(10) +where uf(k) is a sequence of predicted control signals. +B. Lattice-based HE +HE schemes enable addition and/or multiplication on en- +crypted data, which is ensured by a homomorphism between +ciphertext space and plaintext space [18]. HE schemes can +be divided into three categories [16]: partially HE, somewhat +HE and fully HE. Partially HE schemes only support addition +or multiplication. Levelled or somewhat HE schemes extend +the functionality of partially HE and enable both addition and +multiplication, with limited times of computation. Fully HE +schemes allow infinite times of addition and multiplication, +thus support evaluating arbitrary computable functions. Some +levelled HE schemes could be converted to fully HE schemes +with the use of a refresh algorithm called bootstrapping [19]. +In this work, we use CKKS scheme [8], [19], a typical +public key encryption scheme which is levelled homomor- +phic on complex vectors. CKKS scheme supports addition, +finite times element-wise multiplication on real vectors, to +protect the privacy of data-driven control. Besides, CKKS +scheme utilizes key-switching technique to support advanced +operation like element-wise vector rotation and relineariza- +tion after multiplication. Also, CKKS scheme supports ci- +phertext rescaling to control the noise expansion caused by +specific operations. +A brief description of CKKS scheme is shown in Fig. +1. Denote N be power of 2 and QL be a big modulus +that equals to the product of a series of positive integers +{q0, q1, ..., qL}. In CKKS scheme, a complex vector m with +at most N/2 elements is interpolated into a polynomial. +Then the embedded polynomial is multiplied by a large +scaling factor ∆ and truncated to get plaintext p, which is a +polynomial in ZQL [X] /(XN + 1), for further encryption. +Vector +Plaintext Polynomial +Ciphertext +Vector +Plaintext Polynomial +Ciphertext +Interpolation +Evaluation +Encryption +Decryption +Addition +Multiplication +Rotation +…… +�/� +�� +� +� +� +�� +� +� +� +� +�� +� +� +�� +� +�/� +Fig. 1. +A brief description of CKKS scheme. +The plaintext p will be encrypted into the form of cipher- +text c = (c0, c1) such that c0 + c1s = p + e (mod Ql), +where s is the secret key and e is the error. Here, ciphertext +c ∈ Z2 +Ql [X] /(XN + 1) is denoted to be at level l with +Ql = �l +i=0 qi for l = 1, ..., L + 1. The plaintext p could be +encrypted both by the secret key s and the public key but +could be only decrypted with the secret key. The security +properties of CKKS scheme are ensured by the hardness of +the RLWE problem [18]. Specifically, all the public keys +are in the form of RLWE example (−as + e, a), where +random polynomial a and error e safely seal the secret +key s according to the hardness of the RLWE problem. +Besides, extra public keys in CKKS scheme are available to +perform advanced operations like relinearization and rotation +to support the design of elaborated computations. +The noise bound in ciphertexts explodes when performing +multiple homomorphic multiplications since the noise is +exponentially amplified by the extra scaling factor ∆. As +shown in Fig. 2, the multiplication result c at level l could +be rescaled by dividing ql, and the level is consequently +reduced to l − 1. Therefore, the noise bound explosion +could be reduced to linear expansion, which allows more +multiplications to be performed. +� +� +��� +� +Multiplication +& +Relinearization +Rescalation +Fig. 2. +Illustrated procedure of the scale limitation in CKKS scheme. +IV. PRIVACY-PRESERVING DPCC DESIGN WITH +DOB +In DPCC scenarios, we assume that the public cloud envi- +ronment and potential eavesdroppers are honest but curious, +which means that they will perform the specified compu- +tation or communication correctly, but they want to access +the system information to infer the current state and system +dynamics. Therefore, the untrusted part placed in the cloud +should be encrypted. In this process, the encryption module +may introduce new uncertainty. Based on this consideration, +the DOB-based privacy-preserving DPCC solution requires +the cooperation of three general components: public cloud, +trustable edge and plant, respectively. The system design +is shown in Fig. 3. In the public cloud, an encrypted con- +troller is deployed, maintaining some encrypted matrices to +compute encrypted control input sequences. On the trustable +edge platform, the HE module is equipped to encrypt and +decrypt data, along with a DOB to perform control signal +compensation. The plant feeds the modified control input +to the system and returns the current output to the edge +side. The encrypted data in the cloud controller could be +periodically updated to fit the current system dynamics. + +Edge +Plant +Public Cloud +Encrypted Data +Predicted control +inputs (encrypted) +Historical information +(encrypted) +System Dynamics +HE Module +DOB-based Compensator +Encryption +Decryption +Compensated inputs +Trustable +Untrustable +Fig. 3. +Design of privacy-preserving DPCC. +A. Privacy-preserving DPC +The privacy of the system behavior, including input-output +data, should be protected. Similar to [16], an offline privacy- +preserving DPC solution is introduced based on CKKS +homomorphic encryption scheme. +We could observe that the computation of (10) is realized +by specified matrix-vector multiplications. In practice, denote +matrix Mr := (R + L⊤ +u QLu)−1L⊤ +u Q and Mv := (R + +L⊤ +u QLu)−1L⊤ +u QLv, which are 2 terms in (10). Since we +could compute Lv and Lu in advance, Mr and Mv could +be consequently computed offline on a trustable platform, +which could be encrypted and uploaded to the cloud, then +updated periodically. +Then, the cloud receives the ciphertexts of Mr and Mv, +and the control input could be consequently computed: +uf = Mrrf − Mvvp, +(11) +where vp is the same as in (3) and timestamp t is omitted +for convenience. For the efficiency of computation, matrices +Mr and Mv would be reused for a given interval and then +updated, which is a trade-off in the computation overhead. +Consequently, the computing procedure could be reduced +to a matrix-vector multiplication in ciphertext space. Here, +a diagonal computation method is utilized to perform the +computation [19]. To implement the encrypted matrix-vector +computation Mx, the matrix M ∈ RK×L and vector x ∈ RL +should firstly be rewritten in an encryption-friendly way, +which are illustrated in upper part of Fig. 4(a). The modified +matrix Mmod of matrix M and repeated vector xdup = +� +x⊤ x⊤ ... x⊤�⊤ of x are provided, which are encrypted +and sent to the cloud computing component. +Denote the encrypted columns of matrix Mmod ∈ RK×L +as M (i) +mod, and we need to homomorphically compute matrix- +vector multiplication y = Mx in the form of ciphertexts. The +matrix-vector multiplication in ciphertext is shown as below: +y = +L−1 +� +i=0 +M (i) +mod ∗ rot(xdup, i), +(12) +where the function rot(xdup, i) is the rotation operation +supported by the CKKS scheme, meaning that rotating vector +xdup i steps to the left. The computation procedures are +illustrated in Fig. 4(b). +Based on above description, the whole encrypted matrix- +vector computation procedure is described in Algorithm 1. +Algorithm 1 Encryption-friendly matrix-vector multiplica- +tion. +Input: Matrix M ∈ Rm×n, vector x ∈ Rn. +Output: Encrypted result of Mx. +1: Initialization: build a full zero matrix Mmod with the +same shape as M. +2: for i := 0 to n − 1 do +3: +for j := 0 to m − 1 do +4: +Mmod[j][i] = M[j][(i + j) mod n]. +5: +end for +6: end for +7: xdup := Encryption of +� +x⊤ x⊤ ... x⊤�⊤. +8: M (0) +mod, ... M (n−1) +mod +:= Encryption of Mmod’s columns +9: Compute matrix-vector multiplication through (12). +B. DOB and DOB-based cooperative control design +As analyzed in III, CKKS scheme introduces error to pro- +tect its security, meanwhile the amplification and truncation +procedures bring error to the system. Besides, the process +and measurement noise may also impact the control effect. +For reducing the uncertainty and disturbance existed in HE +scheme and system dynamics, we adopt the solution in [5], +which uses a cloud-edge cooperative control design with a +data-driven DOB to estimate the uncertainty and disturbance +brought by the cloud. The estimation result obtained by +data-driven DOB could be added to the control input for +compensation with a proper gain. +Assume that only the first term in the decrypted uf is fed +to the system, which is denoted as uc, as the cloud control +signal. We take the nominal input-output relationship into +consideration without noise and disturbance: +ˆy(k + 1) = +N +� +i=1 +ˆgiy(k + i − N) ++ +N +� +i=1 +ˆhiu(k + i − N) + ˆb(k)uc(k + 1), +(13) +where ˆgi and ˆhis form the first block row of ˆ +Lv and ˆ +Lu, +i.e. the disturbed term of Lv and Lu, respectively. (13) is +actually the first p rows of the HE implementation of (4). +If uncertainty and disturbance are considered, the real +system dynamics should be: +y(k + 1) = +N +� +i=1 +ˆgiy(k + i − N) ++ +N +� +i=1 +ˆhiu(k + i − N) + ˆbuc(k) + ˆb(k)d(k), +(14) +where d(k) = ∆u(k) is the input disturbance. +Then, a DOB is introduced with the form +ˆd(k) = P(k) + Ky(k), +(15) +where the disturbance d(k) is estimated by ˆd(k), K is the +observer amplification matrix to be designed, and P(k) is an + +Repeat & Concatenate +Duplicate +Reform +xdup +Mmod +M0,0 +M0,1 +M1,0 +M1,1 +M2,0 +M2,1 +M3,0 +M3,1 +M0,2 +M1,2 +M2,2 +M3,2 +M0,0 +M0,1 +M1,0 +M1,1 +M2,0 +M2,1 +M3,0 +M3,1 +M0,2 +M1,2 +M2,2 +M3,2 +M0,0 +M0,1 +M1,0 +M1,1 +M2,0 +M2,1 +M3,0 +M3,1 +M0,2 +M1,2 +M2,2 +M3,2 +M0,0 +M0,1 +M1,1 +M1,2 +M2,2 +M2,0 +M3,0 +M3,1 +M0,2 +M1,0 +M2,1 +M3,2 +x0 +x1 +x2 +x0 +x1 +x2 +x0 +x1 +x2 +x0 +x1 +x2 +(a) Reformation of matrix and vector. +Mul +Mul +Mul +Sum +Rotate(1) +Rotate(1) +M0,0 +M0,1 +M1,1 +M1,2 +M2,2 +M2,0 +M3,0 +M3,1 +M0,2 +M1,0 +M2,1 +M3,2 +* +* +* +* +* +* +x2 +x0 +x1 +x2 +* +* +x0 +x1 +x2 +x0 +x1 +x2 +x1 +x2 +x0 +x1 +x2 +* +(Mx)0,1,2,3 +* +(b) Matrix-vector multiplication procedure. +Fig. 4. +Encryption-friendly matrix-vector multiplication: an illustrative example. +auxiliary vector which is updated as below: +P(k + 1) = −K( +N +� +i=1 +ˆgi(k)y(k + i − N) ++ +N +� +i=1 +ˆhi(k)u(k + i − N) ++ˆbuc(k) + ˆb ˆd(k)). +(16) +From (16), one can obtain +ˆd(k + 1) = Kˆb(d(k) − ˆd(k)). +(17) +Now, define the estimation error as ∆d(k) = d(k) − ˆd(k) +and we have the residue system: +∆d(k + 1) = −Kˆb∆d(k) + d(k + 1). +(18) +In this system, the edge-compensated input ue is added to +the cloud control signal uc, i.e. u = uc+ue, to get the DPCC +cloud-edge co-design. Since the uncertainty caused by HE is +viewed as a part of input disturbance, ue is designed to be +ue(k) = − ˆd(k), +(19) +and +ˆd(k) = K +� +y(k) − +N +� +i=1 +ˆgi(k − 1)y(k − N + i − 1) +− +N−1 +� +i=0 +ˆhi(k − 1)u(k − N + i − 1) +− ˆb(k − 1)uc(k − 1) +� +(20) +when k = N + 1, N + 2, .... +When k = 1, 2, ..., N, the DOB-based edge compensator +do not have enough data in the DPC stage, and ue could be +set to 0 in this time interval, i.e. u = uc. +V. NUMERICAL EXAMPLES +We consider a typical 2-order discrete LTI system control +problem with parameters +A = +�2 +−1 +1 +0 +� +, +(21) +B = +�1 +0 +� +, +(22) +and +C = +�0.00014 +0.00014� +. +(23) +The control input u is clipped between -0.15 and 0.15, +and the measure output y is clipped between 0 and 0.4. +The system parameters are: N = 20, j = 1000, K = 62, +λ = 0.009. The system state is initialized at [0 0]⊤ and the +whole control procedure is divided into 2 stages, i.e. data +precollection stage and data-driven control stage. In the data +precollection stage, the system is controlled through a PID +controller with Kp = Kd = 9 and Ki = 3. The control +reference is yr = 0.2 in the first 2N + j = 1040 steps. +In the data-driven control stage, Lw and Lu are computed +and updated periodically every 50 iterations based on newly +collected data. In this stage, the control reference is set to +0.1. +The whole experiment is realized in a standard Hyper +Elastic Cloud Server (HECS) in Huawei Cloud with 2GB +RAM and 1 CPU. We implement the private-preserving +part of the whole algorithm using the RLWE-based HE +library Microsoft SEAL [20]. The security parameter λ is +chosen to be 128-bit, meaning an encryption scheme could +be infiltrated with a probability of 2−128. The ring dimension +is chosen to be 4096, which controls the packing capability +of vectors and multiplication depth. The truncation error, +which is related to the scaling factor and modulus bits, +influences the effect of control. The scaling factor determines +the multiplication level, which is bounded by the 128-bit +security requirement. The multiplication depth is chosen to +be 2, since in this experiment only one multiplication depth is +performed in each step. The scaling factor of CKKS scheme +is chosen to be 222 and 225, based on which the influence of +floating point number truncation is researched. The process +noise and measurement noise are set to be Gaussian with the +variance of 0.0027. +The experiment is performed to show the control effect of +the privacy-preserving DPCC with a DOB-based compen- +sator in three circumstances for comparison, i.e. data-driven +control in plaintext, data-driven control in ciphertext with +and without DOB-based compensator. +The experimental results are illustrated in Fig. 5(a) and +Fig. 5(b). As shown in these figures, the DOB-based com- +pensator effectively removes the error induced by system +uncertainty, encryption error and external noise. Specifically, +in Fig 5(a), the scaling factor is set to be 222, i.e. about + +0 +250 +500 +750 +1000 1250 1500 1750 2000 2250 2500 +Time Step +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +Output +Unencrypted without DOB +Encrypted without DOB +Encrypted with DOB +Switching line +DPC Reference +(a) Control results with 22-bit scaling factor. +0 +250 +500 +750 +1000 1250 1500 1750 2000 2250 2500 +Time Step +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +Output +Unencrypted without DOB +Encrypted without DOB +Encrypted with DOB +Switching line +DPC Reference +(b) Control results with 25-bit scaling factor. +Fig. 5. +Simulation results of the privacy-preserving DPCC. +4 million, which truncates too much information from the +plaintext such that compromises the system performance. +The system is out of control without compensation. In +contrast, DOB-based compensator successfully compensates +the uncertainty and disturbance, which improves the control +quality. In Fig 5(b), the scaling factor is 8 times bigger +than 222, reducing the truncation error by 8 times, which +leads to a similar performance compared to the unencrypted +and uncompensated benchmark. In this case, the uncertainty +mainly appears in encryption and noise, which could be well +estimated and compensated. +VI. CONCLUSION +In this work, we design a privacy-preserving DPCC so- +lution. Based on HE, we implement a privacy-preserving +cloud controller to ensure the data privacy using the CKKS +scheme. Also, the uncertainty and disturbance in HE-based +control systems are considered, a DOB-based compensator +is designed on a trustable edge to estimate and compensate +the uncertainty and disturbance. A numerical example shows +the effect of our proposed privacy-preserving DPCC design. +In the future, the computation efficiency problem of privacy- +preserving cloud control solutions would be studied. +REFERENCES +[1] Y. 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Microsoft Research, Redmond, WA. + diff --git a/DNAyT4oBgHgl3EQfefhw/content/tmp_files/load_file.txt b/DNAyT4oBgHgl3EQfefhw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ee4174bf37989286facc7258d97e4f998e79695d --- /dev/null +++ b/DNAyT4oBgHgl3EQfefhw/content/tmp_files/load_file.txt @@ -0,0 +1,533 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf,len=532 +page_content='Encrypted Data-driven Predictive Cloud Control with Disturbance Observer Qiwen Li, Runze Gao and Yuanqing Xia∗ Abstract— In data-driven predictive cloud control tasks, the privacy of data stored and used in cloud services could be leaked to malicious attackers or curious eavesdroppers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Homomorphic encryption technique could be used to protect data privacy while allowing computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' However, extra errors are intro- duced by the homomorphic encryption extension to ensure the privacy-preserving properties, and the real number truncation also brings uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Also, process and measure noise existed in system input and output may bring disturbance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' In this work, a data-driven predictive cloud controller is developed based on homomorphic encryption to protect the cloud data privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Besides, a disturbance observer is introduced to estimate and compensate the encrypted control signal sequence computed in the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' The privacy of data is guaranteed by encryption and experiment results show the effect of our cloud-edge cooperative design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Index Terms— Cloud Control Systems, Data-Driven Predic- tive Control, Disturbance Observer, Homomorphic Encryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' INTRODUCTION Cloud computing provides enormous computing and stor- age resources for the implementation of control applications, which brings the concept of cloud control systems (CCSs) [1]–[3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' In CCSs, control algorithms are outsourced and executed on cloud platforms to offer control services for local plants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' With the development of CCSs, there is an emerging requirement of cloud control for complex systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' However, the complexity and scale of control systems bring new difficulty in designing model-based cloud control laws, since system models are difficult to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' As a kind of model-free control approach, data-driven predictive control (DPC) [4] directly computes control sequences based on the input-output data of the system, which avoids the process of system modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Therefore, the combination of CCSs and DPC, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=', data-driven predictive cloud control (DPCC) [5]–[7], takes advantage of data storage and computation in the cloud, as well as the model-free manner in control of complex systems, becoming a potential candidate in CCSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' However, in DPCC scenarios, the input-output data and control law of systems are stored and computed in the cloud with no data privacy protection, leading to the risk of privacy leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' To be specific, an eavesdropper could get access to the private system data through communication channel, cloud storage and memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' The eavesdropper could consequently infer the state and model of the system for malicious purposes, such as advanced persistent threat (APT) Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Li, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Gao and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Xia are with School of Automation, Beijing Institute of Technology, Beijing 100081, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' (Corresponding author: Yuanqing Xia).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' E-mail address: penguinlee@bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='cn (Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Li), runze gao@bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='cn (R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Gao), xia yuanqing@bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='cn (Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Xia).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' design and system state tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Thus, the privacy issues in DPCC should be seriously considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' As a solution, we use homomorphic encryption (HE) approaches to protect data privacy while computing the DPCC control law, since HE schemes allow computations on encrypted data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Specifically, we use CKKS scheme [8], which is a RLWE-based HE protocol that ensures the privacy of the scheme through introducing errors to satisfy the hardness of the RLWE problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' In CKKS scheme, complex- number vectors are mapped to integer-coefficient polyno- mials through interpolation, amplification and truncation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Consequently, the addition and multiplication of ciphertext in polynomials are homomorphically equivalent to element- wise addition and multiplication of plaintext in vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' In this work, we design a privacy-preserving DPCC controller based on CKKS scheme to compute control sequences while keeping system information invisible to potential attackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' When performing the privacy-preserving DPCC tasks described above, we should consider the effects on the control quality induced by system noise and uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Firstly, errors are introduced to the privacy-preserving DPCC procedure through HE scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' To be specific, errors are introduced to public keys in CKKS scheme to protect the semantic security properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Moreover, the amplification and truncation procedure bring noises into ciphertexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Besides, measurement noise, process noise and system uncertainty are ubiquitous in control systems, which consequently influence the control effect of data-driven approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Hence, disturbance observer (DOB) [5], [9], [10] is used to guarantee the control accuracy under the uncertainty, including system noise and errors induced by HE scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' The function of DOB is to estimate the effects performed on a system based on an auxiliary system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' If estimated, the system uncertainty could be properly compensated with a suitable magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Motivated by the above reasons, the main contributions of the privacy-preserving DPCC based on HE scheme are listed as follows: We design a private DPCC protocol based on CKKS scheme, which preserves the privacy of sensitive system input-output data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' We apply the DOB technique to estimate and com- pensate for the uncertainty induced by the HE scheme and system noise under the privacy-preserving DPCC scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' A numerical example shows the effectiveness of privacy-preserving DPCC with DOB, compared to un- encrypted non-DOB and encrypted non-DOB condi- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='00322v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='SY] 1 Jan 2023 tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' The remainder of this work is shown as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' DPCC approaches and their privacy issues are briefly surveyed in Section II, based on which we develop a privacy-preserving data-driven control protocol in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' In Section IV, a disturbance observer is proposed to compensate for the error induced by encryption and data noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' In Section V a numerical example of our proposed method is shown to demonstrate its effectiveness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Section VI concludes the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' RELATED WORKS Showing potential in model-free control scenarios, DPC approaches compute the control input directly from the input- output data of the system, and have been widely used in extended situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' [11] propose a model-free approach for linear parameter-varying systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' A data-driven error model is learned with precollected data in [12] to achieve accurate position tracking with a robot arm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' DPC approaches may require extensive data to estimate system models or generate control inputs, in which cases the computation time of system input may become the bottleneck of implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Thus cloud computing and distributed computing are gathering more and more attention in DPC tasks for the possibility of computation acceleration by prop- erly utilizing elastic resources in the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' [6], [7] develop a cloud-edge-endpoint DPC prototype, showing the feasibility of cloud-based control systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' To optimize the effort of subspace identification task, which is the basis of data-driven control, [13] decomposes the identification algorithm to inter- connected containerized tasks through parallel computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' A further implementation of cloud-edge cooperative DPCC [5] uses workflow-based parallel cloud control and edge compensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' The privacy of data and models could be leaked through outsourced tasks, since the communication channel and execution environment could be eavesdropped by untrusted third-parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Therefore, encrypted control approaches have been widely studied since it could simultaneously allow the computation of control signals and preservation of data privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Encrypted linear feedback controllers are realized in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Moreover, the encrypted realization of more ef- ficient and complex control schemes are proposed to fit integrated cloud scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' In [15], a privacy-preserving sub- space identification approach based on partially HE scheme is proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Alexandru et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' [16] offer offline and online encrypted cloud control designs, both based on HE, to protect the input-output data of DPC based on a single cloud server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Subsequently a privacy-preserving distributed alter- nating direction method of multipliers approach is designed to perform the system estimation process in ciphertexts [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' PRELIMINARIES In this section, we sketch the preliminaries of DPC and RLWE-based HE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Implementation of data-driven predictive control We consider a state-space expression of discrete linear time-invariant (LTI) system: x(k + 1) =Ax(k) + Bu(k) + ϵp, y(k) =Cx(k) + ϵs, (1) where x(k) ∈ Rn, u(k) ∈ Rm, y(k) ∈ Rp are the state, input and output vector of the system, ϵp, ϵs are process noise and measure noise of suitable shapes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' In the following statements, vectors are all viewed as column vectors, except for additional specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' In DPC, we cannot access the specific parameter A, B and C mentioned in (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Therefore, data-driven approaches are used to infer the system information and perform control task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Specifically, we have the input-output data series of the system through time: {u(n), y(n), n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=', T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' At every time step k, we use some slices of the input- output data series as prior information of the system for identification, which are denoted as: uf(k) = � ���� u(k) u(k + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' u(k + N − 1) � ���� , yf(k) = � ���� y(k) y(k + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' y(k + N − 1) � ���� , up(k) = � ���� u(k − N) u(k − N + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' u(k − 1) � ���� , yp(k) = � ���� y(k − N) y(k − N + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' y(k − 1) � ���� , (2) and vp(k) = � yp(k) up(k) � , (3) where the subscript ”p” and ”f” indicate ”past” and ”future”, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Based on the slices shown above, we can fit the implicit system expression with linear regression: yf(k) = Lvvp(k) + Luuf(k) + e(k), (4) where Lv and Lu are coefficient matrices to be fit with appropriate shapes that contain system information, e(k) is a noise vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Aiming at sufficiently utilizing prior information, we con- catenate the slices of data into the form of Hankel matrix: Uf(k) = [uf(N) uf(N + 1) · · · uf(N + j − 1)], (5) Yf(k) = [yf(N) yf(N + 1) · · · yf(N + j − 1)], (6) Vp(k) = [vp(N) vp(N + 1) · · · vp(N + j − 1)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' (7) Thus the linear regression problem (4) can be viewed as: Yf(k) = LvVp(k) + LuUf(k) + E(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' (8) After solving this linear regression problem, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Lv, Lu being obtained, we consider an optimal control problem with the loss function J = (rf(k) − yf(k))⊤Q(rf(k) − yf(k)) + uf(k)⊤Ruf(k), (9) where Q and R are positive-definite matrices of appropriate shapes, rf is the reference signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Problem (9) could be solved by taking derivative of J with respect to uf after substituting (4) to (9): uf(k) = (R + L⊤ u QLu)−1L⊤ u Q(rf − Lvvp(k)), (10) where uf(k) is a sequence of predicted control signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Lattice-based HE HE schemes enable addition and/or multiplication on en- crypted data, which is ensured by a homomorphism between ciphertext space and plaintext space [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' HE schemes can be divided into three categories [16]: partially HE, somewhat HE and fully HE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Partially HE schemes only support addition or multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Levelled or somewhat HE schemes extend the functionality of partially HE and enable both addition and multiplication, with limited times of computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Fully HE schemes allow infinite times of addition and multiplication, thus support evaluating arbitrary computable functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Some levelled HE schemes could be converted to fully HE schemes with the use of a refresh algorithm called bootstrapping [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' In this work, we use CKKS scheme [8], [19], a typical public key encryption scheme which is levelled homomor- phic on complex vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' CKKS scheme supports addition, finite times element-wise multiplication on real vectors, to protect the privacy of data-driven control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Besides, CKKS scheme utilizes key-switching technique to support advanced operation like element-wise vector rotation and relineariza- tion after multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Also, CKKS scheme supports ci- phertext rescaling to control the noise expansion caused by specific operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' A brief description of CKKS scheme is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Denote N be power of 2 and QL be a big modulus that equals to the product of a series of positive integers {q0, q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=', qL}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' In CKKS scheme, a complex vector m with at most N/2 elements is interpolated into a polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Then the embedded polynomial is multiplied by a large scaling factor ∆ and truncated to get plaintext p, which is a polynomial in ZQL [X] /(XN + 1), for further encryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Vector Plaintext Polynomial Ciphertext Vector Plaintext Polynomial Ciphertext Interpolation Evaluation Encryption Decryption Addition Multiplication Rotation ……' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' �/� �� � � � �� � � � � �� � � �� � �/� Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' A brief description of CKKS scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' The plaintext p will be encrypted into the form of cipher- text c = (c0, c1) such that c0 + c1s = p + e (mod Ql), where s is the secret key and e is the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Here, ciphertext c ∈ Z2 Ql [X] /(XN + 1) is denoted to be at level l with Ql = �l i=0 qi for l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=', L + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' The plaintext p could be encrypted both by the secret key s and the public key but could be only decrypted with the secret key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' The security properties of CKKS scheme are ensured by the hardness of the RLWE problem [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Specifically, all the public keys are in the form of RLWE example (−as + e, a), where random polynomial a and error e safely seal the secret key s according to the hardness of the RLWE problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Besides, extra public keys in CKKS scheme are available to perform advanced operations like relinearization and rotation to support the design of elaborated computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' The noise bound in ciphertexts explodes when performing multiple homomorphic multiplications since the noise is exponentially amplified by the extra scaling factor ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' 2, the multiplication result c at level l could be rescaled by dividing ql, and the level is consequently reduced to l − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Therefore, the noise bound explosion could be reduced to linear expansion, which allows more multiplications to be performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' � � ��� � Multiplication & Relinearization Rescalation Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Illustrated procedure of the scale limitation in CKKS scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' PRIVACY-PRESERVING DPCC DESIGN WITH DOB In DPCC scenarios, we assume that the public cloud envi- ronment and potential eavesdroppers are honest but curious, which means that they will perform the specified compu- tation or communication correctly, but they want to access the system information to infer the current state and system dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Therefore, the untrusted part placed in the cloud should be encrypted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' In this process, the encryption module may introduce new uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Based on this consideration, the DOB-based privacy-preserving DPCC solution requires the cooperation of three general components: public cloud, trustable edge and plant, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' The system design is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' In the public cloud, an encrypted con- troller is deployed, maintaining some encrypted matrices to compute encrypted control input sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' On the trustable edge platform, the HE module is equipped to encrypt and decrypt data, along with a DOB to perform control signal compensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' The plant feeds the modified control input to the system and returns the current output to the edge side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' The encrypted data in the cloud controller could be periodically updated to fit the current system dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Edge Plant Public Cloud Encrypted Data Predicted control inputs (encrypted) Historical information (encrypted) System Dynamics HE Module DOB-based Compensator Encryption Decryption Compensated inputs Trustable Untrustable Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Design of privacy-preserving DPCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Privacy-preserving DPC The privacy of the system behavior, including input-output data, should be protected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Similar to [16], an offline privacy- preserving DPC solution is introduced based on CKKS homomorphic encryption scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' We could observe that the computation of (10) is realized by specified matrix-vector multiplications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' In practice, denote matrix Mr := (R + L⊤ u QLu)−1L⊤ u Q and Mv := (R + L⊤ u QLu)−1L⊤ u QLv, which are 2 terms in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Since we could compute Lv and Lu in advance, Mr and Mv could be consequently computed offline on a trustable platform, which could be encrypted and uploaded to the cloud, then updated periodically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Then, the cloud receives the ciphertexts of Mr and Mv, and the control input could be consequently computed: uf = Mrrf − Mvvp, (11) where vp is the same as in (3) and timestamp t is omitted for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' For the efficiency of computation, matrices Mr and Mv would be reused for a given interval and then updated, which is a trade-off in the computation overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Consequently, the computing procedure could be reduced to a matrix-vector multiplication in ciphertext space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Here, a diagonal computation method is utilized to perform the computation [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' To implement the encrypted matrix-vector computation Mx, the matrix M ∈ RK×L and vector x ∈ RL should firstly be rewritten in an encryption-friendly way, which are illustrated in upper part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' The modified matrix Mmod of matrix M and repeated vector xdup = � x⊤ x⊤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' x⊤�⊤ of x are provided, which are encrypted and sent to the cloud computing component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Denote the encrypted columns of matrix Mmod ∈ RK×L as M (i) mod, and we need to homomorphically compute matrix- vector multiplication y = Mx in the form of ciphertexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' The matrix-vector multiplication in ciphertext is shown as below: y = L−1 � i=0 M (i) mod ∗ rot(xdup, i), (12) where the function rot(xdup, i) is the rotation operation supported by the CKKS scheme, meaning that rotating vector xdup i steps to the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' The computation procedures are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Based on above description, the whole encrypted matrix- vector computation procedure is described in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Algorithm 1 Encryption-friendly matrix-vector multiplica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Input: Matrix M ∈ Rm×n, vector x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Output: Encrypted result of Mx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' 1: Initialization: build a full zero matrix Mmod with the same shape as M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' 2: for i := 0 to n − 1 do 3: for j := 0 to m − 1 do 4: Mmod[j][i] = M[j][(i + j) mod n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' 5: end for 6: end for 7: xdup := Encryption of � x⊤ x⊤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' x⊤�⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' 8: M (0) mod, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' M (n−1) mod := Encryption of Mmod’s columns 9: Compute matrix-vector multiplication through (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' DOB and DOB-based cooperative control design As analyzed in III, CKKS scheme introduces error to pro- tect its security, meanwhile the amplification and truncation procedures bring error to the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Besides, the process and measurement noise may also impact the control effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' For reducing the uncertainty and disturbance existed in HE scheme and system dynamics, we adopt the solution in [5], which uses a cloud-edge cooperative control design with a data-driven DOB to estimate the uncertainty and disturbance brought by the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' The estimation result obtained by data-driven DOB could be added to the control input for compensation with a proper gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Assume that only the first term in the decrypted uf is fed to the system, which is denoted as uc, as the cloud control signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' We take the nominal input-output relationship into consideration without noise and disturbance: ˆy(k + 1) = N � i=1 ˆgiy(k + i − N) + N � i=1 ˆhiu(k + i − N) + ˆb(k)uc(k + 1), (13) where ˆgi and ˆhis form the first block row of ˆ Lv and ˆ Lu, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' the disturbed term of Lv and Lu, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' (13) is actually the first p rows of the HE implementation of (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' If uncertainty and disturbance are considered, the real system dynamics should be: y(k + 1) = N � i=1 ˆgiy(k + i − N) + N � i=1 ˆhiu(k + i − N) + ˆbuc(k) + ˆb(k)d(k), (14) where d(k) = ∆u(k) is the input disturbance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' a DOB is introduced with the form ˆd(k) = P(k) + Ky(k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' (15) where the disturbance d(k) is estimated by ˆd(k),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' K is the observer amplification matrix to be designed,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' and P(k) is an Repeat & Concatenate Duplicate Reform xdup Mmod M0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='0 M0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='1 M1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='0 M1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='1 M2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='0 M2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='1 M3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='0 M3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='1 M0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='2 M1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='2 M2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='2 M3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='2 M0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='0 M0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='1 M1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='0 M1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='1 M2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='0 M2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='1 M3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='0 M3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='1 M0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='2 M1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='2 M2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='2 M3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='2 M0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='0 M0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='1 M1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='0 M1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='1 M2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='0 M2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='1 M3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='0 M3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='1 M0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='2 M1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='2 M2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='2 M3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='2 M0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='0 M0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='1 M1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='1 M1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='2 M2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='2 M2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='0 M3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='0 M3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='1 M0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='2 M1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='0 M2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='1 M3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='2 x0 x1 x2 x0 x1 x2 x0 x1 x2 x0 x1 x2 (a) Reformation of matrix and vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Mul Mul Mul Sum Rotate(1) Rotate(1) M0,0 M0,1 M1,1 M1,2 M2,2 M2,0 M3,0 M3,1 M0,2 M1,0 M2,1 M3,2 x2 x0 x1 x2 x0 x1 x2 x0 x1 x2 x1 x2 x0 x1 x2 (Mx)0,1,2,3 (b) Matrix-vector multiplication procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Encryption-friendly matrix-vector multiplication: an illustrative example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' auxiliary vector which is updated as below: P(k + 1) = −K( N � i=1 ˆgi(k)y(k + i − N) + N � i=1 ˆhi(k)u(k + i − N) +ˆbuc(k) + ˆb ˆd(k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' (16) From (16), one can obtain ˆd(k + 1) = Kˆb(d(k) − ˆd(k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' (17) Now, define the estimation error as ∆d(k) = d(k) − ˆd(k) and we have the residue system: ∆d(k + 1) = −Kˆb∆d(k) + d(k + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' (18) In this system, the edge-compensated input ue is added to the cloud control signal uc, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' u = uc+ue, to get the DPCC cloud-edge co-design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Since the uncertainty caused by HE is viewed as a part of input disturbance, ue is designed to be ue(k) = − ˆd(k), (19) and ˆd(k) = K � y(k) − N � i=1 ˆgi(k − 1)y(k − N + i − 1) − N−1 � i=0 ˆhi(k − 1)u(k − N + i − 1) − ˆb(k − 1)uc(k − 1) � (20) when k = N + 1, N + 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='. When k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=', N, the DOB-based edge compensator do not have enough data in the DPC stage, and ue could be set to 0 in this time interval, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' u = uc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' NUMERICAL EXAMPLES We consider a typical 2-order discrete LTI system control problem with parameters A = �2 −1 1 0 � , (21) B = �1 0 � , (22) and C = �0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='00014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='00014� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' (23) The control input u is clipped between -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='15 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='15, and the measure output y is clipped between 0 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' The system parameters are: N = 20, j = 1000, K = 62, λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' The system state is initialized at [0 0]⊤ and the whole control procedure is divided into 2 stages, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' data precollection stage and data-driven control stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' In the data precollection stage, the system is controlled through a PID controller with Kp = Kd = 9 and Ki = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' The control reference is yr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='2 in the first 2N + j = 1040 steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' In the data-driven control stage, Lw and Lu are computed and updated periodically every 50 iterations based on newly collected data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' In this stage, the control reference is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' The whole experiment is realized in a standard Hyper Elastic Cloud Server (HECS) in Huawei Cloud with 2GB RAM and 1 CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' We implement the private-preserving part of the whole algorithm using the RLWE-based HE library Microsoft SEAL [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' The security parameter λ is chosen to be 128-bit, meaning an encryption scheme could be infiltrated with a probability of 2−128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' The ring dimension is chosen to be 4096, which controls the packing capability of vectors and multiplication depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' The truncation error, which is related to the scaling factor and modulus bits, influences the effect of control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' The scaling factor determines the multiplication level, which is bounded by the 128-bit security requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' The multiplication depth is chosen to be 2, since in this experiment only one multiplication depth is performed in each step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' The scaling factor of CKKS scheme is chosen to be 222 and 225, based on which the influence of floating point number truncation is researched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' The process noise and measurement noise are set to be Gaussian with the variance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='0027.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' The experiment is performed to show the control effect of the privacy-preserving DPCC with a DOB-based compen- sator in three circumstances for comparison, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' data-driven control in plaintext, data-driven control in ciphertext with and without DOB-based compensator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' The experimental results are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' 5(a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' 5(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' As shown in these figures, the DOB-based com- pensator effectively removes the error induced by system uncertainty, encryption error and external noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Specifically, in Fig 5(a), the scaling factor is set to be 222, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' about 0 250 500 750 1000 1250 1500 1750 2000 2250 2500 Time Step 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='40 Output Unencrypted without DOB Encrypted without DOB Encrypted with DOB Switching line DPC Reference (a) Control results with 22-bit scaling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' 0 250 500 750 1000 1250 1500 1750 2000 2250 2500 Time Step 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='40 Output Unencrypted without DOB Encrypted without DOB Encrypted with DOB Switching line DPC Reference (b) Control results with 25-bit scaling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Simulation results of the privacy-preserving DPCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' 4 million, which truncates too much information from the plaintext such that compromises the system performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' The system is out of control without compensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' In contrast, DOB-based compensator successfully compensates the uncertainty and disturbance, which improves the control quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' In Fig 5(b), the scaling factor is 8 times bigger than 222, reducing the truncation error by 8 times, which leads to a similar performance compared to the unencrypted and uncompensated benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' In this case, the uncertainty mainly appears in encryption and noise, which could be well estimated and compensated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' CONCLUSION In this work, we design a privacy-preserving DPCC so- lution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Based on HE, we implement a privacy-preserving cloud controller to ensure the data privacy using the CKKS scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Also, the uncertainty and disturbance in HE-based control systems are considered, a DOB-based compensator is designed on a trustable edge to estimate and compensate the uncertainty and disturbance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' A numerical example shows the effect of our proposed privacy-preserving DPCC design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' In the future, the computation efficiency problem of privacy- preserving cloud control solutions would be studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' REFERENCES [1] Y.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Techn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=', pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' 360–384, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' [20] “Microsoft seal (release 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='0).” https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content='com/Microsoft/SEAL, Mar 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} +page_content=' Microsoft Research, Redmond, WA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/DNAyT4oBgHgl3EQfefhw/content/2301.00322v1.pdf'} diff --git a/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf b/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..bffa402e18fda7cef8c50a6bca70439727586746 --- /dev/null +++ b/E9FQT4oBgHgl3EQfRDbD/content/2301.13285v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2ea4ee81cff5fca7576352c599312b897201ad2445e6dd33fba5e22e5ed5e160 +size 311059 diff --git a/E9FQT4oBgHgl3EQfRDbD/vector_store/index.faiss b/E9FQT4oBgHgl3EQfRDbD/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..5bb59c3a968b472435994b6898f38c56652d4049 --- /dev/null +++ b/E9FQT4oBgHgl3EQfRDbD/vector_store/index.faiss @@ -0,0 +1,3 @@ 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100644 index 0000000000000000000000000000000000000000..1a7c8eb464d94f956b0834aa94155d5449934690 --- /dev/null +++ b/GdE0T4oBgHgl3EQfhQG7/content/tmp_files/2301.02430v1.pdf.txt @@ -0,0 +1,660 @@ +arXiv:2301.02430v1 [math.GM] 6 Jan 2023 +Some Solitons on Homogeneous Almost +α-Cosymplectic 3-Manifolds and Harmonic +Manifolds +Naeem Ahmad Pundeer, Paritosh Ghosh, Hemangi Madhusudan +Shah and Arindam Bhattacharyya +Abstract. In this paper, we investigate the nature of Einstein solitons, +whether it is steady, shrinking or expanding on almost α-cosymplectic +3-manifolds. We also prove that a simply connected homogeneous al- +most α-cosymplectic 3-manifold, admitting a contact Einstein soliton, +is an unimodular semidirect product Lie group. Finally, we show that a +harmonic manifold admits a Ricci soliton if and only if it is flat. +Mathematics Subject Classification (2010). 53B40, 58B20, 53C25, +53D15. +Keywords. Almost α-cosymplectic manifold, Harmonic manifold, Ricci +soliton, Einstein soliton. +1. Introduction +The study of solitons, in particular Ricci solitons, on Riemannian man- +ifolds play a vital role in understanding the geometry of underlying mani- +fold. It is very interesting to study Ricci and Einstein solitons on almost α- +cosymplectic 3-manifolds. Recently, Jin and Ximin [9] showed that a simply +connected homogeneous almost α-cosymplectic 3-manifold, admitting con- +tact Ricci solitons, is cosymplectic; and the manifold under consideration is +an unimodular semidirect product Lie group R2⋊AR, where A = +� +0 +b +−b +0 +� +, +equipped with a flat left invariant cosymplectic structure. +Motivated by this result we show in this paper that, if a simply con- +nected homogeneous almost α-cosymplectic 3-manifold, with some additional +hypothesis, admits a contact Einstein soliton, then the manifold is an uni- +modular semidirect product Lie group G of type G0bb = R2 ⋊A R, where + +2 +N. A. Pundeer, P. Ghosh, H. M. Shah and A. Bhattacharyya +A = +� +0 +b +−b +0 +� +̸= 0. And also G is the Lie group ˜E2 equipped with its +flat left invariant cosymplectic structure (see Corrollary 3.5). In order to +prove this result, we first obtain a characterization of almost α-cosymplectic +3-manifold admitting contact Einstein solitons, which is the main theorem +(Theorem 3.4) of Section 3. To establish this aforementioned theorem we +derive an identity (Lemma 3.3) involving scalar curvature, Lie derivative of +the metric and Ricci operator on a Riemannian manifold admitting Einstein +soliton. We also give some conditions on α for contact Einstein solitons to +be steady, shrinking or expanding on almost α-cosymplectic 3-manifolds (see +Theorem 3.1). +Another interesting topic in the differential geometry is the geometry +of harmonic manifolds. In 1965, Tashiro [19] showed that if a complete Rie- +mannian manifold admits a Gaussian, then it is either flat or a complete +warped product manifold or a sphere. In this paper, we show that a har- +monic manifold admits a Gaussian if and only if it is flat; thus confirming +Tashiro’s result for harmonic manifolds. Moreover, we prove that flat har- +monic manifold admits Ricci solitons of steady, expanding or shrinking type. +We also determine the corresponding potential function. In fact, Busemann +function on Rn turns to be the potential function in case of steady solitons +(see Theorem 4.1 of Section 4). +The paper is divided into four sections. Section 2 is devoted to the +preliminaries about Ricci soliton, Einstein soliton, almost α-cosymplectic 3- +manifolds and harmonic manifolds. In Section 3, we prove our main results +on almost α-cosymplectic 3-manifold admitting contact Einstein solitons, as +stated above. In the last section, we prove the main flatness result about +harmonic manifolds admitting Ricci solitons. +2. Preliminaries +In this section, we discuss some notions required to prove the results of this +paper. +2.1. Ricci solitons +Ricci solitons are the self similar solutions of the Ricci flow. The concept of +Ricci flow was first introduced by Hamilton [7] in (1982), motivated by the +work of Eells and Sampson [6] on harmonic map and the flow was given by +the equation +∂g +∂t = −2S, +where S is the Ricci tensor. +Ricci solitons are the generalizations of the Einstein metrics and are the +solutions of the equation +Ric(g) + 1 +2LXg = λg, +(1) + +On harmonic and homogeneous almost α-cosymplectic 3-manifolds +3 +where Ric(X, Y ) = S(X, Y ) is the Ricci curvature tensor, LX is the Lie +derivative along the direction of the vector field X and λ is a real constant. +The soliton is said to be shrinking if λ > 0, steady if λ = 0 and expanding if +λ < 0. +Tashiro [15] proved very important result for complete Einstein manifolds +admitting Ricci solitons. +Theorem 2.1. [15] Let (M, g) be a complete Riemannian n-manifold admit- +ting a nontrivial function f such that Hess f = λg, then (M, g) is isometric +to a complete warped product metric and must have one of the three forms: +1. M = R × N, g = dr2 + ρ2(r)gN, +2. M = Rn, g = dr2 + ρ2(r)ds2 +n−1, r ≥ 0, +3. M = Sn, g = dr2 + ρ2(r)ds2 +n−1, r ∈ [a, b]. +2.2. Einstein solitons +The Einstein solitons are the generalization of the Ricci solitons, was first +introduced by Catino and Mazzieri [3] in (2016). They are the solutions of +the equation +LV g + 2S = (2λ + r)g, +(2) +where, Ricci tensor S(X, Y ) = g(X, QY ), Q being the Ricci operator, r is +the scalar curvature, λ ∈ R is a constant and V is known as potential vector +field. +Einstein solitons are the self-similar solutions of the Einstein flow, +∂ +∂tg + 2S = rg. +It is said to be steady if λ = 0, shrinking if λ > 0 and expanding if λ < 0. +2.3. Almost contact metric manifolds +In order to define contact metric manifolds, we need the concept of Reeb +vector field. +Reeb vector field [2]: A global vector field ξ on a contact manifold M 2n+1, +equipped with a global 1-form η, is called Reeb vector field or characteristic +vector field, if any vector field X satisfies η(ξ) = 1 and dη(X, ξ) = 0. +Almost contact manifold [2]: Let M be a Riemannian manifold of di- +mension (2n + 1), n ≥ 1. M 2n+1 is said to have an almost contact structure +(ϕ, ξ, η), if there exists a (1, 1)-tensor ϕ, a global vector field ξ and a 1-form +η such that +ϕ2X = −X + η(X)ξ, η(ξ) = 1, +(3) +for any vector field X on M, where ξ is the Reeb vector field. The manifold +M equipped with the structure (ϕ, ξ, η) is called an almost contact manifold. +Almost contact metric manifold [2]: A Riemannian metric g is said to +be compatible with an almost contact structure (ϕ, ξ, η), if +g(ϕX, ϕY ) = g(X, Y ) − η(X)η(Y ), +(4) + +4 +N. A. Pundeer, P. Ghosh, H. M. Shah and A. Bhattacharyya +holds for any X, Y ∈ χ(M) and (M, ϕ, ξ, η, g) is called an almost contact +metric manifold. +Normal almost contact metric manifold [2]: An almost contact met- +ric manifold is said to be normal, if for any X, Y ∈ χ(M) the tensor field +N = [ϕ, ϕ]+ 2dη ⊗ ξ vanishes everywhere on the manifold, where [ϕ, ϕ] is the +Nijenhuis tensor of ϕ. +Homogeneous almost contact metric manifold [9]: An almost contact +metric manifold (M, ϕ, ξ, η, g) is said to be homogeneous, if there exists a con- +nected Lie group G of isometries acting transitively on M leaving η invariant. +2.4. Cosymplectic manifolds +A (2n + 1)-dimensional manifold is said to be a cosymplectic manifold [10], +if it admits a closed, 1-form η and 2-form Φ such that η ∧ Φn is a volume +element, where Φ(X, Y ) = g(ϕX, Y ) is a 2-form on M 2n+1. +Almost cosymplectic manifold [10]: If η and Φ are not closed but η ∧ Φn +is a volume form, then the manifold is called almost cosymplectic manifold. +α-cosymplectic manifold [12]: An almost cosymplectic manifold is said +to be α-cosymplectic if dη = 0 and dΦ = 2αη ∧ Φ for some constant α. +Almost α-cosymplectic manifold [10]: An almost α-cosymplectic manifold +is defined as an almost contact metric manifold with dη = 0 and dΦ = 2αη∧Φ, +for any constant α. In particular, the almost α-cosymplectic manifold is +• almost α-Kenmotsu if α ̸= 0, +• almost cosymplectic if α = 0, +• almost Kenmotsu if α = 1. +Harmonic vector field [14]: A characteristic vector field ξ on an almost +α-cosymplectic manifold is harmonic if and only if ξ is an eigenvector field +of the Ricci operator Q. +2.5. Almost α-cosymplectic 3-manifold +In this article, we will mainly focus on 3-dimensional almost α-cosymplectic +manifold. In what follows, we will be using the following results. +Theorem 2.2. [12] An almost α-cosymplectic 3-manifold is α-cosymplectic +if and only if Lξh = 0, where h = 1 +2Lξϕ. +Any almost α-cosymplectic 3-manifold satisfies important relationships be- +tween Φ, ξ and h. +Lemma 2.3. [12] Let M 2n+1 be an almost α-cosymplectic 3-manifold, then +we have, +∇ξϕ = 0, ∇ξ = 0, hϕ + ϕh = 0, hξ = 0, +(5) + +On harmonic and homogeneous almost α-cosymplectic 3-manifolds +5 +with +∇Xξ = −αϕ2X − ϕhX. +(6) +We would require some identities on the ϕ-bases [2] and the following table +of the Levi-Civita connection. +Proposition 2.4. [12] On almost α-cosymplectic 3-manifold, there exists +ϕ-bases satisfying +he = σe, hϕe = −σϕe, hξ = 0, +with σ a local smooth eigen-function of h. +Theorem 2.5. [12] The Levi-Civita connection on almost α-cosymplectic +3-manifold are given by, + + + + + +∇ee = −aϕe − αξ, ∇ϕee = −bϕe + σξ, ∇ξe = µϕe, +∇eϕe = ae + σξ, ∇ϕeϕe = be − αξ, ∇ξϕe = −µe, +∇eξ = αe − σϕe, ∇ϕeξ = −σe + αϕe, ∇ξξ = 0, +(7) +where a = g(∇eϕe, e), b = −g(∇ϕee, ϕe) and µ = g(∇ξe, ϕe) are smooth +functions. +The Ricci operator on almost α-cosymplectic 3-manifold is known explicitly +[12]. +Proposition 2.6. [12] The Ricci operator Q on almost α-cosymplectic 3- +manifold is given by, + + + + + +Qξ = −(2α2 + tr h2)ξ + (2bσ − e(σ))ϕe − (2aσ + (ϕe)(σ))e, +Qϕe = (2bσ − e(σ))ξ + (α2 + r +2 + tr h2 +2 ++ 2σµ)ϕe + (ξ(σ) + 2ασ)e, +Qe = −(2aσ + (ϕe)(σ))ξ + (ξ(σ) + 2ασ)ϕe + (α2 + r +2 + tr h2 +2 +− 2σµ)e. +(8) +Furthermore, the scalar curvature r = tr Q is given by +r = −6α2 − tr h2 − 2(a2 + b2) − 2(ϕe)(a) + 2e(b). +(9) +The structure of simply-connected, homogeneous almost α-cosymplectic 3- +manifold, admitting a contact Ricci soliton, is very well known. +Theorem 2.7. [9] Let M be a simply-connected, homogeneous almost α- +cosymplectic 3-manifold admitting a contact Ricci soliton. Then M is an +unimodular semidirect product Lie group G of type G0bb = R2 ⋊A R, where +A = +� +0 +b +−b +0 +� +, equipped with a flat left invariant cosymplectic structure. +Moreover, we have the following: +1. If A = 0, i.e., b = 0, G is the abelian Lie group R3 equipped with its flat +left invariant cosymplectic structure. +2. If A ̸= 0, i.e., b ̸= 0, G is the Lie group ˜E2 equipped with its flat left +invariant cosymplectic structure. + +6 +N. A. Pundeer, P. Ghosh, H. M. Shah and A. Bhattacharyya +2.6. Harmonic manifolds +A complete Riemannian manifold (M n, g) is said to be harmonic, if for any +p ∈ M, the volume density ωp(q) = +� +det(gij(q)) in normal coordinates, +centered at any p ∈ M is a radial function [1]. Thus, +Θ(r) = rn−1� +det(gij(q)) +is density of geodesic sphere, is a radial function. It is known that harmonic +manifolds are Einstein [1]. They are naturally classified as per the sign of the +Ricci constant. Let r be the constant scalar curvature of M. +• If r = 0, then M is flat, that is (M, g) = (Rn, Can) (Lemma 4.4). +• If r > 0, then by Bonnet-Myer’s theorem M is compact with finite +fundamental group. They are compact rank one symmetric spaces by a +well known result of Szabo (cf. [18]). +• If r < 0, then M is non-compact harmonic manifold. They are rank one +symmetric spaces of non-compact type, if dimension of M is atmost 5. +The main result in the theory of harmonic spaces is the Lichnerowicz +Conjecture: Any simply connected, complete harmonic manifold is either flat +or a rank one symmetric space. By the above classification, we see that the +conjecture is resolved for compact harmonic manifolds and is open for non- +compact harmonic manifolds of dimension 6. There are counter examples to +the conjecture when dimension is atleast 7, known as the Damek-Ricci spaces +or NA spaces. See for more details references in [18]. +In the category of non-compact harmonic manifolds, we will be con- +sidering simply connected, complete, non-compact harmonic manifolds. It +follows that, these spaces don’t have conjugate points (cf. [18]). Hence, by +the Cartan-Hadamard theorem, +expp : TpM → M +is a diffeomorphism and every geodesic of M is a line. That is, if γv : R → M +is a geodesic of M with v ∈ SpM, γ′ +v(0) = v, then d(γv(t), γv(s)) = |t − s|. +Busemann function: Let γv be a geodesic line, then the two Busemann +functions associated to γv are defined as [15]: +b+ +v (x) = lim +t→∞ d(x, γv(t)) − t, +b− +v (x) = +lim +t→−∞ d(x, γv(t)) − t. +3. Einstein Solitons on Almost α-Cosymplectic +3-Manifolds +In this section, we examine the nature of a contact Einstein soliton on al- +most α-cosymplectic manifold. We also show that, the characteristic vector +field ξ is harmonic on almost α-cosymplectic 3-manifold admitting a contact + +On harmonic and homogeneous almost α-cosymplectic 3-manifolds +7 +Einstein soliton. Finally, we generalize Theorem 2.7 using these results. +Contact Einstein soliton: Let (M 2n+1, g) be a Riemannian manifold of +dimension 2n + 1 (n ≥ 1). Consider the Einstein soliton (2), with potential +vector field V , on an almost contact metric manifold (M, ϕ, ξ, η, g). Then the +soliton is called contact Einstein soliton, if V = ξ that is, the potential vector +field is the characteristic vector field. +The potential vector field V is called transversal, if it is orthogonal to the +characteristic vector field, that is V ⊥ ξ. +Theorem 3.1. Let (M, ϕ, ξ, η, g) be an almost α-cosymplectic 3-manifold, +admitting a contact Einstein soliton. Then the soliton is: +1. steady, if α2 = σ2 − (a2 + b2) − (ϕe)(a) + e(b), +2. shrinking, if α2 > σ2 − (a2 + b2) − (ϕe)(a) + e(b), +3. expanding, if α2 < σ2 − (a2 + b2) − (ϕe)(a) + e(b). +Proof. If the soliton is contact Einstein soliton, using V = ξ in (2), we have +g(∇Xξ, Y ) + g(X, ∇Y ξ) + 2g(X, QY ) = (2λ + r)g(X, Y ), +(10) +for any vector fields X, Y on M. +Substituting X = Y = ξ in the above equation and using (8), we obtain +λ = −2α2 − 2σ2 − r +2. +(11) +From the expression of r (9), we get +λ = α2 − σ2 + (a2 + b2) + (ϕe)(a) − e(b), +(12) +from which we can conclude the proof. +□ +Theorem 3.2. Let (M, ϕ, ξ, η, g) be an almost α-cosymplectic 3-manifold, +admitting a contact Einstein soliton. Then the characteristic vector field ξ is +harmonic. +Proof. From (10), we get for X = ξ and Y = e, +(ϕe)(σ) = −2aσ. +(13) +And for X = ξ and Y = ϕe, from (10) we have +e(σ) = 2bσ. +(14) +Now, using (13) and (14) in the expression of Qξ in (8), we obtain +Qξ = −(2α2 + 2σ2)ξ, +which shows that ξ is an eigenvector field of the Ricci operator Q concluding +the fact that ξ is harmonic. +□ +We derive the identity involving the Lie derivative of the metric, Ricci oper- +ator, the potential vector field V . + +8 +N. A. Pundeer, P. Ghosh, H. M. Shah and A. Bhattacharyya +Lemma 3.3. Let (M, g) be a Riemannian manifold of scalar curvature r, +admitting an Einstein soliton (2). Then +∥LV g∥2 = 2V (r) + 4 div +�� +λ + r +2 +� +V − QV +� +, +(15) +where Q is the Ricci operator. +Proof. In local coordinate system, (2) leads to +LV gij + Sij = (2λ + r)gij. +Therefore, +∥LV g∥2 = − SijLV gij + (2λ + r)gijLV gij. += − LV r + gijLV Sij − (2λ + r)gijLV gij. +(16) +Now, +gijLV Sij =gij∇V Sij − gij∇αViSαj − gij∇αVjSiα +=2V (r) − 2 div QV. +(17) +Observing that gijLV gij = −2 div V and using (16) and (17), we get the +required result. +□ +Now we derive the main result of this section. +Theorem 3.4. Consider M to be an almost α-cosymplectic 3-manifold, ad- +mitting a contact Einstein soliton. Then the following hold. +1. If σ ̸= 0, then α = a2 + b2 − 2λ2 + (ϕe)(a) − e(b). +2. If σ = 0, then M is cosymplectic. +Proof. Replacing X by e and Y by ϕe, from (10) we get +g(∇eξ, ϕe) + g(e, ∇ϕeξ) + 2g(e, Qϕe) = (2λ + r)g(e, ϕe). +Using (7) and (8), after simplification we acquire, +ξ(σ) = σ − 2ασ. +(18) +Now putting X = e = Y in (10) and using (7), (8), (9) and (12), we get +6α2 + 6σ2 − 4σµ + 2α + r = 0. +(19) +Similarly, putting X = ϕe = Y in (10) and using (7), (8), (9) and (12), we +also obtain +6α2 + 6σ2 + 4σµ + 2α + r = 0. +(20) +So comparing (19) and (20), we have σµ = 0. If σ ̸= 0, then from (20), we +obtain the required result using (9). +Now suppose σ = 0, then M is α-cosymplectic. From [12], recall that an +almost α-cosymplectic manifold M is α-cosymplectic if and only if for any +X ∈ χ(M), +QX = +� +α2 + r +2 +� +X − +� +3α2 + r +2 +� +η(X)ξ. +(21) + +On harmonic and homogeneous almost α-cosymplectic 3-manifolds +9 +Since ∇ξ is symmetric, (10) becomes +g(∇Xξ, Y ) + g(X, QY ) = +� +λ + r +2 +� +g(X, Y ). +(22) +Using (6) and (21), we have from (22), for any X, Y ∈ χ(M), +(α2 + α − λ)g(X, Y ) − +� +3α2 + α + r +2 +� +η(X)η(Y ) = 0, +which implies α2 + α − λ = 0 and 3α2 + α + r +2 = 0. +That is λ = α2 + α and r = −6α2 − 2α = constant, so that, λ + r +2 = −2α2. +Also, from (21), we have Qξ = −2α2ξ which implies (λ + r +2)ξ − Qξ = 0. +Therefore, using Lemma 3.3 (15), we can say that ξ is a Killing vector field, +that is, ∇ξ is skew-symmetric. But in our case ∇ξ is symmetric, which implies +∇ξ = 0, that is, α = 0, proving the fact that M is cosymplectic. +□ +Corollary 3.5. Consider M to be a simply-connected, homogeneous, almost +α-cosymplectic 3-manifold, admitting a contact Einstein soliton with σ = 0. +Then M is an unimodular semidirect product Lie group G of type G0µµ = +R2 ⋊A R, where A = +� +0 +µ +−µ +0 +� +̸= 0, is a real matrix. Moreover, G is the +Lie group ˜E2 equipped with its flat left invariant cosymplectic structure. +Proof. The proof follows from Theorem 2.7 and Theorem 3.4. +□ +4. Ricci Solitons on Harmonic Manifolds +In this section, we study Ricci solitons on complete, simply connected, har- +monic manifolds. We prove a Lichnerowicz type result that, a harmonic man- +ifold admits a Ricci soliton if and only if M is flat. More precisely, we show +that compact harmonic manifolds and non-flat harmonic manifolds do not +admit Ricci solitons. But flat harmonic manifold do admit steady, shrinking, +expanding Ricci solitons. +In the sequel, harmonic manifold means complete, simply connected harmonic +manifold. The main theorem of this section is: +Theorem 4.1. Let (M, g) be a harmonic manifold. Then M admits Ricci +soliton if and only if M is flat. In this case, the steady Ricci soliton is Killing +given by X = ∇bv +−; where b− +v (x) = −⟨x, v⟩, the Busemann function, is a +potential function on M. In case, the Ricci soliton is shrinking or expanding, +the potential function is given by f(x) = λd(x, p)2 +f(p), for constant λ ̸= 0; +and point p is minimum or maximum of f and X = ∇f is the corresponding +Ricci soliton. +We begin with the following important proposition. + +10 +N. A. Pundeer, P. Ghosh, H. M. Shah and A. Bhattacharyya +Proposition 4.2. Every Ricci soliton is a gradient soliton on complete man- +ifold. Hence, in particular on any harmonic manifold Ricci soliton is a gra- +dient soliton. Consequently, any harmonic manifold admits a Gaussian. +Proof. Perelman showed that, Ricci soliton on any complete manifold is al- +ways a gradient soliton [11]. Hence, in this case X = ∇f, for some smooth +function f : M → R. As L∇fg = ∇2f, (1) reduces to +Ric + 1 +2∇2f = λg. +(23) +As (M, g) is harmonic and hence Einstein, then it follows that +∇2f = 2(λ − r)g, +(24) +where r is a constant scalar curvature of M. Thus f is a Gaussian, that is it +satisifes (24). +□ +Remark 4.3. Note that because any harmonic manifold is Einstein, trivial +solitons X = 0 and X a Killing vector field are solutions of (1) with λ = r. +Lemma 4.4. Ricci flat harmonic manifold is flat. +Proof. It can be shown that any harmonic manifold (M, g) is asymptotically +harmonic [18]. That is there exists a constant h ≥ 0 such that +∆bv +± = h. +Let L = ∇2bv ++ denote the second fundamental form of horospheres, b−1 +v (t). +Then L satisfies the Riccati equation, that is for x ∈ v⊥, +L′(x) + L2(x) + R(x, v)v = 0. +Tracing the above equation, we obtain that tr L2 = 0, as Ricci(v, v) = 0. But +L is a symmetric operator on v⊥. This implies that L = 0 for any v ∈ SM. +Consequently, R(x, v)v = 0 for any x ∈ v⊥. Thus (M, g) is flat. +□ +Lemma 4.5. Let X = ∇f be a Killing vector field on compact harmonic +manifold, then X is trivial. Solitons of Killing type do not exist on non- +compact, non-flat harmonic manifold. On flat harmonic manifold, Killing +vector field is X = ∇bv +−, where b− +v (x) = −⟨x, v⟩ is a Busemann function on +Rn. +Proof. Because X = ∇f is a non-trivial Killing vector field, we have +∇2f = 0. +Therefore, ∥∇f∥ = constant ̸= 0, consequently, f has no critical points. +Any Killing vector field of constant norm satisfies (p. 164-167, [15]): +∥∇2f∥ +2 = Ric(∇f, ∇f). + +On harmonic and homogeneous almost α-cosymplectic 3-manifolds +11 +Therefore, +0 =∥∇2f∥ = r∥∇f∥2 +This implies that for f non-constant, r = 0 and therefore Ric ≡ 0 and hence +harmonic manifold must be flat (Lemma 4.4). +We have ∥∇f∥ = constant. We may assume that ∥∇f∥ = 1, therefore f is +distance function which is harmonic function on (Rn, Can). By Proposition +5.1 of [18], it follows that +f(x) = b− +v (x) = −⟨x, v⟩, +is a Busemann function on Rn [15]. +If M is compact, ∇2f = 0 implies that f is a harmonic function. Hence, f +must be a constant function. +□ +Proposition 4.6. Let (M, g) be a compact harmonic manifold, then a Ricci +soliton on M is trivial. +Proof. We have, +∇2f = 2(λ − r)g. +Therefore, ∆f = 2(λ − r)n implies by the Bochner’s formula that, +1 +2∆(∥∇f∥2) = 4(λ − r)2n2 + r(∥∇f∥2). +(25) +Therefore, +4(λ − r)2n2 Vol(M) = −r +� +M +∥∇f∥2 < 0. +This implies that ∥∇f∥ = 0, therefore f is constant. +□ +Lemma 4.7. Let (M, g) be a non-compact, non-flat harmonic manifold. +Then Ricci solitons on M don’t exist. In case, (λ − r) ̸= 0, implies that +M is flat and r = 0. In this case the potential function is given by f(x) = +λd(p, x)2 + f(p), for some p ∈ M. +Proof. We have, +∇2f = 2(λ − r)g. +Therefore, f is either convex or concave function. Consequently, the only +possible critical point of f is either maximum or minimum of f. Suppose +that p is a critical point of f. Note that along any unit speed geodesic of M +starting from p, +f ′′(t) = 2(λ − r). +(26) +Therefore, f ′(t) = 2(λ − r)t + c. Hence, there is exactly one critical point, +and hence c = 0. Thus, f(t) = (λ − r)t2 + f(p), consequently f is a radial +function. This implies that, +∆f = f ′′ + Θ′ +Θ f ′ = 2(λ − r)n. + +12 +N. A. Pundeer, P. Ghosh, H. M. Shah and A. Bhattacharyya +Therefore, +f ′′ + Θ′ +Θ 2(λ − r)t = 2(λ − r)n. +Consequently by (26), +Θ′(t) +Θ(t) = n − 1 +t +. +Comparing with the series expansion (see (4.4) of [18]), +Θ′(t) +Θ(t) = n − 1 +r +− r +3 + · · · , +we obtain r = 0, hence M is flat. Finally, f(x) = λd(p, x)2 + f(p) follows +from section 1 of [4]. +□ +Finally we come to the proof of Theorem 4.1. +Proof of Theorem 4.1: If M is compact, then the Ricci soliton on M is +trivial (Proposition 4.6). If (λ−r) = 0, then M is flat and X = ∇bv +− (Lemma +4.5). If (λ−r) ̸= 0, then M is flat, and X = ∇f, where f(x) = λd(p, x)2+f(p), +for some p ∈ M (Lemma 4.7). +□ +Remark: We have shown that Theorem 4.1 confirms Theorem 2.1 in case of +harmonic manifolds. Also Theorem 4.1 implies that there are no non-trivial +deformation of non-flat harmonic manifolds. This indicates a result support- +ing the conjecture that, there are no non-trivial deformations of harmonic +manifolds; and hence there should be only finitely many classes of harmonic +manifolds. +5. Acknowledgements +Dr. Naeem Ahmad Pundeer would like to thank to U.G.C. for its Dr. D.S. +Kothari Postdoctoral Fellowship. The corresponding author, Mr. Paritosh +Ghosh, thanks UGC Junior Research Fellowship of India. The authors also +would like to thank Mr. Dipen Ganguly for his wishful help in this research. +References +[1] Besse, A.L. Manifolds all of whose geodesics are closed, Berlin Heidel- +berg, Springer-Verlag, (1978). +[2] Blair, D.E. Riemannian geometry of contact and symplectic manifolds, +Progress in Mathematics, Birkh¨auser, New York, (2010). +[3] Catino, G. and Mazzieri, L. Gradient Einstein solitons, Nonlinear +Anal., 132, 66–94, (2016). +[4] Cheeger, J. and Colding, T. Lower bounds on Ricci curvature and +the almost rigidity of warped products, Ann. Math., 144(1), 189-237, +(1996). + +On harmonic and homogeneous almost α-cosymplectic 3-manifolds +13 +[5] Cunha, A.W. and Griffin, E. On non-compact gradient solitons, +arXiv:2207.05822, (2022). +[6] Eells, J. and Sampson, J.H. Harmonic Mappings of Riemannian Man- +ifolds, Amer. J. Math., 86, 109-160, (1964). +[7] Hamilton, R.S. Three manifolds with positive Ricci curvature, J. Diff. +Geom., 17, 255-306, (1982). +[8] Hu, Q., Xu, G. and Yu, C. The rigidity and stability of gradient esti- +mates, J. Geom. Anal., 32, 1-13, (2022). +[9] Li, J. and Liu, X. Ricci solitons on homogeneous almost α-cosymplectic +three-Manifolds, Mediterr. J. Math., 19, 1-12, (2022). +[10] Libermann, P. Sur les automorphismes infinit´esimaux des structures +symplectiques et des structures de contact, Colloque G´eom. Diff. Glob- +ale, 37–59, (1959). +[11] Perelman, G. The entropy formula for the Ricci flow and its geometric +applications, arXiv:math 211159, (2002). +[12] Perrone, D. Classification of homogeneous almost α-coK¨ahler three- +manifolds, Diff. Geom. Appl., 59, 66–90, (2018). +[13] Perrone, D. Classification of homogeneous almost cosymplectic three +manifolds, Diff. Geom. Appl., 30, 49–58, (2012). +[14] Perrone, D. Left-invariant almost α-co K¨ahler structures on 3D semidi- +rect product Lie groups, Int. J. Geom. Meth. Mod. Phys., 16, 1-18, +(2018). +[15] Petersen, P. Riemannian geometry, New York, Springer-Verlag, (2006). +[16] Petersen, P. and Wylie, W. Rigidity of gradient Ricci solitons, Pac. J. +Math., 241, 329-345, (2009). +[17] Ranjan, A. and Shah, H. Harmonic manifolds with minimal horo- +spheres, J. Geom. Anal., 12, 683-694, (2002). +[18] Ranjan, A. and Shah, H. Busemann functions in a harmonic manifold, +Geom. Dedicata, 101, 167-183, (2003). +[19] Tashiro, Y. Complete Riemannian manifolds and some vector fields, +Trans. Amer. Math. Soc., 117, 251-275, (1965). +Naeem Ahmad Pundeer +Department of Mathematics +Jadavpur University +Kolkata-700032, India. +e-mail: pundir.naeem@gmail.com +Paritosh Ghosh +Department of Mathematics +Jadavpur University +Kolkata-700032, India. +e-mail: paritoshghosh112@gmail.com + +14 +N. A. Pundeer, P. Ghosh, H. M. Shah and A. Bhattacharyya +Hemangi Madhusudan Shah +Harish-Chandra Research Institute +A CI of Homi Bhabha National Institute +Chhatnag Road, Jhunsi, Prayagraj-211019, India. +e-mail: hemangimshah@hri.res.in +Arindam Bhattacharyya +Department of Mathematics +Jadavpur University +Kolkata-700032, India +e-mail: bhattachar1968@yahoo.co.in + diff --git a/GdE0T4oBgHgl3EQfhQG7/content/tmp_files/load_file.txt b/GdE0T4oBgHgl3EQfhQG7/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..02ac5edc7c35396b04781e652a17352c37d38c6a --- /dev/null +++ b/GdE0T4oBgHgl3EQfhQG7/content/tmp_files/load_file.txt @@ -0,0 +1,474 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf,len=473 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='02430v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='GM] 6 Jan 2023 Some Solitons on Homogeneous Almost α-Cosymplectic 3-Manifolds and Harmonic Manifolds Naeem Ahmad Pundeer, Paritosh Ghosh, Hemangi Madhusudan Shah and Arindam Bhattacharyya Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' In this paper, we investigate the nature of Einstein solitons, whether it is steady, shrinking or expanding on almost α-cosymplectic 3-manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' We also prove that a simply connected homogeneous al- most α-cosymplectic 3-manifold, admitting a contact Einstein soliton, is an unimodular semidirect product Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Finally, we show that a harmonic manifold admits a Ricci soliton if and only if it is flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Mathematics Subject Classification (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' 53B40, 58B20, 53C25, 53D15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Keywords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Almost α-cosymplectic manifold, Harmonic manifold, Ricci soliton, Einstein soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Introduction The study of solitons, in particular Ricci solitons, on Riemannian man- ifolds play a vital role in understanding the geometry of underlying mani- fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' It is very interesting to study Ricci and Einstein solitons on almost α- cosymplectic 3-manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Recently, Jin and Ximin [9] showed that a simply connected homogeneous almost α-cosymplectic 3-manifold, admitting con- tact Ricci solitons, is cosymplectic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' and the manifold under consideration is an unimodular semidirect product Lie group R2⋊AR, where A = � 0 b −b 0 � , equipped with a flat left invariant cosymplectic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Motivated by this result we show in this paper that, if a simply con- nected homogeneous almost α-cosymplectic 3-manifold, with some additional hypothesis, admits a contact Einstein soliton, then the manifold is an uni- modular semidirect product Lie group G of type G0bb = R2 ⋊A R, where 2 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Pundeer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Ghosh, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Shah and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Bhattacharyya A = � 0 b −b 0 � ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' And also G is the Lie group ˜E2 equipped with its flat left invariant cosymplectic structure (see Corrollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' In order to prove this result, we first obtain a characterization of almost α-cosymplectic 3-manifold admitting contact Einstein solitons, which is the main theorem (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='4) of Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' To establish this aforementioned theorem we derive an identity (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='3) involving scalar curvature, Lie derivative of the metric and Ricci operator on a Riemannian manifold admitting Einstein soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' We also give some conditions on α for contact Einstein solitons to be steady, shrinking or expanding on almost α-cosymplectic 3-manifolds (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Another interesting topic in the differential geometry is the geometry of harmonic manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' In 1965, Tashiro [19] showed that if a complete Rie- mannian manifold admits a Gaussian, then it is either flat or a complete warped product manifold or a sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' In this paper, we show that a har- monic manifold admits a Gaussian if and only if it is flat;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' thus confirming Tashiro’s result for harmonic manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Moreover, we prove that flat har- monic manifold admits Ricci solitons of steady, expanding or shrinking type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' We also determine the corresponding potential function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' In fact, Busemann function on Rn turns to be the potential function in case of steady solitons (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='1 of Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' The paper is divided into four sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Section 2 is devoted to the preliminaries about Ricci soliton, Einstein soliton, almost α-cosymplectic 3- manifolds and harmonic manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' In Section 3, we prove our main results on almost α-cosymplectic 3-manifold admitting contact Einstein solitons, as stated above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' In the last section, we prove the main flatness result about harmonic manifolds admitting Ricci solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Preliminaries In this section, we discuss some notions required to prove the results of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Ricci solitons Ricci solitons are the self similar solutions of the Ricci flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' The concept of Ricci flow was first introduced by Hamilton [7] in (1982), motivated by the work of Eells and Sampson [6] on harmonic map and the flow was given by the equation ∂g ∂t = −2S, where S is the Ricci tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Ricci solitons are the generalizations of the Einstein metrics and are the solutions of the equation Ric(g) + 1 2LXg = λg, (1) On harmonic and homogeneous almost α-cosymplectic 3-manifolds 3 where Ric(X, Y ) = S(X, Y ) is the Ricci curvature tensor, LX is the Lie derivative along the direction of the vector field X and λ is a real constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' The soliton is said to be shrinking if λ > 0, steady if λ = 0 and expanding if λ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Tashiro [15] proved very important result for complete Einstein manifolds admitting Ricci solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' [15] Let (M, g) be a complete Riemannian n-manifold admit- ting a nontrivial function f such that Hess f = λg, then (M, g) is isometric to a complete warped product metric and must have one of the three forms: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' M = R × N, g = dr2 + ρ2(r)gN, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' M = Rn, g = dr2 + ρ2(r)ds2 n−1, r ≥ 0, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' M = Sn, g = dr2 + ρ2(r)ds2 n−1, r ∈ [a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Einstein solitons The Einstein solitons are the generalization of the Ricci solitons, was first introduced by Catino and Mazzieri [3] in (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' They are the solutions of the equation LV g + 2S = (2λ + r)g, (2) where, Ricci tensor S(X, Y ) = g(X, QY ), Q being the Ricci operator, r is the scalar curvature, λ ∈ R is a constant and V is known as potential vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Einstein solitons are the self-similar solutions of the Einstein flow, ∂ ∂tg + 2S = rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' It is said to be steady if λ = 0, shrinking if λ > 0 and expanding if λ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Almost contact metric manifolds In order to define contact metric manifolds, we need the concept of Reeb vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Reeb vector field [2]: A global vector field ξ on a contact manifold M 2n+1, equipped with a global 1-form η, is called Reeb vector field or characteristic vector field, if any vector field X satisfies η(ξ) = 1 and dη(X, ξ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Almost contact manifold [2]: Let M be a Riemannian manifold of di- mension (2n + 1), n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' M 2n+1 is said to have an almost contact structure (ϕ, ξ, η), if there exists a (1, 1)-tensor ϕ, a global vector field ξ and a 1-form η such that ϕ2X = −X + η(X)ξ, η(ξ) = 1, (3) for any vector field X on M, where ξ is the Reeb vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' The manifold M equipped with the structure (ϕ, ξ, η) is called an almost contact manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Almost contact metric manifold [2]: A Riemannian metric g is said to be compatible with an almost contact structure (ϕ, ξ, η), if g(ϕX, ϕY ) = g(X, Y ) − η(X)η(Y ), (4) 4 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Pundeer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Ghosh, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Shah and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Bhattacharyya holds for any X, Y ∈ χ(M) and (M, ϕ, ξ, η, g) is called an almost contact metric manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Normal almost contact metric manifold [2]: An almost contact met- ric manifold is said to be normal, if for any X, Y ∈ χ(M) the tensor field N = [ϕ, ϕ]+ 2dη ⊗ ξ vanishes everywhere on the manifold, where [ϕ, ϕ] is the Nijenhuis tensor of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Homogeneous almost contact metric manifold [9]: An almost contact metric manifold (M, ϕ, ξ, η, g) is said to be homogeneous, if there exists a con- nected Lie group G of isometries acting transitively on M leaving η invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Cosymplectic manifolds A (2n + 1)-dimensional manifold is said to be a cosymplectic manifold [10], if it admits a closed, 1-form η and 2-form Φ such that η ∧ Φn is a volume element, where Φ(X, Y ) = g(ϕX, Y ) is a 2-form on M 2n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Almost cosymplectic manifold [10]: If η and Φ are not closed but η ∧ Φn is a volume form, then the manifold is called almost cosymplectic manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' α-cosymplectic manifold [12]: An almost cosymplectic manifold is said to be α-cosymplectic if dη = 0 and dΦ = 2αη ∧ Φ for some constant α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Almost α-cosymplectic manifold [10]: An almost α-cosymplectic manifold is defined as an almost contact metric manifold with dη = 0 and dΦ = 2αη∧Φ, for any constant α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' In particular, the almost α-cosymplectic manifold is almost α-Kenmotsu if α ̸= 0, almost cosymplectic if α = 0, almost Kenmotsu if α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Harmonic vector field [14]: A characteristic vector field ξ on an almost α-cosymplectic manifold is harmonic if and only if ξ is an eigenvector field of the Ricci operator Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Almost α-cosymplectic 3-manifold In this article, we will mainly focus on 3-dimensional almost α-cosymplectic manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' In what follows, we will be using the following results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' [12] An almost α-cosymplectic 3-manifold is α-cosymplectic if and only if Lξh = 0, where h = 1 2Lξϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Any almost α-cosymplectic 3-manifold satisfies important relationships be- tween Φ, ξ and h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' [12] Let M 2n+1 be an almost α-cosymplectic 3-manifold, then we have, ∇ξϕ = 0, ∇ξ = 0, hϕ + ϕh = 0, hξ = 0, (5) On harmonic and homogeneous almost α-cosymplectic 3-manifolds 5 with ∇Xξ = −αϕ2X − ϕhX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' (6) We would require some identities on the ϕ-bases [2] and the following table of the Levi-Civita connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' [12] On almost α-cosymplectic 3-manifold, there exists ϕ-bases satisfying he = σe, hϕe = −σϕe, hξ = 0, with σ a local smooth eigen-function of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' [12] The Levi-Civita connection on almost α-cosymplectic 3-manifold are given by, \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ∇ee = −aϕe − αξ, ∇ϕee = −bϕe + σξ, ∇ξe = µϕe, ∇eϕe = ae + σξ, ∇ϕeϕe = be − αξ, ∇ξϕe = −µe, ∇eξ = αe − σϕe, ∇ϕeξ = −σe + αϕe, ∇ξξ = 0, (7) where a = g(∇eϕe, e), b = −g(∇ϕee, ϕe) and µ = g(∇ξe, ϕe) are smooth functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' The Ricci operator on almost α-cosymplectic 3-manifold is known explicitly [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' [12] The Ricci operator Q on almost α-cosymplectic 3- manifold is given by, \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 Qξ = −(2α2 + tr h2)ξ + (2bσ − e(σ))ϕe − (2aσ + (ϕe)(σ))e, Qϕe = (2bσ − e(σ))ξ + (α2 + r 2 + tr h2 2 + 2σµ)ϕe + (ξ(σ) + 2ασ)e, Qe = −(2aσ + (ϕe)(σ))ξ + (ξ(σ) + 2ασ)ϕe + (α2 + r 2 + tr h2 2 − 2σµ)e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' (8) Furthermore, the scalar curvature r = tr Q is given by r = −6α2 − tr h2 − 2(a2 + b2) − 2(ϕe)(a) + 2e(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' (9) The structure of simply-connected, homogeneous almost α-cosymplectic 3- manifold, admitting a contact Ricci soliton, is very well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' [9] Let M be a simply-connected, homogeneous almost α- cosymplectic 3-manifold admitting a contact Ricci soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Then M is an unimodular semidirect product Lie group G of type G0bb = R2 ⋊A R, where A = � 0 b −b 0 � , equipped with a flat left invariant cosymplectic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Moreover, we have the following: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' If A = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=', b = 0, G is the abelian Lie group R3 equipped with its flat left invariant cosymplectic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' If A ̸= 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=', b ̸= 0, G is the Lie group ˜E2 equipped with its flat left invariant cosymplectic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' 6 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Pundeer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Ghosh, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Shah and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Bhattacharyya 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Harmonic manifolds A complete Riemannian manifold (M n, g) is said to be harmonic, if for any p ∈ M, the volume density ωp(q) = � det(gij(q)) in normal coordinates, centered at any p ∈ M is a radial function [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Thus, Θ(r) = rn−1� det(gij(q)) is density of geodesic sphere, is a radial function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' It is known that harmonic manifolds are Einstein [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' They are naturally classified as per the sign of the Ricci constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Let r be the constant scalar curvature of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' If r = 0, then M is flat, that is (M, g) = (Rn, Can) (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' If r > 0, then by Bonnet-Myer’s theorem M is compact with finite fundamental group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' They are compact rank one symmetric spaces by a well known result of Szabo (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' If r < 0, then M is non-compact harmonic manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' They are rank one symmetric spaces of non-compact type, if dimension of M is atmost 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' The main result in the theory of harmonic spaces is the Lichnerowicz Conjecture: Any simply connected, complete harmonic manifold is either flat or a rank one symmetric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' By the above classification, we see that the conjecture is resolved for compact harmonic manifolds and is open for non- compact harmonic manifolds of dimension 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' There are counter examples to the conjecture when dimension is atleast 7, known as the Damek-Ricci spaces or NA spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' See for more details references in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' In the category of non-compact harmonic manifolds, we will be con- sidering simply connected, complete, non-compact harmonic manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' It follows that, these spaces don’t have conjugate points (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Hence, by the Cartan-Hadamard theorem, expp : TpM → M is a diffeomorphism and every geodesic of M is a line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' That is, if γv : R → M is a geodesic of M with v ∈ SpM, γ′ v(0) = v, then d(γv(t), γv(s)) = |t − s|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Busemann function: Let γv be a geodesic line, then the two Busemann functions associated to γv are defined as [15]: b+ v (x) = lim t→∞ d(x, γv(t)) − t, b− v (x) = lim t→−∞ d(x, γv(t)) − t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Einstein Solitons on Almost α-Cosymplectic 3-Manifolds In this section, we examine the nature of a contact Einstein soliton on al- most α-cosymplectic manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' We also show that, the characteristic vector field ξ is harmonic on almost α-cosymplectic 3-manifold admitting a contact On harmonic and homogeneous almost α-cosymplectic 3-manifolds 7 Einstein soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Finally, we generalize Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='7 using these results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Contact Einstein soliton: Let (M 2n+1, g) be a Riemannian manifold of dimension 2n + 1 (n ≥ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Consider the Einstein soliton (2), with potential vector field V , on an almost contact metric manifold (M, ϕ, ξ, η, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Then the soliton is called contact Einstein soliton, if V = ξ that is, the potential vector field is the characteristic vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' The potential vector field V is called transversal, if it is orthogonal to the characteristic vector field, that is V ⊥ ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Let (M, ϕ, ξ, η, g) be an almost α-cosymplectic 3-manifold, admitting a contact Einstein soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Then the soliton is: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' steady, if α2 = σ2 − (a2 + b2) − (ϕe)(a) + e(b), 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' shrinking, if α2 > σ2 − (a2 + b2) − (ϕe)(a) + e(b), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' expanding, if α2 < σ2 − (a2 + b2) − (ϕe)(a) + e(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' If the soliton is contact Einstein soliton, using V = ξ in (2), we have g(∇Xξ, Y ) + g(X, ∇Y ξ) + 2g(X, QY ) = (2λ + r)g(X, Y ), (10) for any vector fields X, Y on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Substituting X = Y = ξ in the above equation and using (8), we obtain λ = −2α2 − 2σ2 − r 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' (11) From the expression of r (9), we get λ = α2 − σ2 + (a2 + b2) + (ϕe)(a) − e(b), (12) from which we can conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Let (M, ϕ, ξ, η, g) be an almost α-cosymplectic 3-manifold, admitting a contact Einstein soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Then the characteristic vector field ξ is harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' From (10), we get for X = ξ and Y = e, (ϕe)(σ) = −2aσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' (13) And for X = ξ and Y = ϕe, from (10) we have e(σ) = 2bσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' (14) Now, using (13) and (14) in the expression of Qξ in (8), we obtain Qξ = −(2α2 + 2σ2)ξ, which shows that ξ is an eigenvector field of the Ricci operator Q concluding the fact that ξ is harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' □ We derive the identity involving the Lie derivative of the metric, Ricci oper- ator, the potential vector field V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' 8 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Pundeer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Ghosh, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Shah and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Bhattacharyya Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Let (M, g) be a Riemannian manifold of scalar curvature r, admitting an Einstein soliton (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Then ∥LV g∥2 = 2V (r) + 4 div �� λ + r 2 � V − QV � , (15) where Q is the Ricci operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' In local coordinate system, (2) leads to LV gij + Sij = (2λ + r)gij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Therefore, ∥LV g∥2 = − SijLV gij + (2λ + r)gijLV gij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' = − LV r + gijLV Sij − (2λ + r)gijLV gij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' (16) Now, gijLV Sij =gij∇V Sij − gij∇αViSαj − gij∇αVjSiα =2V (r) − 2 div QV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' (17) Observing that gijLV gij = −2 div V and using (16) and (17), we get the required result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' □ Now we derive the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Consider M to be an almost α-cosymplectic 3-manifold, ad- mitting a contact Einstein soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Then the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' If σ ̸= 0, then α = a2 + b2 − 2λ2 + (ϕe)(a) − e(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' If σ = 0, then M is cosymplectic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Replacing X by e and Y by ϕe, from (10) we get g(∇eξ, ϕe) + g(e, ∇ϕeξ) + 2g(e, Qϕe) = (2λ + r)g(e, ϕe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Using (7) and (8), after simplification we acquire, ξ(σ) = σ − 2ασ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' (18) Now putting X = e = Y in (10) and using (7), (8), (9) and (12), we get 6α2 + 6σ2 − 4σµ + 2α + r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' (19) Similarly, putting X = ϕe = Y in (10) and using (7), (8), (9) and (12), we also obtain 6α2 + 6σ2 + 4σµ + 2α + r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' (20) So comparing (19) and (20), we have σµ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' If σ ̸= 0, then from (20), we obtain the required result using (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Now suppose σ = 0, then M is α-cosymplectic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' From [12], recall that an almost α-cosymplectic manifold M is α-cosymplectic if and only if for any X ∈ χ(M), QX = � α2 + r 2 � X − � 3α2 + r 2 � η(X)ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' (21) On harmonic and homogeneous almost α-cosymplectic 3-manifolds 9 Since ∇ξ is symmetric, (10) becomes g(∇Xξ, Y ) + g(X, QY ) = � λ + r 2 � g(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' (22) Using (6) and (21), we have from (22), for any X, Y ∈ χ(M), (α2 + α − λ)g(X, Y ) − � 3α2 + α + r 2 � η(X)η(Y ) = 0, which implies α2 + α − λ = 0 and 3α2 + α + r 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' That is λ = α2 + α and r = −6α2 − 2α = constant, so that, λ + r 2 = −2α2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Also, from (21), we have Qξ = −2α2ξ which implies (λ + r 2)ξ − Qξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Therefore, using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='3 (15), we can say that ξ is a Killing vector field, that is, ∇ξ is skew-symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' But in our case ∇ξ is symmetric, which implies ∇ξ = 0, that is, α = 0, proving the fact that M is cosymplectic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' □ Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Consider M to be a simply-connected, homogeneous, almost α-cosymplectic 3-manifold, admitting a contact Einstein soliton with σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Then M is an unimodular semidirect product Lie group G of type G0µµ = R2 ⋊A R, where A = � 0 µ −µ 0 � ̸= 0, is a real matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Moreover, G is the Lie group ˜E2 equipped with its flat left invariant cosymplectic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' The proof follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='7 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Ricci Solitons on Harmonic Manifolds In this section, we study Ricci solitons on complete, simply connected, har- monic manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' We prove a Lichnerowicz type result that, a harmonic man- ifold admits a Ricci soliton if and only if M is flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' More precisely, we show that compact harmonic manifolds and non-flat harmonic manifolds do not admit Ricci solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' But flat harmonic manifold do admit steady, shrinking, expanding Ricci solitons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' In the sequel, harmonic manifold means complete, simply connected harmonic manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' The main theorem of this section is: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Let (M, g) be a harmonic manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Then M admits Ricci soliton if and only if M is flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' In this case, the steady Ricci soliton is Killing given by X = ∇bv −;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' where b− v (x) = −⟨x, v⟩, the Busemann function, is a potential function on M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' In case, the Ricci soliton is shrinking or expanding, the potential function is given by f(x) = λd(x, p)2 +f(p), for constant λ ̸= 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' and point p is minimum or maximum of f and X = ∇f is the corresponding Ricci soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' We begin with the following important proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' 10 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Pundeer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Ghosh, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Shah and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Bhattacharyya Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Every Ricci soliton is a gradient soliton on complete man- ifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Hence, in particular on any harmonic manifold Ricci soliton is a gra- dient soliton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Consequently, any harmonic manifold admits a Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Perelman showed that, Ricci soliton on any complete manifold is al- ways a gradient soliton [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Hence, in this case X = ∇f, for some smooth function f : M → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' As L∇fg = ∇2f, (1) reduces to Ric + 1 2∇2f = λg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' (23) As (M, g) is harmonic and hence Einstein, then it follows that ∇2f = 2(λ − r)g, (24) where r is a constant scalar curvature of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Thus f is a Gaussian, that is it satisifes (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Note that because any harmonic manifold is Einstein, trivial solitons X = 0 and X a Killing vector field are solutions of (1) with λ = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Ricci flat harmonic manifold is flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' It can be shown that any harmonic manifold (M, g) is asymptotically harmonic [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' That is there exists a constant h ≥ 0 such that ∆bv ± = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Let L = ∇2bv + denote the second fundamental form of horospheres, b−1 v (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Then L satisfies the Riccati equation, that is for x ∈ v⊥, L′(x) + L2(x) + R(x, v)v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Tracing the above equation, we obtain that tr L2 = 0, as Ricci(v, v) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' But L is a symmetric operator on v⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' This implies that L = 0 for any v ∈ SM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Consequently, R(x, v)v = 0 for any x ∈ v⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Thus (M, g) is flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Let X = ∇f be a Killing vector field on compact harmonic manifold, then X is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Solitons of Killing type do not exist on non- compact, non-flat harmonic manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' On flat harmonic manifold, Killing vector field is X = ∇bv −, where b− v (x) = −⟨x, v⟩ is a Busemann function on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Because X = ∇f is a non-trivial Killing vector field, we have ∇2f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Therefore, ∥∇f∥ = constant ̸= 0, consequently, f has no critical points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Any Killing vector field of constant norm satisfies (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' 164-167, [15]): ∥∇2f∥ 2 = Ric(∇f, ∇f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' On harmonic and homogeneous almost α-cosymplectic 3-manifolds 11 Therefore, 0 =∥∇2f∥ = r∥∇f∥2 This implies that for f non-constant, r = 0 and therefore Ric ≡ 0 and hence harmonic manifold must be flat (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' We have ∥∇f∥ = constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' We may assume that ∥∇f∥ = 1, therefore f is distance function which is harmonic function on (Rn, Can).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' By Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='1 of [18], it follows that f(x) = b− v (x) = −⟨x, v⟩, is a Busemann function on Rn [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' If M is compact, ∇2f = 0 implies that f is a harmonic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Hence, f must be a constant function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' □ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Let (M, g) be a compact harmonic manifold, then a Ricci soliton on M is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' We have, ∇2f = 2(λ − r)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Therefore, ∆f = 2(λ − r)n implies by the Bochner’s formula that, 1 2∆(∥∇f∥2) = 4(λ − r)2n2 + r(∥∇f∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' (25) Therefore, 4(λ − r)2n2 Vol(M) = −r � M ∥∇f∥2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' This implies that ∥∇f∥ = 0, therefore f is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Let (M, g) be a non-compact, non-flat harmonic manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Then Ricci solitons on M don’t exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' In case, (λ − r) ̸= 0, implies that M is flat and r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' In this case the potential function is given by f(x) = λd(p, x)2 + f(p), for some p ∈ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' We have, ∇2f = 2(λ − r)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Therefore, f is either convex or concave function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Consequently, the only possible critical point of f is either maximum or minimum of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Suppose that p is a critical point of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Note that along any unit speed geodesic of M starting from p, f ′′(t) = 2(λ − r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' (26) Therefore, f ′(t) = 2(λ − r)t + c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Hence, there is exactly one critical point, and hence c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Thus, f(t) = (λ − r)t2 + f(p), consequently f is a radial function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' This implies that, ∆f = f ′′ + Θ′ Θ f ′ = 2(λ − r)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' 12 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Pundeer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Ghosh, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Shah and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Bhattacharyya Therefore, f ′′ + Θ′ Θ 2(λ − r)t = 2(λ − r)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Consequently by (26), Θ′(t) Θ(t) = n − 1 t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Comparing with the series expansion (see (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='4) of [18]), Θ′(t) Θ(t) = n − 1 r − r 3 + · · · , we obtain r = 0, hence M is flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Finally, f(x) = λd(p, x)2 + f(p) follows from section 1 of [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' □ Finally we come to the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='1: If M is compact, then the Ricci soliton on M is trivial (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' If (λ−r) = 0, then M is flat and X = ∇bv − (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' If (λ−r) ̸= 0, then M is flat, and X = ∇f, where f(x) = λd(p, x)2+f(p), for some p ∈ M (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' □ Remark: We have shown that Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='1 confirms Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='1 in case of harmonic manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Also Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='1 implies that there are no non-trivial deformation of non-flat harmonic manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' This indicates a result support- ing the conjecture that, there are no non-trivial deformations of harmonic manifolds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' and hence there should be only finitely many classes of harmonic manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Acknowledgements Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Naeem Ahmad Pundeer would like to thank to U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' for its Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Kothari Postdoctoral Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' The corresponding author, Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Paritosh Ghosh, thanks UGC Junior Research Fellowship of India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' The authors also would like to thank Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Dipen Ganguly for his wishful help in this research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' References [1] Besse, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Complete Riemannian manifolds and some vector fields, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=', 117, 251-275, (1965).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Naeem Ahmad Pundeer Department of Mathematics Jadavpur University Kolkata-700032, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' e-mail: pundir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='naeem@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='com Paritosh Ghosh Department of Mathematics Jadavpur University Kolkata-700032, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' e-mail: paritoshghosh112@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='com 14 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Pundeer, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Ghosh, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Shah and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' Bhattacharyya Hemangi Madhusudan Shah Harish-Chandra Research Institute A CI of Homi Bhabha National Institute Chhatnag Road, Jhunsi, Prayagraj-211019, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content=' e-mail: hemangimshah@hri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='in Arindam Bhattacharyya Department of Mathematics Jadavpur University Kolkata-700032, India e-mail: bhattachar1968@yahoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} +page_content='in' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfhQG7/content/2301.02430v1.pdf'} diff --git a/HNE1T4oBgHgl3EQfXgRU/content/tmp_files/2301.03127v1.pdf.txt b/HNE1T4oBgHgl3EQfXgRU/content/tmp_files/2301.03127v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..86b38788ed0b1eb5c4f6ec3c8658d8a5d75c7c10 --- /dev/null +++ b/HNE1T4oBgHgl3EQfXgRU/content/tmp_files/2301.03127v1.pdf.txt @@ -0,0 +1,1010 @@ +Logically at Factify 2023: A Multi-Modal Fact +Checking System Based on Evidence Retrieval +techniques and Transformer Encoder Architecture +Pim Jordi Verschuuren, Jie Gao, Adelize van Eeden, Stylianos Oikonomou and +Anil Bandhakavi +Brookfoot Mills, Brookfoot Industrial Estate, Brighouse, HD6 2RW, United Kingdom +Abstract +In this paper, we present the Logically submissions to De-Factify 2 challenge (DE-FACTIFY 2023) on +the task 1 of Multi-Modal Fact Checking. We describes our submissions to this challenge including +explored evidence retrieval and selection techniques, pre-trained cross-modal and unimodal models, and +a cross-modal veracity model based on the well established Transformer Encoder (TE) architecture which +is heavily relies on the concept of self-attention. Exploratory analysis is also conducted on this Factify +2 data set that uncovers the salient multi-modal patterns and hypothesis motivating the architecture +proposed in this work. A series of preliminary experiments were done to investigate and benchmarking +different pre-trained embedding models, evidence retrieval settings and thresholds. The final system, a +standard two-stage evidence based veracity detection system, yields weighted avg. 0.79 on both val set +and final blind test set on the task 1, which achieves 3rd place with a small margin to the top performing +system on the leaderboard among 9 participants. +Keywords +fact verification, multimodal representation learning, multimodal entailment, text entailment, Multi-head +Attention +1. Introduction +Misinformation and fake news can spread rapidly and cause harm at various levels. One way to +protect ourselves from these negative impacts is through fact-checking and debunking false +information with evidence-based reporting. However, this process can be resource-intensive and +time-consuming. To address this issue, researchers have developed automated fact-checking +systems using deep learning techniques, which can handle tasks such as claim detection, +claim matching, evidence retrieval, and veracity prediction using natural language processing +techniques on textual content. While there has been progress in this area, there is still a need +for multimodal approaches that can handle both text and image inputs. To address this gap, +this paper presents a multimodal veracity prediction system for automated fact-checking and is +De-Factify: Workshop on Multimodal Fact-Checking and Hate Speech Detection, co-located with AAAI 2023. 2023 +Washington DC, USA +� Pim.jv@logically.co.uk (P. J. Verschuuren); jie@logically.co.uk (J. Gao); adelize.ve@logically.co.uk (A. v. Eeden); +stylianos@logically.co.uk (S. Oikonomou); anil@logically.co.uk (A. Bandhakavi) +� https://www.logically.ai/team/leadership/anil-bandhakavi (A. Bandhakavi) +� 0000-0002-3610-8748 (J. Gao) +© 2023 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). +CEUR +Workshop +Proceedings +http://ceur-ws.org +ISSN 1613-0073 +CEUR Workshop Proceedings (CEUR-WS.org) +arXiv:2301.03127v1 [cs.CL] 9 Jan 2023 + +developed as part of the Factify 2 competition organized by De-Factify@AAAI 2023. +The remainder of the paper is structured as follows: Section 2 presents a brief overview of +related work and section 3 describes our general framework and model architecture. Section +4 discusses the dataset supplied by the Factify 2 competition followed by an overview of +our experiments in section 5. Section 6 and 7 present the final results and our conclusions, +respectively. +2. Related Work +As an essential part of automated fact verification, effective techniques for modeling claim- +evidence for veracity prediction have been a hot topic and key research questions in existing +fact-checking methods. Most of the recent work focuses on using textual evidence in veracity +prediction and there are mainly two lines of work. One direction [1, 2, 3] is to use a single +document (such as is provided in the Factify task dataset) with long text evidence and through +leveraging models constructed for long sequences. Examples such as BigBird [4], Longformer[5] +and recent advancements in the ConvNets architecture witnessed in the Long Range Arena +leaderboard (e.g., Mega [6], S5[7]) are seen to obtain top results in a wide range of tasks and +leader boards. The benefits of exploiting long-sequence model at document level is a) the +simplicity of the overall architecture; b) allows to accommodate for more context of the whole +article into modeling and natural language inference. An optimal setup of the maximum length +for both claim (or query) and document sequence, and the document level veracity labels is +commonly required [8, 1, 3]. The advantage of incorporating lots of context into inference is also +seen in modeling question answering (QA) tasks [4, 5], for which the document-level veracity +labels are relatively "cheap" to obtain. The downside of using a simple long-text model technique +at document-level is the lack of interpretability (w.r.t. evidence selection), it is computational +expensive, the limitation in dealing with the complexity of certain (multi-hop) claims [9], and +lack of diversity and scalability when dealing with a large amount of diverse documents in a +real-world application. These constraints were more apparent in open domain fact checking +task that make use of web data extracted with commercial search engines as building blocks +in fact-checking system in order to incorporate more diverse sources. It is worth to note that +long-sequence model can be adapted for the purpose of evidence selection e.g., through framing +the task as a token-level prediction task. For instance, as one of the top systems in SciFact +leaderboard 1, LongChecker [10] used LongFormer [5] for scientific claim verification with +paragraph-level evidence selection. In their method, every sentences is inserted with a [CLS] +token with global attention, which allows the model to predict on this sentence-level token as +evidence. Most of these works focus on a limited context such as a few Wikipedia documents, a +single article and abstracts or text snippets from research literature or a small synthetic corpus. +Another line of work widely adopted and one of the key tasks in FEVER [11, 12] is to involve +evidence retrieval and selection. The framework exploits larger document context to extract +evidentiary (or rationales) passages as first step and veracity prediction is then modeled to +condition on the claim and the selected rationales. The evidentiary passages can be either at +sentence-level or paragraph-level and report the findings to the claim which can be used to +1https://leaderboard.allenai.org/scifact/submissions/public + +justify each veracity label. Despite the revolutionary breakthroughs with Large-Scale Language +Models (LSLMs), such as GPT-3[13] and ChatGPT2, and their impressive generative capabilities, +these large models are still lacking key zero-shot or few-shot learning capabilities needed for +fact checking tasks. This is mainly due to their incorrectly retrieved, incomplete or outdated +knowledge stored in their weights which makes these techniques susceptible to hallucinations +[14, 15], which is conflicting with fact checking tasks that require factuality as an essential +element in modeling. Moreover, an efficient approach to keep LSLMs up-to-date and grounded +to ever-growing factual and new information is imperative but still unresolved to date. Recent +work [15, 16] shows that lightweight methods with fine-tuned and smaller models outperform +these big models in a range of knowledge-intensive NLP tasks including Natural Language +Inference (NLI), Recognizing textual entailment (RTE), Reading Comprehension (RC), QA, etc. +Sentence-BERT (SBERT) [17] is one of the most popular techniques based on the BERT language +model [18] used for evidence selection [19, 20] which can be framed as a sentence-pair regression +task. SBERT models are used to encode contextualized representations for each of the evidence +passages which are then ranked according to their semantic similarity with the contextualized +representation of the corresponding claim. In the final step, top 𝑘 evidentiary passages are +selected for veracity prediction. The challenge of this multi-staged verification framework is +1) the rationales extracted out-of-context may lack information required to make a prediction +(e.g., acronyms, unresolved coreferences); 2) the evidence extraction (through passages ranking) +requires high quality training data that is costly to obtain with domain experts from both closed +and open domain tasks [21]. Various efforts to address the constraints have been undertaken +to explore 1) paragraph level train data from scientific literature with paper title as claim and +abstract as evidence as high-precision heuristics (e.g., SciFact [1]); 2) QA dataset with question +and answer considered as claim and evidence respectively [22]; 3) NLI dataset with hypothesis +as the claim and premise as evidence [23]. We follow a second line of work for which the +evidence retrieval component is implemented in our system following current SoTA methods. +Automated multi- or cross-modal fact checking is an under developed field compared to +text-based techniques. Recent developments have shown that cross-modal pre-trained models +(e.g.,VideoBERT [24], VisualBERT [25], Uniter [26], CLIP [27]) has achieved significant results +in downstream cross-modal tasks [28, 29, 30] with great transferability for zero-shot or few-shot +scenarios. Our work is inspired by [31], which one of the initial explorations in multimodal +fact-checking task. In their proposed method, Contrastive Language–Image Pre-training (CLIP) +model [27]) is adopted as encoder to learn joint language-image embedding between each +image and the input claim text. Top-5 candidate image evidences are taken as input along +with multi-modal claim for multimodal claim verification model with a simple cross-attention +network. It is worth noting that CLIP model allows to model image-text contextual alignment at +coarse-grained contextual (global) level but ignores the compositional matching of disentangled +concepts (i.e., finer-grained cross-modal alignment at region-word level)[30, 32, 32]. +2https://openai.com/blog/chatgpt/ + +3. Methodology +3.1. Problem statement +We frame the Factify 2 problem as a multimodal entailment task as in the previous submission +[3], which considers a multimodal claim 𝑐 = 𝑐𝑡𝑒𝑥𝑡 + 𝑐𝑖𝑚𝑎𝑔𝑒 as hypothesis and a multimodal +document 𝑑 = 𝑑𝑡𝑒𝑥𝑡 + 𝑑𝑖𝑚𝑎𝑔𝑒 as premise. The goal is to learn a function 𝑓(𝑐, 𝑑) that infers one +of the five entailment categories including "Support_Multimodal", "Support_Text", "Refutes", +"Insufficient_Multimodal" and "Insufficient_Text". Additional details on the task can be found in +[33]. +3.2. General Architecture +Our system architecture follows a standard two-stage claim verification approach as established +through various shared tasks in recent years, typically FEVER[34], FEVER 2.0 [35], FEVEROUS +[36] and SCIVER [37]. First, a textual evidence retrieval component identifies from a given +document the evidence passages most relevant to the corresponding claim text. Then, a trans- +former based cross-modal model is trained on all the input across multi-modalities including +selected evidence passages text, claim text, claim image, document image, claim OCR text +and document OCR text to predict five multimodal entailment categories with respect to the +multimodal claim. A pre-trained cross-modal model (i.e. CLIP) and a pre-trained text embedding +model are both employed in the embedding layer in order to learn a cross-modal matching +model using both unified-multimodal and unimodal representations. Overall, the implemented +architecture adopts listwise concatenation strategy [38] which is one of common strategies in +most recent sequence-to-sequence SoTA veracity prediction models. +Figure 1: Logically General System Architecture + +Cross-modal veracity prediction model +Text Encoder + Cross-modal +(CLIP) +Claim text embed +Transformer Encoder + Self-Attention +Evidence Retrieval +max +Multihead +embedding layer +Masked +Semantic Search +Image Encoder +(cosine similarity) +00 +(CLIP) +Sofmax +4 +Re-rank & Concatenate +Text passages dense +Top K evidence +W +representations +candidates +Claim text dense +Text Embedding Layer +representations +Transformer E ncoder +im+Doc text embedding +Self-Attention +M utihead +xe W +Wr2V +Masked + Pooling +SBERT +(MPNet-QA dense retriever) +Doc +Claim text +Claim Image +Doc Image +Claim ORC text +Doc ORC text3.3. Evidence Retrieval +In evidence retrieval, ‘multi-qa-mpnet-base-dot-v1‘ 3 is employed to compute embeddings for +both claim text and document text at passage level. In terms of passage granularity, both +paragraph-level retrieval and sentence-level retrieval are experimented (see Section 5). This +is a SBERT model based on the MPNet architecture [39] and is trained on a Question-Answer +(QA) dataset with 215M QA pairs from diverse sources. The model was tuned for a semantic +search using a dot-product score function in order to find relevant passages corresponding to a +given query. The model encodes text into a 768-d vector and supports 512 maximum number of +tokens. +Regarding the similarity computation and semantic search, we use a simple dot product with +the normalised SBERT embeddings (as proxy to cosine similarity) which enables a quick and +efficient passage ranking and scalability of up to about 1 Million entries. +Top 𝐾 passages obtained from the semantic search are then re-ranked based on their relevancy +to the claim text and concatenated into a longer text snippet before being fed into the cross-modal +veracity prediction model. +3.4. Embedding Layer +Our embedding layer consists of a cross-modal encoder and a unimodal text encoder. We +hypothesize that modeling solely on text-to-text interaction (i.e., text premise and hypothesis) +can supplement the modeling solely on cross-modal premise and hypothesis interaction and +vice versa. This architecture facilitates the measuring of multi-modal semantic relatedness in +this multi-modal fact checking task by mapping more textual alignment signals into subse- +quent semantic space. This considers that text specific models can capture more accurate and +semantically meaningful word-level or sentence level alignment. +The cross-modal encoder is implemented with a pre-trained CLIP model that aims to map +visual and text embeddings into a common space. The ViT-B/32 variant (ViT-Base with patch +size 32) is chosen in this work because of its smaller amount of parameters, less FLOPS and +greater inference speed. ViT-B/32 consists of a text encoder and an image encoder which +are used to encode text inputs (including claim text, evidentiary passage and two images +OCR text) and image inputs (including claim image and document image) respectively before +concatenating into a 6 × 512 matrix as a single input to subsequent transformer encoder. +The CLIP architecture allows for a maximum input text length of 77 tokens. The pre-trained +Word2vec model ("Word2vec Google News 300") [40] is adopted as a unimodal text encoder. It +encodes the concatenated text sequence of claim and document evidentiary passage text, and +obtains a 300-D feature vector for each token. Zero-padding is applied to match the longest +sentence in the training set. Both the pre-trained CLIP and Word2Vec embedding model were +not fine-tuned. +3The model is available in on the Hugging Face hub and accessible via https://huggingface.co/sentence- +transformers/multi-qa-mpnet-base-dot-v1 + +3.5. Cross-modal veracity prediction +The second component of veracity prediction is based on the well established Transformer +Encoder (TE) architecture, which heavily relies on the concept of self-attention [41] to effectively +model higher-order interactions and context in an input. Recent research has shown that multi- +head self-attention mechanisms and transformer architectures are computationally efficient and +accurate in this regard. The self-attention mechanisms of the TE encoder allows for simple but +powerful reasoning that can identify hidden relationships between vector entities, regardless of +whether they are visual or textual in nature. Therefore, our cross-modal veracity prediction +model is implemented based on self-attention mechanisms to learn the joint distribution of +text representations of claim-document text pair and cross-modal feature representations of all +modalities contained in claim and document . +Specifically, the claim and document embeddings of joint input by CLIP and text input by +text embedding layer are passed through two separate transformer encoder [41] consisting +of 𝑁 identical sequential blocks of a multi-head attention (MHA) and a fully connected feed- +forward network (FFN). Within each transformer encoder, multiple blocks allows for a deeper +understanding of the inputs. For each block the input 𝑥 is passed through a multi-head attention +layer of which the output is added to the initial input such that. Passing on both the initial input +and the output ensures that information in the initial sequence is not lost. Layer normalization is +applied to the output to allow for faster training and small regularization i.e. 𝑥 = LayerNorm(𝑥+ +MHA(𝑥)). The output is then passed to a feed-forward network to allow for more model +complexity. The output is again added to the original input and layer normalization is applied i.e. +𝑥 = LayerNorm(𝑥 + FFN(𝑥)). The output of the final block (i.e., the output of each transformer +encoder in the diagram) is passed through an adaptive max pooling layer to reduce the output +dimensions. The output of two separate transformer encoders are then concatenated before +feeding into a MLP classifier for five categories prediction. The five categories probabilities are +obtained from the final output softmax layer. +4. Factify Dataset +4.1. Dataset Description +The Factify 2 dataset created and supplied by the organisers covers a train, validation, and test +set. The train set contains 35000 data pairs, while the validation and test sets each contain 7500 +data pairs. Each data pair consists of a claim and a document, each of which comprises an image, +a text, and OCR text extracted from the image. The data pairs are annotated with one label from +5 categories including Support_Multimodal, Support_Text, Refute, Insufficient_Multimodal, or +Insufficient_Text. +4.2. Text Length Distribution +The training set text and OCR text length distributions are represented in Figures 2 and 3. The +text length distribution varies between the claim and document text, with the document text +that tends to be much longer. This is expected as it is to be used to verify the claim. From Figure +2 (a), we can can see that claim text is much shorter and less varied for the Refute category + +than for the rest of the categories, which all have similar claim text length distributions. Figure +2 (b) shows that the Support_Multimodal and Support_Text categories have the larger spread +of document text lengths and also the longest document text lengths. The two Insufficient +categories have a smaller document text length distribution, and Refute has the smallest variance +and maximum length in document text length. +Considering the claim OCR length we see from Figure 3a that the Refute category has a much +larger claim OCR length distribution and maximum length than any other category. The second +largest claim OCR length distributions are the Support_Text and the Insufficient_Text categories, +which then leaves the two Multimodal categories with the shortest claim OCR text lengths. The +document OCR length distribution is very similar to that of the claim OCR, from Figure 3b we +see the only real difference is that the two Text categories have a smaller document OCR length +distrubution than that of the claim OCR. +(a) Claim Text Length +(b) Document Text Length +Figure 2: Boxplot of Text Length of all Categories +4.3. Image Similarity Distribution +An image similarity investigation was conducted in order to gain an intuition of the similarity +between the claim and document images for each category. Using image pairwise CLIP embed- +dings we calculate a similarity score and analyse it per category. Figures 4a and 4b illustrate +that the similarity between the claim and document image is comparatively higher within +the categories for Support_Multimodal and Insufficient_Multimodal than the other categories. +The label correlation with similarity of image pairs has been largely increased compared to + +350 +300 +Length of Text +250 +200 +150 +100 +50 +0 +Refute +Support_Multi +Support_Text +Multi +Text +Insuf_ +Insuf_ +Categories40000 +30000 +Length of Text +20000 +10000 +Refute +Multi +_Text +_Multi +Insuf_Text, +R + Support_ +Insuf_ +S +Categories(a) Claim OCR Text Length +(b) Document OCR Text Length +Figure 3: Boxplot of OCR Text Length of all Categories +factity 1 dataset [3] last year. This further indicates that there is explicit correlation multimodal +categories which can be leveraged to learn and verify multimodal entailment categories. +4.4. Multimodal Similarity Distribution +The multimodal CLIP similarity between claim-text pairs is explored to investigate our hypothe- +sis that Image should contain content that is related to claim in order to entail either support or +refute veracity decision. Figures 5a and 5b depict the cosine similarity scores between the claim +text and document image. From the figures, there is no clear indicator of the entailment between +doc image and claim text. However, it is noticeable that “Support_Multimodal” presents the +highest pairwise similarity correlation between label and claim-evidence pair. “Insufficient text” +have the lowest pairwise similarity correlation, although our initial hypothesis was that “Insuf- +ficient_Multimodal” should have the lowest value. This analysis suggests that differentiating +between the different categories based on the claim text and document image correlation could +be challenging. +In terms of correlation between the claim image and document text, due to the maximum +text sequence constraints with CLIP, text access maximum length is truncated. Consequently, +longer context of document text is not incorporated in this analysis. As shown in Figure 6a and +6c, there is low degree of similarity correlation across 5 categories, among which the "Refute" +category shows highest similarity correlation. +Lastly, Figure 6b and Figure 6d about the similarity correlation between the claim image and +the claim text show no significant deviation in similarity scores of different categories when + +800 +Length of Text +600 +400 +200 +工 +工 +0 +Refute + Support_Text +Multi - +Insuf_Text +Insuf_ +Categories800 +Length of Text +600 +400 +200 +工 +0 +Refute +Support_Text +Multi - +Insuf_Text +Insuf_ +Categories(a) Claim Image and Document Image Similarity Score Histogram +(b) Claim Image and Document Image Similarity Boxplot +Figure 4: Claim Image and Document Image Similarity Scores +the claim image and claim text are compared to each other. For the purpose of this task and this +dataset, we hypothesis that the claim image provides supplementary information to the claim +text. + +Occurence +150 +Refute +Support Multi +100 +Support_Text +Insuf Multi +Insuf Text +50 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Similaritv Score1.0 +0.8 +Score +0.6 +S +0.4 +0.2 +Refute +Multi +Text +Insuf Multi +Insuf_Text - +Support_ +Categories(a) Claim Text and Document Image Similarity Score Histogram +(b) Claim Text and Document Image Similarity Boxplot +Figure 5: Claim Image and Document Image Similarity Scores +5. Experiments +5.1. Model settings +To validate and optimal the effect of evidence retrieval, We attempt to experiment our model +with 1) with or without evidence selection; 2) vary length of evidence doc text sorted by evidence +retriever; 3) passage ranking at paragraph level versus sentence level; 4) text-to-text alignment +with SBERT versus cross-modal alignment with CLIP. Both SBERT and CLIP is used to rank +evidence doc with paragraph and sentence level; 5) if SBERT model trained on QA dataset +perform better than general purpose SBERT model. Note that ranking at paragraph level on + +Occurence +Refute +150 +Support_Multi +Support_Text +100 +of +Insuf Multi +Frequency +Insuf Text +50 +0 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.400.40 +0.35 +0.30 +0.25 +0.20 +0.15 +0.10 +Refute +Support_Multi +Support_Text +Insuf_Multi +Insuf_Text +Categories(a) Claim Image and Document Text Similarity Score His- +togram +(b) Claim Image and Claim Text Similarity Score His- +togram +(c) Claim Image and Document Text Similarity Score +Boxplot +(d) Claim Image and Claim Text Similarity Score Box- +plot +Figure 6: Similarity Scores +top <5 or sentence level on top <5 is only option to CLIP due to its maximum allowed length +restriction. +For two transformer encoders, we choose an empirical setting of four heads in two MHAs. +The number of sequential MHA and feed-forward network blocks per embedding input is +𝑁𝑏𝑙𝑜𝑐𝑘𝑠 = 2. All our experiments are trained on 3-layered MLP and number of nodes per layer +are set to 3072, 1024 and 5 respectively. A dropout of 0.5 and ReLU activations are applied +between the MLP layers. +Preliminary experiments conducted in this work are elaborated in details as follows: +• "model_w/o_ER": to validate the effectiveness with evidence retrieval, we remove evidence +retrieval in our system and provide original document text to "Cross-modal veracity +prediction model". + +Occurence +150 +Refute +Support Multi +100 +Support Text +of +Insuf Multi +Frequency +Insuf Text +50 +0 +0.0 +0.1 +0.2 +0.3 +0.4 +Similarity ScoreOccurence +200 +Refute +150 +Support Multi +Support Text +a + 100 +Insuf Multi +Frequency +Insuf Text +50 +0 +0.0 +0.1 +0.2 +0.3 +0.4 +Similarity Score0.5 +0.4 +Similarity Score +0.3 +0.2 +0.1 +0.0 +Refute +Support_Multi +Support_Text +Insuf_Multi +Insuf_Text +Categories0.5 +0.4 +Score +0.3 +Similarity s +0.2 +0.1 +0.0 +Refute +Support_Multi +Support_Text +Insuf_Multi +Insuf_Text +Categories• "SBERT_sentence_ER_top5": One of the "top" 4 performing general purpose SBERT model +("all-MiniLM-L6-v2") is chosen in our experiment. This is a all-round model tuned for +many use-cases and 5 times faster while offering good quality compared to best all-round +model "all-mpnet-base-v2". The model is trained on a large and diverse dataset of over +1 billion training pairs and also fine-tuned for dot-product score function suitable for +cosine similarity. The use of all-round model allows us to evaluate the value of adopting +QA fine-tuned counterpart that we hypothesis as optimal solution. Top 5 sorted sentence +sorted by all-round SBERT model is configured in this setting. +• "SBERT_sentence_ER_top10": Top 10 sorted sentence sorted by all-round SBERT model +is configured in this setting. +• "SBERT_sentence_ER_top15": Top 15 sorted sentence sorted by all-round SBERT model +is configured in this setting. +• "SBERT-QA_paragraph_ER_top5": SBERT QA dataset fine-tuned model (as described +in 3.3) is adopted in this setting to obtain top 5 paragraphs as evidentiary passages for +veracity inference in this setting. +• "SBERT-QA_sentence_ER_top5": Top 5 sentences sorted by SBERT QA model and selected +as evidentiary passages in this setting. +• "BigBird_w/o_ER": To evaluate the value of evidence selection against the long context +modeling solution, the Google’s BigBird pre-trained model fine-tuned on Factity dataset +from last year [3] is used in replace of Word2Vec model in "Text Embedding layer" with this +setting. This BigBird model allows a maximum 1396 tokens and contextual representation +of text is adopted in this setting. +5.2. Training and validation +For our experiment, the model was trained up to 80 epochs with early stopping on minimum +validation loss by minimizing the cross-entropy loss function, the adaptive AdamW optimizer +[42] with initial learning rate 𝛾 = 1e−4 and epsilon 𝜖 = 1e−8 with batch size 𝑁𝑏𝑎𝑡𝑐ℎ = 16. +Early stopping patience is set to 5. A linear decreasing learning rate scheduler was used including +𝑁𝑠𝑡𝑒𝑝𝑠 = 438 warming up training steps during which the learning rate increased linearly to +the chosen learning rate. +We have found that data scraping error leads to invalid doc text content in the development +dateset provided by organiser with 463 and 114 invalid samples in train and val set respectively. +There also are 112 invalid samples in test set. This results in document text containing only +"We’ve detected that JavaScript is disabled in this browser ...". The invalid samples are removed +from our training data. +6. Results and Discussion +The best model results in preliminary experiments described in section 5 are presented in Table +1, Table 2 and Table 3 respectively. +4The best performing general purpose model is selected with a sorted list of model performances and use cases +recommended provided by SBERT, accessible via https://www.sbert.net/docs/pretrained_models.html + +Table 1 +5-way Classification Results of experiments without ER on val set +Categories +model_w/o_ER +BigBird_w/o_ER +P +R +F1 +P +R +F1 +Support_Multimodal +0.73 +0.79 +0.76 +0.73 +0.81 +0.77 +Support_Text +0.71 +0.61 +0.66 +0.77 +0.59 +0.67 +Insufficient_Multimodal +0.66 +0.66 +0.66 +0.64 +0.70 +0.67 +Insufficient_Text +0.71 +0.75 +0.73 +0.73 +0.75 +0.74 +Refute +0.99 +0.98 +0.98 +0.98 +0.98 +0.98 +Weighted Avg. +0.76 +0.76 +0.76 +0.77 +0.77 +0.77 +Table 2 +5-way Classification Results of experiments with all-round SBERT + ER on val set +Categories +SBERT_sentence_ER_top5 +SBERT_sentence_ER_top10 +SBERT_sentence_ER_top15 +P +R +F1 +P +R +F1 +P +R +F1 +Support_Multimodal +0.72 +0.85 +0.78 +0.74 +0.78 +0.76 +0.75 +0.77 +0.76 +Support_Text +0.63 +0.73 +0.68 +0.71 +0.61 +0.66 +0.71 +0.62 +0.66 +Insufficient_Multimodal +0.70 +0.64 +0.67 +0.66 +0.67 +0.66 +0.65 +0.67 +0.66 +Insufficient_Text +0.80 +0.58 +0.67 +0.70 +0.77 +0.74 +0.71 +0.76 +0.73 +Refute +0.96 +0.99 +0.97 +0.96 +0.99 +0.97 +0.98 +0.98 +0.98 +Weighted Avg. +0.76 +0.76 +0.75 +0.76 +0.76 +0.76 +0.76 +0.76 +0.76 +Firstly, the Table 1 shows that our veracity model without ER exhibits a reasonably good +performance and utilising long sequence model (BigBird) for text embedding improves the +base model with a small margin, by 1% for all categories except "Refute". As comparison, +further experiments with ER are conducted in Table 2 and Table 3. The results in Table 2 +indicates that all-round SBERT based evidence selection do not provide obvious performance +improvement based on current preliminary exploration covering three top K sentences settings +(K=5, 10, 15). In contrast, SERT-QA based model achieves big marginal improvement at both +paragraph and sentence level. Our experiments covers both top 5 paragraphs and sentences, +which improves best base model (without ER) by 1% and 2% respectively. Final results across 7 +different experiment setup shows that combining SBERT-QA at top K sentence-level evidence +passage retrieval achieves optimal performance compared to the base model without ER and +the use of all-round SBERT model. The best model "SBERT-QA_sentence_ER_top5" obtains 0.79 +weighted avg. F1 with 20th epochs. +6.1. Competition Result +Final test set results and competition leaderboard are presented in Table 4. The results shows +that top 3 participating systems achieves similar performance and our system is ranked at 3rd +place with a small margin (by 0.028) to the top performing system. Please refer to [43] for the + +Table 3 +5-way Classification Results of experiments with SBERT-QA + ER on val set +Categories +SBERT-QA_paragraph_ER_top5 +SBERT-QA_sentence_ER_top5 +P +R +F1 +P +R +F1 +Support_Multimodal +0.80 +0.77 +0.78 +0.79 +0.83 +0.81 +Support_Text +0.70 +0.68 +0.69 +0.70 +0.69 +0.70 +Insufficient_Multimodal +0.66 +0.72 +0.69 +0.71 +0.72 +0.73 +Insufficient_Text +0.76 +0.72 +0.74 +0.74 +0.72 +0.73 +Refute +0.96 +1.00 +0.98 +0.99 +0.98 +0.98 +Weighted Avg. +0.78 +0.78 +0.78 +0.79 +0.79 +0.79 +Table 4 +Factify Official Leaderboard +Rank +Team +Support_Text +Support_Multi. +Insufficient_Text +Insufficient_Multi. +Refute +Final +1 +Triple-Check +0.828 +0.914 +0.852 +0.892 +1.0 +0.818 +2 +INO +0.812 +0.9 +0.888 +0.852 +0.999 +0.808 +3 +Logically +0.804 +0.905 +0.844 +0.856 +0.985 +0.79 +4 +Zhang +0.766 +0.879 +0.816 +0.879 +0.999 +0.774 +5 +gzw +0.785 +0.863 +0.814 +0.833 +1.0 +0.761 +6 +coco +0.773 +0.865 +0.815 +0.83 +1.0 +0.757 +7 +Noir +0.771 +0.873 +0.785 +0.816 +0.997 +0.745 +8 +Yet +0.707 +0.826 +0.786 +0.719 +1.0 +0.691 +9 +TeamX +0.582 +0.709 +0.537 +0.556 +0.698 +0.456 +- +BASELINE +0.5 +0.827 +0.802 +0.759 +0.988 +0.65 +competition details. +7. Conclusion +In this research, we present our multimodal fact checking system that is submitted to the De- +Factify 2023 competition. The system consists of various components, including a multimodal +fact checking dataset, a QA-enhanced evidence passage retrieval component, and a Transformer- +based cross-modal sequence-to-sequence veracity prediction model. Our findings from the +De-Factify 2023 competition show that recent advances in pre-trained cross-modal models, such +as CLIP, have strong zero-shot or few-shot capabilities and can be effectively transferred to a +variety of downstream tasks, including multimodal fact checking. However, there is still a need +for more effective techniques for multimodal modeling and explainability, particularly in regards +to learning finer-grained cross-modal representations by jointly modeling intra- and inter- +modality relationships and aligning vision regions with sentence words or entities. 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Kumar, Findings of factify 2: multimodal +fake news detection, in: proceedings of defactify 2: second workshop on Multimodal +Fact-Checking and Hate Speech Detection, CEUR, 2023. + diff --git a/HNE1T4oBgHgl3EQfXgRU/content/tmp_files/load_file.txt b/HNE1T4oBgHgl3EQfXgRU/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d7cf833c59a177bf8360e4b1dd014de6db8bfc09 --- /dev/null +++ b/HNE1T4oBgHgl3EQfXgRU/content/tmp_files/load_file.txt @@ -0,0 +1,995 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf,len=994 +page_content='Logically at Factify 2023: A Multi-Modal Fact Checking System Based on Evidence Retrieval techniques and Transformer Encoder Architecture Pim Jordi Verschuuren, Jie Gao, Adelize van Eeden, Stylianos Oikonomou and Anil Bandhakavi Brookfoot Mills, Brookfoot Industrial Estate, Brighouse, HD6 2RW, United Kingdom Abstract In this paper, we present the Logically submissions to De-Factify 2 challenge (DE-FACTIFY 2023) on the task 1 of Multi-Modal Fact Checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' We describes our submissions to this challenge including explored evidence retrieval and selection techniques, pre-trained cross-modal and unimodal models, and a cross-modal veracity model based on the well established Transformer Encoder (TE) architecture which is heavily relies on the concept of self-attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Exploratory analysis is also conducted on this Factify 2 data set that uncovers the salient multi-modal patterns and hypothesis motivating the architecture proposed in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' A series of preliminary experiments were done to investigate and benchmarking different pre-trained embedding models, evidence retrieval settings and thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The final system, a standard two-stage evidence based veracity detection system, yields weighted avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='79 on both val set and final blind test set on the task 1, which achieves 3rd place with a small margin to the top performing system on the leaderboard among 9 participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Keywords fact verification, multimodal representation learning, multimodal entailment, text entailment, Multi-head Attention 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Introduction Misinformation and fake news can spread rapidly and cause harm at various levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' One way to protect ourselves from these negative impacts is through fact-checking and debunking false information with evidence-based reporting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' However, this process can be resource-intensive and time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' To address this issue, researchers have developed automated fact-checking systems using deep learning techniques, which can handle tasks such as claim detection, claim matching, evidence retrieval, and veracity prediction using natural language processing techniques on textual content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' While there has been progress in this area, there is still a need for multimodal approaches that can handle both text and image inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' To address this gap, this paper presents a multimodal veracity prediction system for automated fact-checking and is De-Factify: Workshop on Multimodal Fact-Checking and Hate Speech Detection, co-located with AAAI 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' 2023 Washington DC, USA � Pim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='jv@logically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='uk (P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Verschuuren);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' jie@logically.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Gao) © 2023 Copyright for this paper by its authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Use permitted under Creative Commons License Attribution 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='0 International (CC BY 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' CEUR Workshop Proceedings http://ceur-ws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='org ISSN 1613-0073 CEUR Workshop Proceedings (CEUR-WS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='org) arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='03127v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='CL] 9 Jan 2023 developed as part of the Factify 2 competition organized by De-Factify@AAAI 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The remainder of the paper is structured as follows: Section 2 presents a brief overview of related work and section 3 describes our general framework and model architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Section 4 discusses the dataset supplied by the Factify 2 competition followed by an overview of our experiments in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Section 6 and 7 present the final results and our conclusions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Related Work As an essential part of automated fact verification, effective techniques for modeling claim- evidence for veracity prediction have been a hot topic and key research questions in existing fact-checking methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Most of the recent work focuses on using textual evidence in veracity prediction and there are mainly two lines of work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' One direction [1, 2, 3] is to use a single document (such as is provided in the Factify task dataset) with long text evidence and through leveraging models constructed for long sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Examples such as BigBird [4], Longformer[5] and recent advancements in the ConvNets architecture witnessed in the Long Range Arena leaderboard (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=', Mega [6], S5[7]) are seen to obtain top results in a wide range of tasks and leader boards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The benefits of exploiting long-sequence model at document level is a) the simplicity of the overall architecture;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' b) allows to accommodate for more context of the whole article into modeling and natural language inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' An optimal setup of the maximum length for both claim (or query) and document sequence, and the document level veracity labels is commonly required [8, 1, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The advantage of incorporating lots of context into inference is also seen in modeling question answering (QA) tasks [4, 5], for which the document-level veracity labels are relatively "cheap" to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The downside of using a simple long-text model technique at document-level is the lack of interpretability (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' evidence selection), it is computational expensive, the limitation in dealing with the complexity of certain (multi-hop) claims [9], and lack of diversity and scalability when dealing with a large amount of diverse documents in a real-world application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' These constraints were more apparent in open domain fact checking task that make use of web data extracted with commercial search engines as building blocks in fact-checking system in order to incorporate more diverse sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' It is worth to note that long-sequence model can be adapted for the purpose of evidence selection e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=', through framing the task as a token-level prediction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' For instance, as one of the top systems in SciFact leaderboard 1, LongChecker [10] used LongFormer [5] for scientific claim verification with paragraph-level evidence selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' In their method, every sentences is inserted with a [CLS] token with global attention, which allows the model to predict on this sentence-level token as evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Most of these works focus on a limited context such as a few Wikipedia documents, a single article and abstracts or text snippets from research literature or a small synthetic corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Another line of work widely adopted and one of the key tasks in FEVER [11, 12] is to involve evidence retrieval and selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The framework exploits larger document context to extract evidentiary (or rationales) passages as first step and veracity prediction is then modeled to condition on the claim and the selected rationales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The evidentiary passages can be either at sentence-level or paragraph-level and report the findings to the claim which can be used to 1https://leaderboard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='allenai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='org/scifact/submissions/public justify each veracity label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Despite the revolutionary breakthroughs with Large-Scale Language Models (LSLMs), such as GPT-3[13] and ChatGPT2, and their impressive generative capabilities, these large models are still lacking key zero-shot or few-shot learning capabilities needed for fact checking tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' This is mainly due to their incorrectly retrieved, incomplete or outdated knowledge stored in their weights which makes these techniques susceptible to hallucinations [14, 15], which is conflicting with fact checking tasks that require factuality as an essential element in modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Moreover, an efficient approach to keep LSLMs up-to-date and grounded to ever-growing factual and new information is imperative but still unresolved to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Recent work [15, 16] shows that lightweight methods with fine-tuned and smaller models outperform these big models in a range of knowledge-intensive NLP tasks including Natural Language Inference (NLI), Recognizing textual entailment (RTE), Reading Comprehension (RC), QA, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Sentence-BERT (SBERT) [17] is one of the most popular techniques based on the BERT language model [18] used for evidence selection [19, 20] which can be framed as a sentence-pair regression task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' SBERT models are used to encode contextualized representations for each of the evidence passages which are then ranked according to their semantic similarity with the contextualized representation of the corresponding claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' In the final step, top 𝑘 evidentiary passages are selected for veracity prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The challenge of this multi-staged verification framework is 1) the rationales extracted out-of-context may lack information required to make a prediction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=', acronyms, unresolved coreferences);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' 2) the evidence extraction (through passages ranking) requires high quality training data that is costly to obtain with domain experts from both closed and open domain tasks [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Various efforts to address the constraints have been undertaken to explore 1) paragraph level train data from scientific literature with paper title as claim and abstract as evidence as high-precision heuristics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=', SciFact [1]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' 2) QA dataset with question and answer considered as claim and evidence respectively [22];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' 3) NLI dataset with hypothesis as the claim and premise as evidence [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' We follow a second line of work for which the evidence retrieval component is implemented in our system following current SoTA methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Automated multi- or cross-modal fact checking is an under developed field compared to text-based techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Recent developments have shown that cross-modal pre-trained models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=',VideoBERT [24], VisualBERT [25], Uniter [26], CLIP [27]) has achieved significant results in downstream cross-modal tasks [28, 29, 30] with great transferability for zero-shot or few-shot scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Our work is inspired by [31], which one of the initial explorations in multimodal fact-checking task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' In their proposed method, Contrastive Language–Image Pre-training (CLIP) model [27]) is adopted as encoder to learn joint language-image embedding between each image and the input claim text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Top-5 candidate image evidences are taken as input along with multi-modal claim for multimodal claim verification model with a simple cross-attention network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' It is worth noting that CLIP model allows to model image-text contextual alignment at coarse-grained contextual (global) level but ignores the compositional matching of disentangled concepts (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=', finer-grained cross-modal alignment at region-word level)[30, 32, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' 2https://openai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='com/blog/chatgpt/ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Methodology 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Problem statement We frame the Factify 2 problem as a multimodal entailment task as in the previous submission [3], which considers a multimodal claim 𝑐 = 𝑐𝑡𝑒𝑥𝑡 + 𝑐𝑖𝑚𝑎𝑔𝑒 as hypothesis and a multimodal document 𝑑 = 𝑑𝑡𝑒𝑥𝑡 + 𝑑𝑖𝑚𝑎𝑔𝑒 as premise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The goal is to learn a function 𝑓(𝑐, 𝑑) that infers one of the five entailment categories including "Support_Multimodal", "Support_Text", "Refutes", "Insufficient_Multimodal" and "Insufficient_Text".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Additional details on the task can be found in [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' General Architecture Our system architecture follows a standard two-stage claim verification approach as established through various shared tasks in recent years, typically FEVER[34], FEVER 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='0 [35], FEVEROUS [36] and SCIVER [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' First, a textual evidence retrieval component identifies from a given document the evidence passages most relevant to the corresponding claim text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Then, a trans- former based cross-modal model is trained on all the input across multi-modalities including selected evidence passages text, claim text, claim image, document image, claim OCR text and document OCR text to predict five multimodal entailment categories with respect to the multimodal claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' A pre-trained cross-modal model (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' CLIP) and a pre-trained text embedding model are both employed in the embedding layer in order to learn a cross-modal matching model using both unified-multimodal and unimodal representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Overall, the implemented architecture adopts listwise concatenation strategy [38] which is one of common strategies in most recent sequence-to-sequence SoTA veracity prediction models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='Figure 1: Logically General System Architecture ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='Cross-modal veracity prediction model ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='representations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='Transformer E ncoder ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='im+Doc text embedding ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='Self-Attention ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='M utihead ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='xe W ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='Wr2V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='Masked ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='Pooling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='SBERT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='(MPNet-QA dense retriever) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='Doc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='Claim text ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='Claim Image ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='Doc Image ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='Claim ORC text ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='Doc ORC text3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Evidence Retrieval In evidence retrieval, ‘multi-qa-mpnet-base-dot-v1‘ 3 is employed to compute embeddings for both claim text and document text at passage level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' In terms of passage granularity, both paragraph-level retrieval and sentence-level retrieval are experimented (see Section 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' This is a SBERT model based on the MPNet architecture [39] and is trained on a Question-Answer (QA) dataset with 215M QA pairs from diverse sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The model was tuned for a semantic search using a dot-product score function in order to find relevant passages corresponding to a given query.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The model encodes text into a 768-d vector and supports 512 maximum number of tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Regarding the similarity computation and semantic search, we use a simple dot product with the normalised SBERT embeddings (as proxy to cosine similarity) which enables a quick and efficient passage ranking and scalability of up to about 1 Million entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Top 𝐾 passages obtained from the semantic search are then re-ranked based on their relevancy to the claim text and concatenated into a longer text snippet before being fed into the cross-modal veracity prediction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Embedding Layer Our embedding layer consists of a cross-modal encoder and a unimodal text encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' We hypothesize that modeling solely on text-to-text interaction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=', text premise and hypothesis) can supplement the modeling solely on cross-modal premise and hypothesis interaction and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' This architecture facilitates the measuring of multi-modal semantic relatedness in this multi-modal fact checking task by mapping more textual alignment signals into subse- quent semantic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' This considers that text specific models can capture more accurate and semantically meaningful word-level or sentence level alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The cross-modal encoder is implemented with a pre-trained CLIP model that aims to map visual and text embeddings into a common space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The ViT-B/32 variant (ViT-Base with patch size 32) is chosen in this work because of its smaller amount of parameters, less FLOPS and greater inference speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' ViT-B/32 consists of a text encoder and an image encoder which are used to encode text inputs (including claim text, evidentiary passage and two images OCR text) and image inputs (including claim image and document image) respectively before concatenating into a 6 × 512 matrix as a single input to subsequent transformer encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The CLIP architecture allows for a maximum input text length of 77 tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The pre-trained Word2vec model ("Word2vec Google News 300") [40] is adopted as a unimodal text encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' It encodes the concatenated text sequence of claim and document evidentiary passage text, and obtains a 300-D feature vector for each token.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Zero-padding is applied to match the longest sentence in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Both the pre-trained CLIP and Word2Vec embedding model were not fine-tuned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' 3The model is available in on the Hugging Face hub and accessible via https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='co/sentence- transformers/multi-qa-mpnet-base-dot-v1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Cross-modal veracity prediction The second component of veracity prediction is based on the well established Transformer Encoder (TE) architecture, which heavily relies on the concept of self-attention [41] to effectively model higher-order interactions and context in an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Recent research has shown that multi- head self-attention mechanisms and transformer architectures are computationally efficient and accurate in this regard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The self-attention mechanisms of the TE encoder allows for simple but powerful reasoning that can identify hidden relationships between vector entities, regardless of whether they are visual or textual in nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Therefore, our cross-modal veracity prediction model is implemented based on self-attention mechanisms to learn the joint distribution of text representations of claim-document text pair and cross-modal feature representations of all modalities contained in claim and document .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Specifically, the claim and document embeddings of joint input by CLIP and text input by text embedding layer are passed through two separate transformer encoder [41] consisting of 𝑁 identical sequential blocks of a multi-head attention (MHA) and a fully connected feed- forward network (FFN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Within each transformer encoder, multiple blocks allows for a deeper understanding of the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' For each block the input 𝑥 is passed through a multi-head attention layer of which the output is added to the initial input such that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Passing on both the initial input and the output ensures that information in the initial sequence is not lost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Layer normalization is applied to the output to allow for faster training and small regularization i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' 𝑥 = LayerNorm(𝑥+ MHA(𝑥)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The output is then passed to a feed-forward network to allow for more model complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The output is again added to the original input and layer normalization is applied i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' 𝑥 = LayerNorm(𝑥 + FFN(𝑥)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The output of the final block (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=', the output of each transformer encoder in the diagram) is passed through an adaptive max pooling layer to reduce the output dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The output of two separate transformer encoders are then concatenated before feeding into a MLP classifier for five categories prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The five categories probabilities are obtained from the final output softmax layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Factify Dataset 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Dataset Description The Factify 2 dataset created and supplied by the organisers covers a train, validation, and test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The train set contains 35000 data pairs, while the validation and test sets each contain 7500 data pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Each data pair consists of a claim and a document, each of which comprises an image, a text, and OCR text extracted from the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The data pairs are annotated with one label from 5 categories including Support_Multimodal, Support_Text, Refute, Insufficient_Multimodal, or Insufficient_Text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Text Length Distribution The training set text and OCR text length distributions are represented in Figures 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The text length distribution varies between the claim and document text, with the document text that tends to be much longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' This is expected as it is to be used to verify the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' From Figure 2 (a), we can can see that claim text is much shorter and less varied for the Refute category than for the rest of the categories, which all have similar claim text length distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Figure 2 (b) shows that the Support_Multimodal and Support_Text categories have the larger spread of document text lengths and also the longest document text lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The two Insufficient categories have a smaller document text length distribution, and Refute has the smallest variance and maximum length in document text length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Considering the claim OCR length we see from Figure 3a that the Refute category has a much larger claim OCR length distribution and maximum length than any other category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The second largest claim OCR length distributions are the Support_Text and the Insufficient_Text categories, which then leaves the two Multimodal categories with the shortest claim OCR text lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The document OCR length distribution is very similar to that of the claim OCR, from Figure 3b we see the only real difference is that the two Text categories have a smaller document OCR length distrubution than that of the claim OCR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' (a) Claim Text Length (b) Document Text Length Figure 2: Boxplot of Text Length of all Categories 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Image Similarity Distribution An image similarity investigation was conducted in order to gain an intuition of the similarity between the claim and document images for each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Using image pairwise CLIP embed- dings we calculate a similarity score and analyse it per category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Figures 4a and 4b illustrate that the similarity between the claim and document image is comparatively higher within the categories for Support_Multimodal and Insufficient_Multimodal than the other categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The label correlation with similarity of image pairs has been largely increased compared to 350 300 Length of Text 250 200 150 100 50 0 Refute Support_Multi Support_Text Multi Text Insuf_ Insuf_ Categories40000 30000 Length of Text 20000 10000 Refute Multi _Text _Multi Insuf_Text, R Support_ Insuf_ S Categories(a) Claim OCR Text Length (b) Document OCR Text Length Figure 3: Boxplot of OCR Text Length of all Categories factity 1 dataset [3] last year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' This further indicates that there is explicit correlation multimodal categories which can be leveraged to learn and verify multimodal entailment categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Multimodal Similarity Distribution The multimodal CLIP similarity between claim-text pairs is explored to investigate our hypothe- sis that Image should contain content that is related to claim in order to entail either support or refute veracity decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Figures 5a and 5b depict the cosine similarity scores between the claim text and document image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' From the figures, there is no clear indicator of the entailment between doc image and claim text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' However, it is noticeable that “Support_Multimodal” presents the highest pairwise similarity correlation between label and claim-evidence pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' “Insufficient text” have the lowest pairwise similarity correlation, although our initial hypothesis was that “Insuf- ficient_Multimodal” should have the lowest value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' This analysis suggests that differentiating between the different categories based on the claim text and document image correlation could be challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' In terms of correlation between the claim image and document text, due to the maximum text sequence constraints with CLIP, text access maximum length is truncated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Consequently, longer context of document text is not incorporated in this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' As shown in Figure 6a and 6c, there is low degree of similarity correlation across 5 categories, among which the "Refute" category shows highest similarity correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Lastly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Figure 6b and Figure 6d about the similarity correlation between the claim image and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='the claim text show no significant deviation in similarity scores of different categories when ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='Length of Text ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='工 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='工 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='Refute ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='Support_Text ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='Multi - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='Insuf_Text ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='Insuf_ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='Categories800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='Length of Text ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='工 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='Refute ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='Support_Text ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='Multi - ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='Insuf_Text ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='Insuf_ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='Categories(a) Claim Image and Document Image Similarity Score Histogram ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='(b) Claim Image and Document Image Similarity Boxplot ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='Figure 4: Claim Image and Document Image Similarity Scores ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='the claim image and claim text are compared to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' For the purpose of this task and this dataset, we hypothesis that the claim image provides supplementary information to the claim text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Occurence 150 Refute Support Multi 100 Support_Text Insuf Multi Insuf Text 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='0 Similaritv Score1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='8 Score 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='6 S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='2 Refute Multi Text Insuf Multi Insuf_Text - Support_ Categories(a) Claim Text and Document Image Similarity Score Histogram (b) Claim Text and Document Image Similarity Boxplot Figure 5: Claim Image and Document Image Similarity Scores 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Experiments 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Model settings To validate and optimal the effect of evidence retrieval, We attempt to experiment our model with 1) with or without evidence selection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' 2) vary length of evidence doc text sorted by evidence retriever;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' 3) passage ranking at paragraph level versus sentence level;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' 4) text-to-text alignment with SBERT versus cross-modal alignment with CLIP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Both SBERT and CLIP is used to rank evidence doc with paragraph and sentence level;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' 5) if SBERT model trained on QA dataset perform better than general purpose SBERT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Note that ranking at paragraph level on Occurence Refute 150 Support_Multi Support_Text 100 of Insuf Multi Frequency Insuf Text 50 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='400.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='10 Refute Support_Multi Support_Text Insuf_Multi Insuf_Text Categories(a) Claim Image and Document Text Similarity Score His- togram (b) Claim Image and Claim Text Similarity Score His- togram (c) Claim Image and Document Text Similarity Score Boxplot (d) Claim Image and Claim Text Similarity Score Box- plot Figure 6: Similarity Scores top <5 or sentence level on top <5 is only option to CLIP due to its maximum allowed length restriction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' For two transformer encoders, we choose an empirical setting of four heads in two MHAs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The number of sequential MHA and feed-forward network blocks per embedding input is 𝑁𝑏𝑙𝑜𝑐𝑘𝑠 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' All our experiments are trained on 3-layered MLP and number of nodes per layer are set to 3072, 1024 and 5 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' A dropout of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='5 and ReLU activations are applied between the MLP layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Preliminary experiments conducted in this work are elaborated in details as follows: "model_w/o_ER": to validate the effectiveness with evidence retrieval, we remove evidence retrieval in our system and provide original document text to "Cross-modal veracity prediction model".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Occurence 150 Refute Support Multi 100 Support Text of Insuf Multi Frequency Insuf Text 50 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='4 Similarity ScoreOccurence 200 Refute 150 Support Multi Support Text a 100 Insuf Multi Frequency Insuf Text 50 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='4 Similarity Score0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='4 Similarity Score 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='0 Refute Support_Multi Support_Text Insuf_Multi Insuf_Text Categories0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='4 Score 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='3 Similarity s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='0 Refute Support_Multi Support_Text Insuf_Multi Insuf_Text Categories• "SBERT_sentence_ER_top5": One of the "top" 4 performing general purpose SBERT model ("all-MiniLM-L6-v2") is chosen in our experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' This is a all-round model tuned for many use-cases and 5 times faster while offering good quality compared to best all-round model "all-mpnet-base-v2".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The model is trained on a large and diverse dataset of over 1 billion training pairs and also fine-tuned for dot-product score function suitable for cosine similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The use of all-round model allows us to evaluate the value of adopting QA fine-tuned counterpart that we hypothesis as optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Top 5 sorted sentence sorted by all-round SBERT model is configured in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' "SBERT_sentence_ER_top10": Top 10 sorted sentence sorted by all-round SBERT model is configured in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' "SBERT_sentence_ER_top15": Top 15 sorted sentence sorted by all-round SBERT model is configured in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' "SBERT-QA_paragraph_ER_top5": SBERT QA dataset fine-tuned model (as described in 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='3) is adopted in this setting to obtain top 5 paragraphs as evidentiary passages for veracity inference in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' "SBERT-QA_sentence_ER_top5": Top 5 sentences sorted by SBERT QA model and selected as evidentiary passages in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' "BigBird_w/o_ER": To evaluate the value of evidence selection against the long context modeling solution, the Google’s BigBird pre-trained model fine-tuned on Factity dataset from last year [3] is used in replace of Word2Vec model in "Text Embedding layer" with this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' This BigBird model allows a maximum 1396 tokens and contextual representation of text is adopted in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Training and validation For our experiment, the model was trained up to 80 epochs with early stopping on minimum validation loss by minimizing the cross-entropy loss function, the adaptive AdamW optimizer [42] with initial learning rate 𝛾 = 1e−4 and epsilon 𝜖 = 1e−8 with batch size 𝑁𝑏𝑎𝑡𝑐ℎ = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Early stopping patience is set to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' A linear decreasing learning rate scheduler was used including 𝑁𝑠𝑡𝑒𝑝𝑠 = 438 warming up training steps during which the learning rate increased linearly to the chosen learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' We have found that data scraping error leads to invalid doc text content in the development dateset provided by organiser with 463 and 114 invalid samples in train and val set respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' There also are 112 invalid samples in test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' This results in document text containing only "We’ve detected that JavaScript is disabled in this browser .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The invalid samples are removed from our training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Results and Discussion The best model results in preliminary experiments described in section 5 are presented in Table 1, Table 2 and Table 3 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' 4The best performing general purpose model is selected with a sorted list of model performances and use cases recommended provided by SBERT, accessible via https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='sbert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='net/docs/pretrained_models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='html Table 1 5-way Classification Results of experiments without ER on val set Categories model_w/o_ER BigBird_w/o_ER P R F1 P R F1 Support_Multimodal 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='76 Firstly, the Table 1 shows that our veracity model without ER exhibits a reasonably good performance and utilising long sequence model (BigBird) for text embedding improves the base model with a small margin, by 1% for all categories except "Refute".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' As comparison, further experiments with ER are conducted in Table 2 and Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The results in Table 2 indicates that all-round SBERT based evidence selection do not provide obvious performance improvement based on current preliminary exploration covering three top K sentences settings (K=5, 10, 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' In contrast, SERT-QA based model achieves big marginal improvement at both paragraph and sentence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Our experiments covers both top 5 paragraphs and sentences, which improves best base model (without ER) by 1% and 2% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Final results across 7 different experiment setup shows that combining SBERT-QA at top K sentence-level evidence passage retrieval achieves optimal performance compared to the base model without ER and the use of all-round SBERT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The best model "SBERT-QA_sentence_ER_top5" obtains 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='79 weighted avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' F1 with 20th epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Competition Result Final test set results and competition leaderboard are presented in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The results shows that top 3 participating systems achieves similar performance and our system is ranked at 3rd place with a small margin (by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='028) to the top performing system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Please refer to [43] for the Table 3 5-way Classification Results of experiments with SBERT-QA + ER on val set Categories SBERT-QA_paragraph_ER_top5 SBERT-QA_sentence_ER_top5 P R F1 P R F1 Support_Multimodal 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='988 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content='65 competition details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Conclusion In this research, we present our multimodal fact checking system that is submitted to the De- Factify 2023 competition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The system consists of various components, including a multimodal fact checking dataset, a QA-enhanced evidence passage retrieval component, and a Transformer- based cross-modal sequence-to-sequence veracity prediction model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Our findings from the De-Factify 2023 competition show that recent advances in pre-trained cross-modal models, such as CLIP, have strong zero-shot or few-shot capabilities and can be effectively transferred to a variety of downstream tasks, including multimodal fact checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' However, there is still a need for more effective techniques for multimodal modeling and explainability, particularly in regards to learning finer-grained cross-modal representations by jointly modeling intra- and inter- modality relationships and aligning vision regions with sentence words or entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Additionally, more focus should be placed on real-world challenges that involve handling large amounts of textual and multimodal information from multiple sources and domains for claim verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' There is also a need for techniques that can effectively handle more complex and nuanced real-world scenarios, such as those involving sarcasm, irony, and misleading context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' The difficulties in creating large and high-quality multimodal fact checking datasets that accurately reflect real-world scenarios, as identified in our last year work, remain a significant challenge.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Sheth, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Chinnakotla, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Ekbal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} +page_content=' Kumar, Findings of factify 2: multimodal fake news detection, in: proceedings of defactify 2: second workshop on Multimodal Fact-Checking and Hate Speech Detection, CEUR, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNE1T4oBgHgl3EQfXgRU/content/2301.03127v1.pdf'} diff --git a/HdA0T4oBgHgl3EQfB_8L/vector_store/index.pkl b/HdA0T4oBgHgl3EQfB_8L/vector_store/index.pkl new file mode 100644 index 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[math.FA] 5 Jan 2023 +RECURRENT AND (STRONGLY) RESOLVABLE +GRAPHS +DANIEL LENZ, SIMON PUCHERT, AND MARCEL SCHMIDT +Abstract. We develop a new approach to recurrence and the +existence of non-constant harmonic functions on infinite weighted +graphs. The approach is based on the capacity of subsets of metric +boundaries with respect to intrinsic metrics. The main tool is a +connection between polar sets in such boundaries and null sets of +paths. This connection relies on suitably diverging functions of +finite energy. +1. Introduction +Potential theory on infinite weighted graphs (sometimes called net- +works) studies the induced graph energy functional and derived quanti- +ties (e.g. harmonic functions, random walks, resistances, Laplacians,...). +As already noted in the pioneering works by Yamasaki [27, 16, 28], so- +called null sets of infinite paths play an important role in this theory. +For a comprehensive account of Yamasaki’s work (and beyond) we refer +to the book [22]. +Another approach to potential theory on weighted graphs is via +Dirichlet forms. +The graph energy induces a Dirichlet form, which +in turn leads to the notion of capacity of sets. In this language sets of +capacity zero, so-called polar sets, are key to understanding properties +of the Dirichlet form. In contrast to the situation on graphs, for general +Dirichlet forms the concept of a path may be meaningless. Instead, for +many applications it turned out fruitful to consider intrinsic metrics as +geometric input, see e.g. [23, 24, 25, 3, 10]. +In this paper we follow the intrinsic metric line of thinking. Our main +observation relates polar sets in certain metric boundaries of graphs to +null sets of paths, see Theorem 3.6. This sheds a new light on classical +results formulated in terms of null sets of paths and allows us to prove +new theorems. In this text we focus on consequences for recurrence +and the existence of non-constant harmonic functions of finite energy: +We prove a new characterization of recurrence in terms of the ex- +istence of intrinsic metrics with finite balls, see Theorem 4.2. This in +turn leads to an alternative proof of a classical characterization of re- +currence due to Yamasaki, see Corollary 4.3 and a characterization of +recurrence in terms of metric boundaries being polar, see Corollary 4.4 +and Theorem 4.5. +1 + +2 +LENZ, PUCHERT, AND SCHMIDT +We then turn to the problem of the existence of non-constant har- +monic functions. Again, we tackle this problem by means of capacity +on the boundary. Specifically, we introduce the notion of strong resolv- +ability, which is essentially a path-free and capacity-based version of +the notion of resolvability studied in [2]. We show that strongly resolv- +able transient graphs admit non-constant harmonic functions of finite +energy, see Corollary 5.8 and Corollary 5.9. Since strong resolvability +is stronger than resolvability (this is a consequence of our main obser- +vation mentioned previously), we also prove that locally finite planar +graphs of bounded geometry, the main class of examples of resolvable +graphs, are even strongly resolvable. +This allows us to recover one +of the main results from [2] that transient locally finite planar graphs +of bounded geometry have non-constant harmonic functions of finite +energy. Note that recently more precise descriptions of the space of +harmonic functions of planar graphs were obtained, see [1, 5, 9], which +are beyond the scope of our theory. Another consequence of planar +graphs of bounded geometry being strongly resolvable is that they are +never canonically compactifiable, see Theorem 6.4. The latter class of +graphs was introduced and studied in [6]. +Our results show that two basic issues in the theory of recurrence viz +characterization of recurrence and existence of non-constant harmonic +functions can naturally be understood in terms of capacities of (suit- +able) metric boundaries: Recurrence means that metric boundaries +are negligible in the sense of having capacity zero whereas existence +of non-constant harmonic functions is implied by some richness in the +structure of such a boundary in the sense of the positive capacity of +the whole boundary not being concentrated on a single point. +Since our methods do not rely on paths but only on intrinsic metrics, +they are not limited to locally finite graphs as is sometimes the case in +the classical setting. Moreover, they can be adapted to more general +Dirichlet spaces and even non-linear energies. Both directions will be +investigated in upcoming works. +Parts of this text are based on Simon Puchert’s master’s thesis. +Acknowledgments: Partial support of DFG, in particular, within +the Priority programme ’Geometry at infinity’ is gratefully acknowl- +edged. +2. Preliminaries +In this section we introduce the notation and the objects that are +used throughout the text. For a, b ∈ R we let a ∧ b = min{a, b} and +a ∨ b = max{a, b}. Moreover, a+ = a ∨ 0 and a− = (−a)+. We extend +this notation pointwise to real-valued functions. +2.1. Graphs and Dirichlet energy. Our study of graphs is based +on an analytic tool given by the Dirichlet energy. + +RECURRENT AND (STRONGLY) RESOLVABLE GRAPHS +3 +A graph G = (X, b) consists of a nonempty countable set X, whose +elements are called nodes or vertices, and a symmetric edge weight +function b : X × X → [0, ∞), satisfying the following conditions: The +edge weight b vanishes on the diagonal, i.e. b(x, x) = 0 for all x ∈ X, +and the weighted vertex degree +deg(x) := +� +y∈X +b(x, y) +must be finite for all x ∈ X. If the weighted vertex degree is bounded, +we say that the graph has bounded geometry. If the function b takes +only values in {0, 1}, the graph (X, b) is called combinatorial. +Two vertices x, y ∈ X are said to be connected by the edge (x, y) +if b(x, y) > 0. +In this case, we write x ∼ y. +Note that since b is +symmetric, we have x ∼ y if and only if y ∼ x. The set of all (oriented) +edges of G is denoted by +E := {(x, y) ∈ X × X | x ∼ y}. +We say that a graph is locally finite if for all x ∈ X the set of its +neighbors {y ∈ X | x ∼ y} is finite. A locally finite graph is called a +bounded valence graph if the cardinality of the set of neighbors of each +vertex is bounded by a universal constant. +A path in G is a (finite or infinite) sequence (x1, x2, . . .) of nodes +such that xi ∼ xi+1, for i = 1, 2, . . .. We say that two points x, y ∈ X +are connected if there is a finite path (x = x1, . . . , xn = y). +This +defines an equivalence relation on the set of vertices and the resulting +equivalence classes are called connected components. From now on we +will generally assume that the graph G is connected without +mentioning this explicitly. This is not a real restriction because all +our considerations in this text can be reduced to connected components. +We equip X with the discrete topology and write C(X) for all real- +valued functions on X (the continuous functions on X) and Cc(X) +for the finitely supported real-valued functions on X (the continuous +functions of compact support). Any function m : X → (0, ∞) induces +a Radon measure of full support on all subsets of X via +m(A) := +� +x∈A +m(x), +A ⊆ X. +In what follows we do not distinguish between such measures and +strictly positive functions and simply call them measures on X. +Every weighted graph G = (X, b) gives rise to a quadratic form +Q: C(X) → [0, ∞] that assigns to any function f : X → R its Dirichlet +energy +Q(f) := 1 +2 +� +x,y∈X +b(x, y)|f(x) − f(y)|2. + +4 +LENZ, PUCHERT, AND SCHMIDT +The space of functions of finite energy is +D(G) := {f : X → R | Q(f) < ∞}, +on which Q acts as a bilinear form by polarization, namely +Q(f, g) = 1 +2 +� +x,y∈X +b(x, y)(f(x) − f(y))(g(x) − g(y)). +Here we abuse notation so that Q(f) = Q(f, f) for every f ∈ D(G). +The form Q has the following fundamental semi-continuity property, +which is a direct consequence of Fatou’s lemma. +Proposition 2.1 (Semicontinuity of Q). Let (X, b) be a graph. Let (fn) +be a sequence of functions on X converging pointwise to the function +f. Then, +Q(f) ≤ lim inf +n→∞ Q(fn), +where the value ∞ is allowed. +A map C : R → R is called contraction if |C(s)−C(t)| ≤ |s−t| holds +for all s, t ∈ R. The Dirichlet energy has the important property that +it is reduced by contractions. Specifically, the following proposition is +a direct consequence of the definition. +Proposition 2.2 (Fundamental contraction property). Let (X, b) be +a graph. For all f ∈ D(G) and all contractions C : R → R we have +C ◦ f ∈ D(G) and +Q(C ◦ f) ≤ Q(f). +Remark. Two useful families of contractions that will be of use later +are the following: For a, b ∈ R we define the clamping function C[a,b] : R → +R by +C[a,b](x) := (x ∧ b) ∨ a +and for c ∈ R we define the slicing function Sc : R → R by +Sc(x) := (x − c)+ ∧ 1. +For o ∈ X we define the (pseudo-)norm ∥·∥o : D(G) → [0, ∞) by +∥f∥o := +� +Q(f) + |f(o)|2. +We denote the closure of the space of functions of compact support +Cc(X) with respect to ∥·∥o by D0(G). The following well-known lemma +shows that ∥·∥o is indeed a norm and that the space D0(G) does not +depend on the choice of o ∈ X, see e.g. [19, Proposition 1.6]. Its proof +relies on the connectedness of G, which we always assume in this text +(see above). +Lemma 2.3. Let G = (X, b) be a graph and let o ∈ X. +(a) ∥·∥o is a norm and (D(G), ∥·∥o) is a Banach space. +(b) fn → f with respect to ∥·∥o implies fn → f pointwise. + +RECURRENT AND (STRONGLY) RESOLVABLE GRAPHS +5 +(c) For o′ ∈ X the norms ∥·∥o and ∥·∥o′ are equivalent. +A measure m on X induces the Hilbert space +ℓ2(X, m) = {f ∈ C(X) | +� +x∈X +m(x)|f(x)|2 < ∞} +with inner product +⟨f, g⟩m = +� +x∈X +m(x)f(x)g(x) +and corresponding norm ∥·∥m. We denote by H1(G, m) := ℓ2(X, m) ∩ +D(G) the corresponding first-order Sobolev space. Equipped with the +inner product +⟨f, g⟩Q,m := ⟨f, g⟩m + Q(f, g) +it is a Hilbert space. We denote the associated norm by ∥·∥Q,m. For a +proof of the completeness of this space we refer the reader to [19, Sec- +tion 1.3], where the completeness of H1(G, m) is discussed as closedness +of the Dirichlet form Q(N) (in the notation used there). +2.2. Intrinsic metrics. In this section we introduce the (pseudo)metrics +relevant for our considerations and discuss their properties. +A symmetric function σ : X × X → [0, ∞) is called a pseudometric +if it satisfies the triangle inequality, i.e. if for all x, y, z ∈ X it satisfies +σ(x, y) ≤ σ(x, z) + σ(z, y). +For r ≥ 0 and x ∈ X we denote the corresponding ball of radius r +around x by +Bσ +r (x) := {y ∈ X | σ(x, y) ≤ r}. +For x ∈ X the distance σU from a nonempty subset U ⊆ X is defined +by +σU(x) := σ(x, U) := inf +y∈U σ(x, y) +and the diameter of U with respect to σ is +diamσ(U) := sup +x,y∈U +σ(x, y). +A function f on X is called Lipschitz-function with respect to the +pseudometric σ if there exists a C > 0 with |f(x) − f(y)| ≤ Cσ(x, y) +for all x, y ∈ X. We then also say that f is a C-Lipschitz function. The +set of all Lipschitz-functions with respect to σ is denoted by Lipσ(X). +For a graph G = (X, b) a pseudometric σ is called intrinsic with +respect to the measure m if for all x ∈ X it satisfies +1 +2 +� +y∈X +b(x, y)σ(x, y)2 ≤ m(x). + +6 +LENZ, PUCHERT, AND SCHMIDT +We write M(G) for the set of pseudometrics that are intrinsic with +respect to a finite measure. Clearly, a pseudometric σ belongs to M(G) +if and only if +� +x,y∈X +b(x, y)σ(x, y)2 < ∞ +holds. +Remark (Background on intrinsic metrics). Intrinsic metrics have long +proven to be a useful tool in spectral geometry of manifolds and, more +generally, for strongly local Dirichlet spaces, see e.g. Sturm’s seminal +work [23, 24]. For general Dirichlet spaces, including graphs, a system- +atic approach was developed in [3]. A key point in [3] is a Rademacher +type theorem. In the context of graphs this theorem says that a pseudo- +metric σ is intrinsic if and only if for all 1-Lipschitz functions f : X → R +with respect to σ we have |∇f|2 ≤ 1. Here, for f ∈ C(X) and x ∈ X +the quantity +|∇f|2(x) := +1 +2m(x) +� +y∈X +b(x, y)(f(x) − f(y))2 +can be interpreted as the square of the norm of the discrete gradient +of f at x (with respect to the measure m). For graphs with measure +m for which the scaled degree deg /m is uniformly bounded the com- +binatorial metric is an intrinsic metric (up to a constant). For graphs +with unbounded degree this is not the case anymore. For such graphs, +intrinsic metrics (rather than the combinatorial metric) have turned +out to be the right metrics for various questions, see e.g. the survey +[10]. The present article can also be seen as a point in case. +There are strong ties between functions of finite Dirichlet energy and +intrinsic pseudometrics with respect to a finite measure. These will be +of relevance for some of our theorems below. +Lemma 2.4 (From M(G) to D(G)). Let G = (X, b) be a graph. Let σ +be an intrinsic pseudometric with respect to the finite measure m. Let +U be a subset of X. Then, the following statements hold: +(a) Any function f that is C-Lipschitz with respect to σ and constant +on U satisfies +Q(f) ≤ C2 min{m(X), 2m(X \ U)}. +(b) The inequality +Q(σU) ≤ min{m(X), 2m(X \ U)} +is valid and, in particular, σU belongs to D(G). +(c) The inequality +Q(f) ≤ C2m(X) +holds for any C-Lipschitz function f with respect to σ. In particu- +lar, any Lipschitz function with respect to σ belongs to D(G). + +RECURRENT AND (STRONGLY) RESOLVABLE GRAPHS +7 +Proof. Clearly, both (b) and (c) are immediate consequences of (a). +Thus, we only show (a). It suffices to consider the case C = 1 as for +general C > 0 the function g = 1 +Cf is 1-Lipschitz with Q(g) = +1 +C2Q(f). +So, let f be a 1-Lipschitz function vanishing on U. The bound Q(f) ≤ +m(X) follows easily from the estimate +Q(f) = 1 +2 +� +x,y +b(x, y)(f(x) − f(y))2 ≤ +� +x∈X +� +1 +2 +� +y∈X +b(x, y)σ(x, y)2 +� +and the fact that σ is intrinsic with respect to m. The other estimate +can be shown as follows: Using +• f(x) − f(y) = 0 for all x, y ∈ U and +• |f(x) − f(y)| ≤ σ(x, y) for all x, y ∈ X +we infer +Q(f) = 1 +2 +� +x,y∈X +b(x, y)(f(x) − f(y))2 += 1 +2 +� +(x,y)∈X2\U2 +b(x, y)(f(x) − f(y))2 +≤ 1 +2 +� +(x,y)∈X2\U2 +b(x, y)σ(x, y)2 += +� +y∈X\U +� +1 +2 +� +x∈X +b(x, y)σ(x, y)2 +� ++ +� +x∈X\U +� +1 +2 +� +y∈U +b(x, y)σ(x, y)2 +� +≤ 2m(X \ U). +□ +Remark. Clearly, the estimates given in the previous lemma trivially +continue to hold if m is not a finite measure. +Lemma 2.5 (From to D(G) to M(G)). Let G = (X, b) be a graph. +Then, for any function f of finite energy the function +σf : X × X → [0, ∞), +σf(x, y) := |f(x) − f(y)| +is an intrinsic pseudometric with respect to the finite measure mf that +is given by +mf(x) = 1 +2 +� +y∈X +b(x, y)σf(x, y)2. +The function f is 1-Lipschitz with respect to σf and mf(X) = Q(f). +Proof. This is already shown in [6, Proposition 3.11]. +□ +By the preceding lemma any f of finite energy comes with a pseudo- +metric σf that is intrinsic with respect to a finite measure. In general, +this σf will not be a metric (as values of f in different points need +not be distinct). However, this can easily be achieved by an arbitrarily +small perturbation as the next proposition shows. + +8 +LENZ, PUCHERT, AND SCHMIDT +Proposition 2.6 (Small perturbation). Let G = (X, b) be a graph. +(a) For any ε > 0 there exist sx > 0, x ∈ X, such that any function +g : X → R with g(x) ∈ (−sx, sx) for all x ∈ X satisfies Q(g) < ε. +(b) For any f ∈ D(G) and any ε > 0 there exists a function fε ∈ D(G) +with fε(x) ̸= fε(y) for all x, y ∈ X with x ̸= y and +sup +x∈X +|f(x) − fε(x)| < ε and |Q(f − fε)| < ε. +Proof. (a): We write 1x for the characteristic function of x ∈ X. Then +Q(1x) = deg(x) < ∞. For x ∈ X we choose sx > 0 with +� +x∈X +sxQ(1x)1/2 < √ε. +Choose a sequence (Fn) of finite subsets of X with Fn ⊆ Fn+1 and +X = � +n Fn. Then, any g : X → R with g(x) ∈ (−sx, sx) is the point- +wise limit of the functions gn := � +x∈Fn g(x)1x. The pointwise lower +semicontinuity of Q together with Cauchy-Schwarz inequality yield +Q(g) ≤ lim inf +n→∞ Q(gn) ≤ lim inf +n→∞ +� +x,y∈Fn +|g(x)||g(y)|Q(1x)1/2Q(1y)1/2 +≤ +�� +x∈X +sxQ(1x)1/2 +�2 +< ε. +(b): This follows from (a). Let ε > 0 be given and chose sx, x ∈ X, +according to (a). Without loss of generality we can assume sx < ε +2. Now +let real numbers ux, x ∈ X, be given such that ux − uy is irrational for +any x ̸= y. Then, for any x ∈ X we can choose an tx ∈ (−sx, sx) such +that f(x) − ux − tx is rational. Then, fε with +fε(x) = f(x) − tx +for all x ∈ X satisfies supx |f(x) − fε(x)| ≤ supx sx < ε as well as +Q(f − fε) < ε. Moreover, the values of fε are pairwise different as for +x ̸= y we have +(f(x)−tx)−(f(y)−ty) = (ux −uy)+(f(x)−ux −tx)−(f(y)−uy −ty) +can not vanish (as it is the sum of an irrational number and a rational +number.) +□ +For us a special class of pseudometrics will be particularly useful. +They will be introduced next. Given a symmetric function w: X×X → +[0, ∞) and a (possibly infinite) path γ = (x1, x2, . . .) in G we define the +length of γ with respect to w by +Lw(γ) := +� +i +w(xi, xi+1) ∈ [0, ∞]. + +RECURRENT AND (STRONGLY) RESOLVABLE GRAPHS +9 +Since we always assume connectedness, this induces the path pseudo- +metric dw on X via +dw(x, y) = inf{Lw(γ) | γ a path from x to y}. +We say that σ is a path pseudometric on X if σ = dw for some symmetric +function w. A symmetric function w: X × X → [0, ∞) is called edge +weight if w(x, y) > 0 for all (x, y) ∈ E. +A symmetric function w is called adapted with respect to the graph +G = (X, b) and the measure m if for all x ∈ X it satisfies +1 +2 +� +y∈X +b(x, y)w(x, y)2 ≤ m(x). +The following lemma summarizes some elementary properties of path +pseudometrics. +Lemma 2.7 (Path pseudometrics). Let G = (X, b) be a graph and let +w: X ×X → [0, ∞) be a symmetric function. Then dw is a pseudomet- +ric that satisfies dw(x, y) ≤ w(x, y) if x ∼ y. Moreover, the following +are satisfied. +(a) If w is a pseudometric, then dw ≥ w. In particular, dw(x, y) = +w(x, y) for all x, y ∈ X with x ∼ y. +(b) If w is adapted with respect to the measure m, then dw is intrinsic +with respect to the measure m. +(c) For σ = dw the equality dσ = σ holds. +Proof. The trivial path (x, y) is one of the paths over which the infimum +in the definition of dw is taken. As Lw((x, y)) = w(x, y), the inequality +dw(x, y) ≤ w(x, y) is immediate. +(a): Given a path γ = (x = x1, . . . , xn = y), an iteration of the +triangle inequality w(x1, xk+1) ≤ w(x1, xk) + w(xk, xk+1) yields +w(x, y) ≤ +n−1 +� +i=1 +w(xi, xi+1) = Lw(γ). +(b): This is an immediate consequence of the inequality dw(x, y) ≤ +w(x, y) for all x, y ∈ X with x ∼ y. +(c): As σ = dw is a pseudometric, (a) gives σ ≤ dσ and, for x, y ∈ X +with x ∼ y, even σ(x, y) = dσ(x, y). +For arbitrary x, y ∈ X let a +path γ = (x = x1, . . . , xn = y) be given. Then a short computation +involving what we have shown already and the triangle inequality gives +Lw(γ) = +n−1 +� +j=1 +w(xj, xj+1) ≥ +n−1 +� +j=1 +σ(xj, xj+1) = +n +� +j=1 +dσ(xj, xj+1) ≥ dσ(x, y). +Taking the infimum over all γ we find σ(x, y) ≥ dσ(x, y). +□ +We note the following consequence of our considerations: If f is +a function of finite energy on the graph (X, b), then σf (defined in + +10 +LENZ, PUCHERT, AND SCHMIDT +Lemma 2.5) is an intrinsic pseudometric with respect to mf. +Now, +we can also consider σf as a symmetric function (adapted to mf). +This induces the path pseudometric df := dσf. The preceding lemma +immediately gives the following. +Corollary 2.8. Let G = (X, b) be a graph and let f ∈ D(G). Then, +df = dσf is an intrinsic metric with respect to mf and +df(x, y) = |f(x) − f(y)| +holds for all x, y ∈ X with b(x, y) > 0. +Proof. By the preceding lemma we have df = dσf ≤ σf as well as +df(x, y) = σf(x, y) = |f(x)−f(y)| for all x, y ∈ X with b(x, y) > 0. +□ +Remark. Let a graph G = (X, b) be given. Define �Q on the set of +symmetric functions w: X × X → [0, ∞) by +�Q(w) := 1 +2 +� +x,y +b(x, y)w(x, y)2 ∈ [0, ∞]. +Then, part of our considerations can be understood in terms of �Q. As +this may be instructive we give a brief discussion in the present remark: +For a symmetric w: X × X → [0, ∞) we define +mw : X → [0, ∞], +mw(x) = 1 +2 +� +y∈X +b(x, y)w(x, y)2 +and mw(X) := � +x∈X mw(x). Finally, for f : X → R define the sym- +metric function σf : X × X → [0, ∞) with σf(x, y) := |f(x) − f(y)|. +Then, the following holds: +(a) Let w be a symmetric weight. Then, �Q(w) = mw(X), where the +value ∞ is allowed. If w is actually a pseudometric, then �Q(dw) = +�Q(w) holds. +(b) Let σ be a pseudometric on X. Then, mσ is a finite measure if +and only if σ belongs to M(G). If mσ is a finite measure it is the +smallest measure with respect to which σ is an intrinsic metric. +(c) For f : X → R the equality Q(f) = �Q(σf) holds, where the value +∞ is allowed. Moreover, f belongs to D(G) if and only if σf belongs +to M(G). +(d) The function f : X → R is 1-Lipschitz with respect to the pseudo- +metric σ if and only if σf ≤ σ holds. In this case, Q(f) ≤ �Q(σf) is +valid. +As mentioned already we think of the space X underlying the graph +(X, b) as equipped with discrete topology. Thus, metrics compatible +with the discrete topology are of particular relevance for us. The fol- +lowing lemma ensures the existence of such metrics in M(G). + +RECURRENT AND (STRONGLY) RESOLVABLE GRAPHS +11 +Lemma 2.9. Let G = (X, b) be a graph. Then, there exists a metric +in M(G) that induces the discrete topology. +Proof. Let N → X, n �→ xn, be an enumeration of X. We define +f : X → (0, ∞), +f(xn) = +1 +� +2ndeg(xn) +, +and +σ: X × X → [0, ∞), +σ(x, y) = +� +max{f(x), f(y)} +for x ̸= y +0 +else +. +It is readily verified that σ is a metric (and even an ultrametric). By +σ(x, y)2 ≤ f(x)2 + f(y)2, the symmetry of b and Fubini’s theorem we +find +� +x∈X +b(x, y)σ(x, y)2 +≤ +� +x,y∈X +b(x, y)(f(x)2 + f(y)2) += +2 +� +x,y∈X +b(x, y)f(x)2 += +2 +� +x∈X +deg(x)f(x)2. +Now, the definition of f gives +2 +� +x∈X +deg(x)f(x)2 ≤ 2 +and it follows that σ is an intrinsic metric with respect to a finite +measure. +The metric σ induces the discrete topology as the distance from any +point to x ∈ X is bounded from below by f(x) > 0. +□ +2.3. Boundaries of graphs. As outlined in the introduction comple- +tions and boundaries of graphs will be most relevant for our consider- +ations. Here we introduce the corresponding notions. +Let X be a countable set. +Let σ be a pseudometric on X. +The +completion of X with respect to σ is defined as the set of equivalence +classes of σ-Cauchy sequences in X, where two such sequences (an) and +(bn) are considered to be equivalent if +lim +n→∞ σ(an, bn) = 0. +This set is denoted by X +σ and contains a quotient of the vertex set +X as the classes of the constant sequences. learly, σ can be extended +to a pseudometric on X and this extension will - by a slight abuse of +notation - also denoted by σ. Subsequently, the boundary is defined as +∂σX = X +σ \ (X/ ≃), + +12 +LENZ, PUCHERT, AND SCHMIDT +where x ≃ y if σ(x, y) = 0. A graph is called metrically complete with +respect to a pseudometric if the boundary is empty. Clearly, if σ is a +metric then X +σ contains a copy of X, this copy is dense, and and our +definition of metric completeness agrees with the usual definition (that +any Cauchy-sequence converges). +There are further notions of completeness relevant to us. Let G = +(X, b) be a graph and let w be an edge weight. +The pseudometric +space (X, dw) is called geodesically complete if every infinite path has +infinite length with respect to w. +For later purposes we recall the +following discrete Hopf-Rinow type theorem that characterizes geodesic +completeness, see [8, Theorem A.1] and, for further generalizations, see +also [13]. +Theorem 2.10 (Hopf-Rinow type theorem). Let G = (X, b) be a lo- +cally finite graph and let w be an edge weight. Then dw is a metric that +induces the discrete topology on X. Moreover, the following assertions +are equivalent: +(i) (X, dw) is a complete metric space. +(ii) (X, dw) is geodesically complete. +(iii) Every distance ball is finite. +(iv) Every bounded and closed set is compact. +Boundaries of graphs can not only arise from metric completions but +also from compactifications. In fact, they can arise whenever the set +X underlying the graph is suitable extended. We finish this section +with a short discussion of this aspect. Let X be a countable set. Let +Y be a topological Hausdorff space. We say that X embeds densely +in the topological space Y if Y contains a copy of X, the restriction +of the topology of Y on X is the discrete topology, and X is dense in +Y . Clearly, Y must be separable whenever X embeds densely in it. +Whenever X embeds densely in Y we define the boundary ∂Y X of X +in Y by +∂Y X := Y \ X. +The complement (in Y ) of any finite subset of X is open in Y (as any +finite set is compact and then must be closed due to Hausdorffness). +Hence, any such a complement is an open neighborhood of ∂Y X. In +particular, any function h with finite support on X can be extended +(by zero) to a continuous function on Y . +Clearly, X embeds densely in X +σ whenever σ is a metric on X in- +ducing the discrete topology. This is what we have discussed above. +Then, +∂σX = ∂X +σX +holds. +If X embeds densely in a compact Y , then Y is called a compactifica- +tion of X. In this case the open neighborhoods of ∂X are exactly given + +RECURRENT AND (STRONGLY) RESOLVABLE GRAPHS +13 +by the complements of finite sets of X (as the complement of any open +neighborhood of ∂Y X must be a closed, and hence, compact subset of +X). A particular instance is given by the one-point-compactification. +It is given by the set Y = X ∪{pt}, where pt is an arbitrary additional +point, and this set is equipped with topology given by the family of +all subsets of Y that are either subsets of X or whose complement is +finite. In this case the boundary of ∂Y X is just pt. +3. Capacity of sets in the boundary and infinite paths +In this section we introduce the capacity and study the capacity of +sets in the boundary with respect to an intrinsic metric. +Let G = (X, b) be a graph and let m: X → (0, ∞) be a measure. +The capacity of a subset U ⊆ X is defined by +capm(U) := inf{∥f∥2 +Q,m | f ∈ H1(G, m) with f ≥ 1 on U}, +with the convention that capm(U) = ∞ if the set in the above definition +is empty. Using the fundamental contraction property, we can assume +0 ≤ f ≤ 1 in this definition, since (f ∧1)+ satisfies the same constraints +as f but reduces the ∥·∥Q,m-norm compared to f. +Whenever X embeds densely in Y we can extend the capacity to +subsets of Y by setting +capm(A) := inf{capm(O ∩ X) | O open in Y with A ⊆ O}. +Since by assumption every subset A ⊆ X is open in Y (as the topology +of Y induces the discrete topology on X) both definitions of capacity +on X are compatible. This definition can in particular be applied to the +completion X +σ, whenever the metric σ induces the discrete topology +on X. +The capacity is an outer measure on the power set of Y with m(A) ≤ +capm(A) for all A ⊆ X and capm(Y ) ≤ m(X), see e.g. [4, Theorem +2.1.1 and Theorem A.1.2]. +Next we discuss how the vanishing of the capacity of subsets of the +boundary can be characterized with limits of functions of finite energy. +Definition 3.1 (Limes inferior). Let X be a countable set, let σ be a +metric on X that induces the discrete topology and let f : X → R. For +A ⊆ X +σ the limes inferior of f at A with respect to σ is defined by +lim inf +x→A f(x) := sup{inf{f(x) | x ∈ U∩X} | U open in X +σ with A ⊆ U}. +Moreover, we define the limes inferior at infinity by +lim inf +x→∞ f(x) := sup{inf{f(x) | x ∈ X \ F} | F ⊆ X finite}. +Remark. For us the case where lim inf equals ∞ and the case of com- +pact X +σ is is particularly relevant. In this context we note the following. +(a) We have lim infx→A f(x) = ∞ if and only if limn→∞ f(xn) = ∞ for +each sequence (xn) in X with σ(xn, A) → 0, n → ∞. + +14 +LENZ, PUCHERT, AND SCHMIDT +(b) We have lim infx→∞ f(x) = ∞ if and only if limn→∞ f(xn) = ∞ for +any sequence (xn) converging to pt in the one-point-compactification +of X. In fact, this easily shows that lim infx→∞ f(x) = ∞ if and +only if limn→∞ f(xn) = ∞ for any sequence (xn) converging to +some y ∈ ∂Y X, where Y is a compactification of X. +(c) If σ induces the discrete topology on X and (X, σ) is pre-compact, +then every every open neighborhood U of ∂σX in X +σ has the form +U = X \ F for some finite F ⊆ X. Hence, in this case +lim inf +x→∞ f(x) = lim inf +x→∂σX f(x). +In this sense, the limes inferior at infinity is the limes inferior at +the boundary for metric compactifications of X. +It turns out that lim infx→∞ governs lim infx→∂σX in the following +sense. +Proposition 3.2. Let X be a countable set. Let f : X → R be given. +Then, +lim inf +x→∂σX f(x) ≥ lim inf +x→∞ f(x) +for any metric σ on X that induces the discrete topology. +Proof. Any finite set F in X is compact in X +σ and, hence, closed. +Thus, for any finite set F in X the set X +σ \F is an open neighborhood +of ∂σX. We obtain +lim inf +x→∂σX f(x) = sup{inf{f(x) | x ∈ U ∩ X} | U open with ∂σX ⊆ U} +≥ sup{inf{f(x) | x ∈ X \ F ∩ X} | F ⊆ X finite} +≥ lim inf +x→∞ f(x). +□ +With the help of the limes inferior we can characterize sets of capacity +zero in the boundary. +Lemma 3.3 (Characterization of zero capacity sets in the boundary). +Let G = (X, b) be an infinite graph and σ be a metric on G that induces +the discrete topology. Further, let A ⊆ ∂σX. The following assertions +are equivalent: +(i) For one finite measure m on X we have capm(A) = 0. +(ii) For all finite measures m on X we have capm(A) = 0. +(iii) There exists f ∈ D(G) with lim inf +x→A f(x) = ∞. +Proof. (i) ⇒ (iii): The statement capm(A) = 0 implies the existence of +sequences of open sets Un ⊇ A and functions fn ≥ 1Un that satisfy +lim +n→∞ ∥fn∥Q,m = lim +n→∞ +� +Q(fn) + ∥fn∥2 +m = 0. + +RECURRENT AND (STRONGLY) RESOLVABLE GRAPHS +15 +By restricting to a subsequence we can assume without loss of generality +that +� +n∈N +∥fn∥Q,m < ∞. +This implies that the sum f := � +n∈N fn converges in the Hilbert space +H1(G, m). In particular, f ∈ D(G). By the choice of the fn we have +f ≥ N on the set �N +n=1 Un, which is an open set that contains A. This +proves that lim infx→A f(x) is at least N. +Since this is true for all +N ∈ N, the supremum has to be infinite. +(iii) ⇒ (ii): Let f ∈ D(G) satisfy (iii). Without loss of generality +we assume f ≥ 0, for otherwise we can replace f by |f|, which also has +finite energy due to Theorem 2.2. We slice f into the parts +fn := (f − n)+ ∧ 1. +First, we observe that 0 ≤ fn ≤ 1 and f = �∞ +n=0 fn pointwise. More- +over, Theorem 2.2 yields fn ∈ D(G) and since m is finite and the fn +are bounded, we have fn ∈ H1(G, m). Because lim infx→A f(x) = ∞, +for every n ∈ N0 there is an open set Un ⊇ A such that f ≥ n on +Un ∩X. By construction, for x ∈ X the inequality f(x) ≥ n+1 implies +fn(x) = 1. Combining these observations we conclude fn ≥ 1X∩Un+1. +Altogether, this shows that the functions fn are usable in the defini- +tion of the capacity of A and +capm(A) ≤ ∥fn∥Q,m . +We prove ∥fn∥Q,m → 0, as n → ∞. +It is readily verified that for n ̸= m and x, y ∈ X the product +(fm(x) − fm(y))(fn(x) − fn(y)) is always nonnegative, so that +Q(fn, fm) = 1 +2 +� +x,y∈X +b(x, y)(fm(x) − fm(y))(fn(x) − fn(y)) ≥ 0. +Recall that for N ∈ N we defined C[0,N]f = f+ ∧ N. Using Proposi- +tion 2.2, the definition of fn and the previous observation, we obtain +N−1 +� +n=0 +Q(fn) ≤ +N−1 +� +n=0 +Q(fn) + +� +0≤m,n≤N−1 +m̸=n +Q(fm, fn) += Q +�N−1 +� +n=0 +fn +� += Q(C[0,N]f) ≤ Q(f). +Since N was arbitrary and Q(f) < ∞, we arrive at limn→∞ Q(fn) = 0. +The convergence limn→∞ ∥fn∥m = 0 follows from Lebesgue’s dominated +convergence theorem. +This leads to limn→∞ ∥fn∥Q,m = 0 and thus, +capm(A) = 0. +(ii) ⇒ (i): This is clear. +□ + +16 +LENZ, PUCHERT, AND SCHMIDT +Remark. (a) The lemma shows that having capacity zero does not +depend on the choice of the finite measure. Indeed, we do not even +need to assume that m is strictly positive. If we only assume that +m(o) > 0 for one o ∈ X, the space H1(X, m) continuously embeds +into (D(G), ∥·∥o) and the proof can be carried out in the space +(D(G), ∥·∥o). The advantage of working in H1(G, m) is that it is +related to intrinsic metrics with respect to m. +(b) The inequality used in the proof of the implication (iii) ⇒ (ii) can +be extended to a more general form. Let f : X → R be a function +of finite energy and let C1, C2 : R → R be two monotone increasing +1-Lipschitz functions. Then +Q(C1 ◦ f + C2 ◦ f) ≥ Q(C1 ◦ f) + Q(C2 ◦ f). +In the above proof this observation is applied to the monotone +increasing contractions Sn, n ∈ N0, with Sn(x) = (x − n)+ ∧ 1. +One can understand the preceding result also as saying that the sets +A ⊆ ∂σX with zero capacity are infinitely far away from any finite set. +More specifically, the following holds. +Corollary 3.4 (Capacity zero sets in the boundary have infinite dis- +tance). Let G = (X, b) be an infinite graph and σ a metric on G that +induces the discrete topology. Further, let A ⊆ ∂σX. Then, ∂σA has +zero capacity (with respect to any finite measure) if and only if there +exists an intrinsic metric ̺ ∈ M(G) such that for any finite F in X +and any r > 0 there exists an open neighborhood U of A with +̺(U ∩ X, F) := inf{̺(z, x) | z ∈ U ∩ X, x ∈ F} ≥ r. +Proof. Assume that A has capacity zero (with respect to any finite +measure). +By the previous lemma, there exists an f ∈ D(G) with +lim infx→A f(x) = ∞. Without loss of generality we can assume f(x) ̸= +f(y) for all x, y ∈ X with x ̸= y (else we could add an arbitrary small +perturbation by Proposition 2.6). Then, ̺ := σf is an intrinsic metric +with respect to a finite measure. By lim infx→A f(x) = ∞ the metric ̺ +has the given property. +Assume now that there exists an intrinsic metric ̺ with respect to +the finite measure m that has the given property. Let an arbitrary +finite set F be given. Then, there exists an open neighborhood U of A +with ̺(U ∩ X, F) ≥ 1. Hence, gF := ̺F ∧ 1 satisfies 0 ≤ gF ≤ 1, equals +0 on F and equals 1 on U. Hence, +capm(A) ≤ Q(gF) + |gF|2 +m ≤ Q(̺F) + m(X \ F) ≤ 3m(X \ F) +holds, where we used Lemma 2.4 in the last step. As this holds for +arbitrary F we infer capm(A) = 0. By the previous lemma this implies +that the capacity of A vanishes with respect to any finite measure. +□ +Remark. Replacing ̺ by d̺ ≤ ̺ we can even take the intrinsic metric +in the preceding corollary to be a path metric. + +RECURRENT AND (STRONGLY) RESOLVABLE GRAPHS +17 +Remark. Lemma 3.3 and its corollary deal with metric completions +of X. However, they can directly be extended to any topological Haus- +dorff space Y in which X embeds densely. Indeed, both the definition +of lim infx→A and the proofs of the lemma and its corollary carry ver- +batim over to this more general situation. This means in particular +that these considerations also holds for compactifications of X. +Next we discuss how the capacity of sets in the metric boundary is +related to infinite paths. We recall the following standard notion for +sets of infinite paths in a graph. +Definition 3.5 (Null set of paths). A set of infinite paths Γ in G = +(X, b) is called null if there exists an edge weight w with +� +x,y∈X +b(x, y)w(x, y)2 < ∞ +such that Lw(γ) = ∞ for all γ ∈ Γ. +Let σ be a metric on X that induces the discrete topology. +For +A ⊆ ∂σX we denote by ΓA,σ the set of infinite paths which have at +least one accumulation point with respect to σ lying in A. +With the help of our characterization of sets of capacity zero in the +boundary we obtain the following relation between sets of capacity zero +in the boundary and sets of paths with accumulation point in this set. +This is the key observation relating our approach to the classical +approach to recurrence by means of null sets of paths. As noted in the +introduction, this observation was our motivation to write this paper. +Theorem 3.6 (Capacity and null sets of paths). Let G = (X, b) be +an infinite graph and let σ be an intrinsic metric with respect to a +finite measure m that induces the discrete topology. Let A ⊆ ∂σX with +capm(A) = 0. Then ΓA,σ is null. +Proof. According to the previous lemma there is f ∈ D(G) such that +lim infx→A f(x) = ∞. We consider the function w: X × X → R, +w(x, y) = |f(x) − f(y)|. Without loss of generality we can assume +w(x, y) > 0 whenever (x, y) is an edge (else at each vertex x add a +small quantity to f(x) if necessary, see Proposition 2.6). Then +� +x,y∈X +b(x, y)w(x, y)2 ≤ 2Q(f) < ∞. +Let γ = (x1, x2, . . .) be an infinite path with an accumulation point in +A. We obtain +|f(xn) − f(x1)| ≤ +n−1 +� +k=1 +|f(xk) − f(xk+1)| ≤ Lw(γ). +Since lim infx→A f(x) = ∞, the left hand side of this inequality diverges +along a suitable subsequence and so we obtain Lw(γ) = ∞. Hence, ΓA,σ +is null. +□ + +18 +LENZ, PUCHERT, AND SCHMIDT +The converse seems not to hold due to the complicated behavior of +paths at metric boundaries of general graphs. For trees however we +have the following converse for path metrics. Recall that (X, b) is a +tree if it does not have non-trivial cycles (injective paths (x1, . . . , xn) +with x1 ∼ xn). +Proposition 3.7. Let G = (X, b) be a tree and let σ be a path metric +that induces the discrete topology on X and is intrinsic with respect to +a finite measure m. If for A ⊆ ∂σX the set of paths ΓA,σ is null, then +capm(A) = 0. +Proof. Let w be an edge weight for ΓA,σ as in the definition of null +sets of paths. Fix o ∈ X and for x ∈ X let γx be the unique shortest +path with respect to the combinatorial distance connecting o and x +(uniqueness follows from (X, b) being a tree). We define +f : X → R, +f(x) = Lw(γx). +Since (X, b) is a tree, for neighbors x, y ∈ X we have |f(x) − f(y)| = +w(x, y) showing f ∈ D(G). +Let (xn) be a sequence in X with limit in x ∈ A. We construct a +monotone path γ = (y1, y2, . . .) (i.e. the combinatorial distance of y1 +and yn+1 is larger or equal than the combinatrial distance of y1 and yn) +starting in o such that yn → x and +f(xk) = f(ynk) + dw(xk, ynk) ≥ f(ynk) +for a suitable subsequence (ynk). The monotonicity of γ and that (X, b) +is a tree imply +lim inf +k→∞ f(xk) ≥ lim inf +k→∞ f(ynk) = Lw(γ) = ∞. +Construction of γ: We consider o as a root for the graph and denote +by |x| the combinatorial distance of x to o. We say that y is an ancestor +of c if all paths from x to o pass through y. Since (X, b) is a tree, every +A ⊆ X has a unique greatest common ancestor, i.e., there exists and +element y ∈ X with: +• y is an ancestor of every element of A. +• For every x ∈ X with |x| > |y| there exists an a ∈ A such that +x is not an ancestor of a. +We let zn be the greatest common ancestor of {xn, xn+1, . . .}. This +sequence is monotone as zn is an ancestor of {xn+1, xn+2, . . .} and hence +an ancestor of zn+1. For every n ∈ N there exists N > n such that +the greatest common ancestor of {xn, xN} is zn (otherwise zn would +not be a greatest common ancestor). Every path from xn to xN passes +through zn. Since σ is a path metric, this implies +σ(xn, xN) = σ(xn, zn) + σ(zn, xN) + +RECURRENT AND (STRONGLY) RESOLVABLE GRAPHS +19 +and we obtain +σ(xn, zn) ≤ sup{σ(xn, xN) | N > n}. +Hence, (zn) also converges to x but it need not be a path. We make it +a path by inserting monotone paths from zn to zn+1 (these exist since +yn is an ancestor of yn+1). Using that σ is a path metric yields that +any such additional point z lying between zn and zn+1 satisfies +σ(zn, zm) = σ(zn, z) + σ(z, zm) ≥ σ(zn, z). +Hence, also the so-constructed monotone path (yn) converges to x. We +choose nk such that ynk = zk. Using that zk is an ancestor of xk, we +obtain +f(xk) = f(zk) + Lw(γxk) − Lw(γzk) = f(ynk) + dw(ynk, xk). +□ +Remark. In the previous proof we used the following observation uti- +lizing that (X, b) is a tree and σ is a path metric: For every x ∈ ∂σX +and every sequence (xn) in X converging to x there exists a monotone +path (yn) converging to x such that any xn has an element from the +path (yn) as an ancestor. +4. Recurrence and intrinsic metrics +In this section we use similar technics as in Section 3 to give a new +characterization of recurrence in terms of intrinsic metrics and to study +the relation of recurrence to the vanishing of the capacity of the bound- +ary. Moreover, we provide an alternative proof for a classical charac- +terization of recurrence due to Yamasaki. For general background on +recurrence we refer the reader to the textbooks [12, 26]. +The word recurrence stems from the stochastic perspective. In this +perspective the graph gives rise to a Markov process modeling a par- +ticle jumping between the points of X. Recurrence then describes the +phenomenon that the particle comes back to any point of X again and +again. In the analytic description, which is our concern here, this is +encoded by various forms of irrelevance of what is happening far away +(i.e. outside of finite sets). We will see precise versions as we go along +and this is the main topic of this section. +Definition 4.1 (Recurrence). A graph G = (X, b) is called recurrent +if the constant function 1 is contained in D0(G). Graphs that are not +recurrent are called transient. +Remark. (a) The definition of recurrence means that there exists a +sequence of functions (fn) in Cc(X) with fn → 1 pointwise and +Q(fn) = Q(fn −1) → 0, n → ∞. As the fn have finite support this +can be seen as an instance of how the behaviour outside of compact +sets (in this case the supports of the fn) becomes irrelevant. + +20 +LENZ, PUCHERT, AND SCHMIDT +(b) Recurrence is equivalent to D0(G) = D(G), i.e., Cc(X) being dense +in D(G) with respect to the norm ∥·∥o, see e.g.[22, Theorem 3.63]. +(c) For disconnected graphs transience is a stronger property than not +being recurrent. Since all the graphs in this paper are assumed to +be connected, we may well use the above definition. For further +background on recurrence we refer the reader to [19]. +Next we connect recurrence, vanishing of the capacity and finiteness +of metric balls, to the existence of certain unbounded functions of finite +energy. +Theorem 4.2 (Characterization of recurrence). Let G = (X, b) be an +infinite graph. The following conditions are equivalent: +(i) G is recurrent. +(ii) There is a function of finite energy f ∈ D(G) that satisfies +lim inf +x→∞ f(x) = ∞. +(iii) There is an intrinsic metric σ ∈ M(G) that induces the discrete +topology on X such that distance balls with respect to σ are finite. +(iii)’ There exists a finite measure m and an edge weight w adapted to +it such that the distance balls with respect to dw are finite. +(iv) For one (every) finite measure m on X and one (every) compact- +ification Y of X the equality capm(∂Y X) = 0 holds. +(v) One (every) finite measure has the following feature: For any ε > +there exists a finite set F in X with capm(X \ F) = 0. +Proof. (i) ⇒ (iv): Let m be an arbitrary finite measure on X and Y +a compactification of X. By (i) there exists a sequence (fn) in Cc(X) +with fn → 1 pointwise and Q(fn) → 0. Replacing fn by (fn ∨ 0) ∧ 1 +we can assume without loss of generality 0 ≤ fn ≤ 1 for each n. Then, +0 ≤ 1 − fn ≤ 1 holds and 1 − fn is 1 outside the finite support of fn. +Hence, +capm(∂Y X) ≤ Q(1 − fn) + |1 − fn|2 +m +holds for each n. It suffices to show that both terms on the right hand +side converge to zero. The first term satisfies Q(1 − fn) = Q(fn) → +0, n → ∞. +The second term satisfies |1 − fn|2 +m → 0, n → ∞ by +Lebesgue theorem on dominated convergence (as 0 ≤ 1 − fn ≤ 1 holds +and 1 − fn converges pointwise to 0 and m is a finite measure). +(iv) ⇒ (ii): : This follows by a straightforward adaption of the proof +of (i)⇒ (iii) of Lemma 3.3. +(ii) ⇒ (iii): Let f be a function satisfying (ii). By Proposition 2.6 we +can assume without loss of generality that the values of f are pairwise +distinct. Set σ(x, y) = σf(x, y) = |f(x) − f(y)| for all x, y ∈ X. This +yields a pseudo metric that is intrinsic with respect to a finite measure, +see Lemma 2.4. +In fact, it is even a metric as the values of f are + +RECURRENT AND (STRONGLY) RESOLVABLE GRAPHS +21 +pairwise distinct. Its distance balls are given by +Bσ +r (o) = {x ∈ X | f(o) − r ≤ f(x) ≤ r + f(o)}. +Since lim infx→∞ f(x) = ∞, they are finite. In particular, this metric +induces the discrete topology. +(iii) ⇒ (i): Let an intrinsic metric σ with respect to a finite measure +m be given according to (iii). Hence, σ induces the discrete topology +and its distance balls are finite. Now, let F be an arbitrary finite set. +Then, σF := σ(·, F) satisfies +• σF = 0 on F. +• σF ≥ 1 outside of F1 := {x ∈ X : σ(F, x) < 1} and F1 is finite +(as F is a finite set and distance balls with respect to σ are +finite). +Define gF := (1 − σF)+. Then gF equals to 1 on F (by the first +bullet point) and has finite support contained in F1 (by the second +bullet point). +Moreover, as Q is compatible with contractions and +Q(1) = 0 holds we find from Lemma 2.4 the estimate +Q(gF) ≤ Q(1 − σF) = Q(σF) ≤ 2m(X \ F). +So, choosing an increasing sequence (Fn) of finite sets with ∪nFn = X +we obtain a sequence fn := gFn in Cc(X) converging pointwise to 1 +with +Q(fn) ≤ 2m(X \ Fn) → 0, n → ∞. +This shows (i). +The equivalence between (iv) and (v) is clear. +(iii)’ ⇒ (iii): By Lemma 2.7 the metric dw is intrinsic with respect +to the finite measure m. To show that it induces the discrete topology +we note that finiteness of dw-balls implies that for all R > 0 and x ∈ X +the set +{y ∈ X | y ∼ x with w(x, y) < R} +is finite (otherwise the R-ball around x would contain infinitely many +points). This is known as essential local finiteness of w and, according +to [13, Lemma 2.2], implies that dw induces the discrete topology. +(iii) ⇒ (iii)’: We choose w := σ. Then, w is adapted to a finite +measure m and dσ is then an intrinsic metric with respect to m with +σ ≤ dσ by Lemma 2.7. +In particular, balls with respect to dw are +contained in the corresponding balls with respect to σ and are, hence, +finite. +□ +Remark. (a) The equivalence between (i) and (iv) can be seen as a +special instance of the recurrence theory developed by the third +author in his (unpublished) PhD thesis [21]. +(b) Clearly, a metric with finite distance balls must induce the discrete +topology. + +22 +LENZ, PUCHERT, AND SCHMIDT +(c) In the proof of (ii) ⇒ (iii) we have seen that for f of finite energy +with lim infx→∞ f(x) = ∞ the intrinsic (pseudo)metric σf has finite +distance balls. There is a converse of sorts to this: Let σ be a metric +and define for x ∈ X the function fx by fx(y) := σ(y, {x}). Then, +the distance balls around one x ∈ X are finite if and only if the +distance balls around any x ∈ X are finite and this holds if and +only if lim infy→∞ fx(y) = ∞ holds for one (all) x ∈ X. +(d) The proof of (iii) ⇒ (i) only uses that the balls of radius 1 are +finite. In fact, the number 1 is irrelevant. It suffices that there is +an r > 0 such that all balls of radius r are finite. However, if σ is +an intrinsic metric all of whose distance balls of radius r are finite +then for any sequence Fn of finite sets in X with Fn ⊆ Fn+1 and +∪Fn = X we can define f := �∞ +n=1 σ(Fn, ·). Then, f will be well- +defined with lim infx→∞ f(x) = ∞ (by finiteness of r-balls). With +a suitable choice of Fn then f will have finite energy and σ + σf +will be an intrinsic metric with respect to a finite measure that has +finite distance balls. +(e) The existence of an intrinsic metric with respect to a (finite) mea- +sure m that has finite distance balls has strong consequences. In +particular, as observed in [8], it implies that associated graph Lapla- +cians on ℓ2(X, m) (and more general magnetic Schr¨odinger opera- +tors [7, 20]) are essentially self-adjoint. It is somewhat surprising +that recurrence implies essential self-adjointness for a particular fi- +nite measure, as in general recurrence is strictly weaker than essen- +tial self-adjointness for all finite measures. We refer to discussion +after Theorem 11.6.15 in the survey [20]. This survey contains a +version of the previous theorem, which was first obtained in the +second author’s master’s thesis [17]. +We are now going to derive some consequence of the preceding the- +orem. As a a first consequence of it we obtain an alternative proof for +the (by now) classical recurrence criterion of Yamasaki [28]. +Corollary 4.3 (Yamasaki’s criterion). Let G = (X, b) be a locally finite +graph. Then G is recurrent if and only if the set of all infinite paths is +null. +Proof. Let G be recurrent. According to Theorem 4.2 there exists a +finite measure m and a weight w adapted to m such that the intrinsic +path metric dw has finite distance balls. As w is adapted to m we have +� +x,y +b(x, y)w(x, y)2 ≤ 2m(X) < ∞. +It suffices to show that the length Lw(γ) of any infinite path is ∞. We +consider two cases: + +RECURRENT AND (STRONGLY) RESOLVABLE GRAPHS +23 +Case 1: The path γ leaves any finite set. Then, the path leaves in +particular any ball of finite radius (w.r.t. dw). Hence, the path must +have infinite length (as the metric is a path metric). +Case 2: The path γ stays within a fixed finite set. Then, it must +have infinite length anyway. +Suppose that the set of all infinite paths is null and let w be a +corresponding edge weight. The summability condition on w implies +that +mw(x) := 1 +2 +� +y∈X +b(x, y)w(x, y)2 +is a finite measure. We consider the path metric dw induced by w. +Lemma 2.7 (a) ensures that it is intrinsic with respect to mw. Thus, +dw is an intrinsic metric with respect to a finite measure. Since all +infinite paths have infinite length, Theorem 2.10 implies that dw has +finite distance balls. This yields recurrence by Theorem 4.2. +□ +Remark. For proving nullity of the set of all paths on recurrent graphs +we did not use local finiteness. +Another consequence of our characterization of recurrence is van- +ishing of the capacity of all boundaries of metric completions in the +recurrent case. +Corollary 4.4. Let G = (X, b) be a recurrent infinite graph. For any +finite measure m on X and any metric σ on X that induces the discrete +topology we have +capm(∂σX) = 0. +Proof. This follows immediately from (v) of the previous theorem as +∂σX is contained in X \ F for any finite F. +□ +In some cases the vanishing of the capacity of the boundary is equiv- +alent to recurrence. For this one needs that σ is intrinsic with respect +to the finite measure m and some more geometric data. In the following +theorem we discuss two situations where this is the case. +Theorem 4.5 (Capacity criterion). Let G = (X, b) be a graph. Let +σ be a metric that induces the discrete topology and is intrinsic with +respect to a finite measure m. Then G is recurrent if capm(∂σX) = 0 +and one of the following conditions is satisfied: +(a) G is locally finite. +(b) (X, σ) is totally bounded. +Proof. (a): According to Theorem 4.2 it suffices to construct an intrin- +sic metric e with respect to a finite measure that has finite distance +balls and induces the discrete topology on X. The metric e that we + +24 +LENZ, PUCHERT, AND SCHMIDT +construct is a path metric. Since G is locally finite, it automatically in- +duces the discrete topology. By the discrete Hopf-Rinow theorem, The- +orem 2.10, the finiteness of distance balls is equivalent to the complete- +ness of (X, e). According to Lemma 3.3 the assumption capm(∂σX) = 0 +yields a function f ∈ D(G) with lim infx→∂σX f(x) = ∞. We let +e := dσ+σf be the path metric that is induced by the weight σ + σf +with σf(x, y) = |f(x)−f(y)|. Lemma 2.5 shows that the pseudometric +σf is intrinsic with respect to a finite measure and so σ +σf is intrinsic +with respect to a finite measure. We infer from Lemma 2.7 that also +the induced path metric e is intrinsic with respect to a finite measure. +It remains to show the completeness of (X, e). Let (xn) be Cauchy +with respect to e. Lemma 2.7 yields e ≥ σ + σf ≥ σ, so that (xn) must +also be a Cauchy sequence with respect to σ. Due to completeness it +has a limit x ∈ X +σ. We show that x ∈ X and that (xn) also converges +to x with respect to e by considering two cases: +Case 1: x ∈ ∂σX: lim inf +y→∂σX f(y) = ∞ yields lim inf +n→∞ f(xn) = ∞, so that +for each m ∈ N +e(xm, xn) ≥ σf(xn, xm) = |f(xn) − f(xm)| +is unbounded in n. In particular, this contradicts the assumption that +(xn) is Cauchy with respect to e. +Case 2: x ∈ X: Since σ induces the discrete topology on X, con- +vergence with respect to σ to some point in X yields that (xn) must +eventually be constant. Hence, it also converges with respect to e. +(b): By assumption X +σ is compact. Hence, vanishing capacity of +∂σX implies recurrence by Theorem 4.2. +□ +Remark. (a) Part (a) of this Theorem is a generalization of [8, Theo- +rem 3], which only treats certain path metrics. Note that for finite +underlying measures the equality D(Q) = D(Qmax) discussed in +this reference is equivalent to recurrence, see e.g. [19, Theorem 6.5]. +(b) The condition of (X, σ) being totally bounded means that it can +be isometrically embedded into a compact metric space. Below we +consider examples of bounded discrete X ⊆ R2 equipped with the +Euclidean metric. +(c) For general graphs it remains an open question whether or not the +previous theorem is true. +(d) For locally finite G = (X, b) we established the equivalence of the +following assertions. +(i) G is recurrent. +(ii) For one/all intrinsic metrics σ with respect to a finite measure +m that induce the discrete topology we have +capm(∂σX) = 0. +(iii) The set of all infinite paths is null. + +RECURRENT AND (STRONGLY) RESOLVABLE GRAPHS +25 +The implication (iii) ⇒ (ii) can be seen as a sort of converse to +Theorem 3.6 when considering the set A = ∂σX. +5. Resolvable graphs and harmonic functions +In this section we turn to the study of transient graphs. By the re- +sults of the last section, transience is characterized by positivity of the +capacity of (suitable) boundaries. Here, we turn to a different aspect +based on a characterization of recurrence and transience in terms of su- +perharmonic functions (read on for the precise definition). Specifically, +a graph is transient if and only if it admits non-constant superhar- +monic functions of finite energy (see e.g. [12]). In general, these super- +harmonic functions will not be harmonic. This stimulates interest in +those (transient) graphs which admit non-constant harmonic functions +of finite energy. The aim of this section is to derive a capacity based +sufficient condition for existence of such functions. To this end, we +introduce strong resolvability for graphs, which is a somewhat stronger +property than resolvability that was introduced in [2]. We prove that +(most) transient strongly resolvable graphs admit harmonic functions. +Definition 5.1 (Resolvability). A graph G = (X, b) is called resolvable +if there is an edge weight w with +� +x,y∈X +b(x, y)w(x, y)2 < ∞ +such that for every point x ∈ ∂dwX the set of paths converging to x +with respect to dw is null. In this case, w is called a resolving weight +for G. +The previous definition relied on the concept of path. We now aim +at a path-free definition which captures essentially the same concept. +This yields the following definition. +Definition 5.2 (Strong resolvability). A graph G = (X, b) is called +strongly resolvable, if there exists an intrinsic metric σ with respect +to a finite measure m that induces the discrete topology such that +capm({x}) = 0 for all x ∈ ∂σX. In this case, σ is called a resolving +metric for G. +Proposition 5.3. A strongly resolvable graph is resolvable. +Proof. Let σ be a resolving metric for G that is intrinsic with respect +to the finite measure m. By Lemma 2.7, the path metric dσ induced +by σ satisfies σ ≤ dσ. Hence, there is a continuous map ι: X +dσ → X +σ +that extends the identity on X. +Let x ∈ ∂dσX. We first show that ι(x) ∈ ∂σX. Suppose that this +is not the case, i.e. ι(x) ∈ X. We choose a sequence (xn) in X with +xn → x with respect to dσ. Since xn → ι(x) with respect to σ and σ + +26 +LENZ, PUCHERT, AND SCHMIDT +induces the discrete topology, (xn) must be eventually constant. Hence, +x = ι(x) ∈ X, a contradiction. +Any path converging to x with respect to dσ converges to ι(x) with +respect to σ. Hence, the set of all such paths is contained in Γ{ι(x)},σ, +the set of paths having ι(x) as an accumulation point with respect to σ. +It therefore suffices to show that the latter set is null. Since ι(x) ∈ ∂σX, +we have capm({ι(x)}) = 0 by assumption. Theorem 3.6 implies that +Γ{ι(x)},σ is null. +□ +Remark. (a) Strong resolvability transfers the geometric notion of re- +solvability introduced in [2] to a notion of potential theory. This +has two advantages. With strong resolvability one can also treat +non-locally finite graphs, as potential theory does not distinguish +between locally finite and non-locally finite graphs. +This is an +advantage of potential theory. Indeed, for notions invoking infi- +nite paths in general the non-locally finite case poses problems, as +e.g. the discrete Hopf-Rinow theorem 2.7 does not hold on non- +locally finite graphs, see the discussion in [8, Appendix A]. More- +over, strong resolvability is also available on more general spaces +that admit a potential theory, e.g. Riemannian manifolds, fractals +or metric graphs. +(b) As discussed after Theorem 3.6 we believe that capm({x}) = 0 for +all x ∈ ∂σX is strictly stronger than Γ{x},σ being null for all x ∈ +∂σX. Hence, resolvability seems strictly stronger that resolvability +(even though we do not have concrete examples). However, below +we shall see that planar graphs, the main examples for resolvable +graphs in [2], are also strongly resolvable. +Definition 5.4 ((Super)Harmonic functions). Let G = (X, b) be a +graph. A function f : X → C is called superharmonic if for all x ∈ X +it satisfies +f(x) ≥ +1 +deg(x) +� +y∈X +b(x, y)f(y), +where we assume absolute convergence of the sum on the right side of +the equation. The function f is called harmonic if both f and −f are +superharmonic. We write H(G) for the space of harmonic functions. +An important property of these functions is that on transient graphs +functions of finite energy are uniquely represented as sums of harmonic +functions of finite energy and functions in D0(G), see Theorem 6.3 in +[22] for reference. +Theorem 5.5 (Royden decomposition). Let G = (X, b) be a transient +graph. For all f ∈ D(G) there exists a unique f0 ∈ D0(G) and a unique +harmonic fh ∈ D(G) such that +f = f0 + fh + +RECURRENT AND (STRONGLY) RESOLVABLE GRAPHS +27 +and +Q(f) = Q(f0) + Q(fh). +The function fh is the unique function in D(G) that satisfies +Q(fh) = inf{Q(f − g) | g ∈ D0(G)} = inf{Q(f − g) | g ∈ Cc(X)}. +Moreover, if f is bounded, then f0 and fh are bounded as well. +Resolvability was introduced to prove the existence of non-constant +harmonic functions on transient locally finite resolvable graphs. This +result of Benjamini and Schramm carries over to strongly resolvable +graphs that need not be locally finite. Before we prove this we need a +result on harmonic functions induced by Lipschitz functions. +Let σ ∈ M(G) which is intrinsic with respect to the finite measure +m and suppose now that (X, b) is transient. Using the Royden decom- +position we define the map +Φ: D(G) ∩ Cb(X +σ) → D(G) ∩ H(G), +f �→ fh. +Since the Royden decomposition preserves boundedness and m is finite, +we even obtain that Φ maps to H1(G, m). The following is the main +observation in this section, which will be used to construct many non- +constant hamronic functions of finite energy. +Lemma 5.6. Let (X, b) be a graph and let σ be an intrinsic metric +with respect to a finite measure m that induces the discrete topology +such that capm(∂σX) > 0. Moreover, let +S = {x ∈ ∂σX | capm(U ∩ ∂σX) > 0 all open x ∈ U ⊆ X +σ}. +Then +ker Φ ⊆ {f ∈ D(G) ∩ Cb(X +σ) | f|S = 0}. +In particular, if Φ(f) is constant, then f is constant on S. +Proof. Let f ∈ D(G) ∩ Cb(X +σ) with f|S ̸= 0. We show fh = Φ(f) ̸= 0. +Without loss of generality there exists ε > 0 such that f(x) ≥ ε for +some x ∈ S. Using Theorem 5.5 we choose a sequence (gn) in Cc(X) +with Q(fh) = infn≥1 Q(f − gn) = limn→∞ Q(f − gn). As noted above +we have fh ∈ H1(G, m). +Claim: The sequence (gn) can be chosen such that f − gn → fh in +H1(G, m). +Proof of the claim. It suffices to show that (gn) can be chosen such +that f − gn → fh in ℓ2(X, m). We can assume ∥gn∥∞ ≤ 2 ∥f∥∞, as +otherwise we could write +((f − gn) ∧ ∥f∥∞) ∨ (− ∥f∥∞) = f − hn +with appropriate hn ∈ Cc(X). These satisfy ∥hn∥∞ ≤ 2 ∥f∥∞ and, +using the compatibility of Q with contractions, also +Q(fh) ≤ Q(f − hn) ≤ Q(f − gn). + +28 +LENZ, PUCHERT, AND SCHMIDT +Now assume (gn) is chosen with ∥gn∥∞ ≤ 2 ∥f∥∞. +The Royden +decomposition shows Q(f0 − gn) → 0. +Moreover, our assumption +capm(∂σX) > 0 implies transience of the graph, see Corollary 4.4. +On transient graphs convergence on D0(G) with respect to Q implies +pointwise convergence, see e.g. [11, Theorem B.2]. Hence, we obtain +f0 − gn → 0 pointwise, which implies f − gn → fh pointwise. Since +the functions f − gn are uniformly bounded by 3 ∥f∥∞ and since m +is a finite measure, Lebesgue’s dominated convergence theorem yields +f − gn → fh in ℓ2(X, m), which shows the claim. +Let now (gn) be a sequence as in the claim. Since σ induces the +discrete topology, the compactly supported function gn can be contin- +uously extended to X +σ by letting gn = 0 on ∂σX. Then f − gn is +continuous on X +σ with f − gn = f on ∂σX. By the continuity of f +there exists a relatively open neighborhood Ux ⊆ ∂σX of x in ∂σX +with f − gn = f ≥ ε/2 on Ux for all n ∈ N. Using that x ∈ S we +obtain capm(Ux) > 0. +By the continuity of f − gn there exists an +open On ⊆ X +σ with Ux ⊆ On and f − gn ≥ ε/4 on On such that +4(f −gn)/ε ≥ 1 on X ∩On. The way the capacity is defined for subsets +of the boundary yields +16 +ε2 ∥f − gn∥2 +Q,m ≥ capm(X ∩ On) ≥ capm(Ux). +Using f − gn → fh in H1(G, m), we obtain +∥fh∥2 +Q,m = lim +n→∞ ∥f − gn∥2 +Q,m ≥ ε2 +16capm(Ux) > 0 +and arrive at fh ̸= 0. +For the ’In particular’-part assume that Φ(f) is constant equal to +C. Since the harmonic part of a constant function is just the constant +function itself, we obtain Φ(f − C) = Φ(f) − C = 0. Hence, what we +previously proved shows f = C on S. +□ +Remark. The set S in this lemma is the support of the outer measure +capm restricted to subsets of ∂σX. +Assume σ ∈ M(G) is intrinsic with respect to the finite measure m. +We denote the set of bounded Lipschitz functions with respect to σ +by Lipb(X) = Lipb,σ(X). If f ∈ Lipb(X), then Lemma 2.4 shows that +f ∈ H1(G, m) (the lemma implies f ∈ D(G) and the boundedness of +f yields f ∈ ℓ2(X, m)). Moreover, f can be uniquely extended to a +Lipschitz function X +σ, which we also denote by f with a slight abuse +of notation. Hence, Lipb(X) ⊆ D(G) ∩ C(X +σ). This observation is +used in the proof of the following theorem. +Theorem 5.7. Let (X, b) be a graph and let σ be an intrinsic metric +with respect to a finite measure m that induces the discrete topology + +RECURRENT AND (STRONGLY) RESOLVABLE GRAPHS +29 +such that capm(∂σX) > 0. Moreover, let +S = {x ∈ ∂σX | capm(U ∩ ∂σX) > 0 all open x ∈ U ⊆ X +σ}. +Then dim(D(G)∩H(G)) ≥ |S|. In particular, if |S| ≥ 2, then the graph +admits a non-constant harmonic function of finite energy. +Proof. Without loss of generality we can assume |S| ≥ 2 for oth- +erwise the statement is trivial because constant functions belong to +D(G) ∩ H(G). Let f1, . . . , fn ∈ Lipb(X). The previous lemma shows +that if f1|S, . . . , fn|S are linearly independent, then Φ(f1), . . . , Φ(fn) are +linearly independent in D(G)∩H(G). With this at hand the statement +follows from dim Lipb(X) ≥ dim Lipb(S) ≥ |S| (use that any bounded +Lipschitz function on S can be extended to a bounded Lipschitz func- +tion on X +σ). +□ +Corollary 5.8 (Existence of non-constant harmonic functions). Let +G = (X, b) be a strongly resolvable graph and let σ be a resolving metric +that is intrinsic with respect to the finite measure m. If capm(∂σX) > 0, +then the space of bounded harmonic functions of finite energy is infinite +dimensional. +Proof. Let S be the support of the capacity on the boundary introduced +in the previous theorem. Its complement is given by +∂σX\S = {x ∈ ∂σX | ex. open x ∈ U ⊆ X +σ with capm(U∩∂σX) = 0}. +Since (X +σ, σ) is separable, its topology has a countable basis. Hence, +the σ-sub-additivity of capm yields capm(∂σX \ S) = 0. +Using the +subadditivity of the capacity again shows +capm(S) = capm(∂σX) > 0. +By our assumption every point in S has capacity 0. Hence, S must be +uncountable for otherwise σ-subadditivity would imply capm(S) = 0, +which contradicts our previous considerations. With this at hand the +claim follows from the previous theorem. +□ +This theorem is a version of [2, Theorem 3.1] for strongly resolvable +but possibly non-locally finite graphs. We had to replace the transience +assumption of [2] by the stronger capm(∂σX) > 0. As discussed in The- +orem 4.5, for some classes of graphs transience implies this condition. +We mention these situations in the following corollary. +Corollary 5.9. Let G = (X, b) be a transient, strongly resolvable graph +and let one of the following conditions be fulfilled: +(a) G is locally finite. +(b) σ is a resolving metric and (X, σ) is totally bounded. +Then the space of bounded harmonic functions of finite energy is infinite +dimensional. + +30 +LENZ, PUCHERT, AND SCHMIDT +Proof. By Theorem 4.5 both conditions imply capm(∂σX) > 0 with +respect to a resolving metric σ. +Hence, the claim follows from the +previous corollary. +□ +Constructing harmonic functions from functions on a potential theo- +retic boundary (the support of the capacity on the metric boundary) is +reminiscent of solving the Dirichlet problem. Under suitable additional +conditions on the graph and on the function on the boundary this can +be made precise. +Remark (Solving the Dirichlet problem on S for uniformly transient +graphs). Let (X, b) be a graph with D0(G) ⊆ C0(X), where C0(X) +denotes the uniform closure of Cc(X). Graphs with this property are +called uniformly transient. As the name suggests uniformly transient +graphs are transient, see [11] for this fact and further background on +uniform transience. Let σ ∈ M(G) be intrinsic with respect to the +finite measure m and let S denote the support of the capacity on the +boundary discussed above. Then for any bounded Lipschitz function +ϕ: S → R the Dirichlet problem +� +h ∈ H(G) ∩ D(G) +h ∈ Cb +� +X +σ� +with h|S = ϕ +has a unique solution. +Uniqueness: This follows directly from Lemma 5.6. +Existence: The bounded Lipschitz function ϕ: S → R can be ex- +tended to a bounded Lipschitz function f : X +σ → R. +Consider the +Royden decomposition f = f0 + fh with f0 ∈ D0(G) ⊆ C0(X) and +harmonic fh ∈ D(G). Any sequence in X converging to a point in +∂σX must eventually leave any finite set. Hence, f0 can be extended +to a continuous function on X +σ by letting f0 = 0 on ∂σX. This shows +that also fh = f − f0 has a bounded continuous extension to X +σ with +fh|∂σX = f|∂σX. By constructions this yields fh|S = ϕ. +6. Planar and canonically compactifiable graphs +In this section we show that circle packings of bounded geometry +and hence locally finite planar graphs of bounded geometry are always +strongly resolvable. Moreover, we prove that canonically compactifiable +graphs are never strongly resolvable showing that planar graphs of +bounded geometry can never be canonically compactifiable. +First we recall the notion of circle packings and their contact graphs. +For an extensive background on these topics we refer to the book [15]. +Definition 6.1 (Circle packing and subordinated graphs). A circle +packing is a set X ̸= ∅ and two maps r: X → (0, ∞) and ϕ: X → R2 +such that the collection of closed circles Cx = Br(x)(ϕ(x)), x ∈ X, in R2 +satisfies C◦ +x ∩ C◦ +y = ∅ whenever x ̸= y. It is called bounded if � +x∈X Cx + +RECURRENT AND (STRONGLY) RESOLVABLE GRAPHS +31 +is a bounded set. An edge weight b on X is called subordinate to the +circle packing if b(x, y) > 0 implies Cx ∩ Cy ̸= ∅. +Remark. In what follows we simply write Cx, x ∈ X, to denote a circle +packing. The contact graph or nerve of a circle packing Cx, x ∈ X, is +the combinatorial graph on X with x ∼ y if Cx ∩ Cy ̸= ∅. Hence, an +edge weight b on X is subordinate to the circle packing if and only if +the induced discrete graph is a subgraph of the contact graph. +The following is our main observation in this section. +Theorem 6.2. Let Cx, x ∈ X, be a bounded circle packing and suppose +(X, b) is subordinate to the circle packing and has bounded geometry. +Then (X, b) is strongly resolvable. In particular, if (X, b) is transient, +then it possesses a non-constant harmonic function of finite energy. +Proof. Let Ω = supx∈X deg(x). +Then Ω < ∞ due to (X, b) having +bounded geometry. As above we let r, ϕ denote the maps inducing the +circle packing. We consider the metric σ on X defined by σ(x, y) = +|ϕ(x) − ϕ(y)|. We first show that σ is an intrinsic metric with respect +to a finite measure inducing the discrete topology. +The assumption Cx ∩ Cy ̸= ∅ for all x ̸= y implies +σ(x, y) ≥ r(x) + r(y) > r(x) +for all y ̸= x. +Hence, σ induces the discrete topology. +Since b is +subordinate to the circle packing, we also have Cx ∩ Cy ̸= ∅ whenver +x ∼ y. For x ∼ y this implies σ(x, y) = r(x) + r(y). We infer +� +x,y∈X +b(x, y)σ(x, y)2 ≤ 2 +� +x,y∈X +b(x, y)(r(x)2 + r(y)2) +≤ 4Ω +� +x∈X +r(x)2 +≤ 4Ω +π λ(A) < ∞, +with A = � +x∈X Cx and λ the Lebesgue measure. This shows that σ is +intrinsic with respect to a finite measure m. +Using that ϕ is an isometry we identify (X, σ) with ϕ(X) in R2. +In particular, the boundary with respect to σ is just the Euclidean +boundary. Given w ∈ ∂X we show capm({w}) = 0. For r > 0 we +consider the function +fr : X → R, +fr(x) = +� +2 − |x − w| +r +� ++ +∧ 1. +It satisfies fr = 1 on Br(w) ∩ X and fr = 0 on X \ B2r(w). Moreover, +for x ∼ y we have +|fr(x) − fr(y)|2 ≤ |x − y|2 +r2 += (r(x) + r(y))2 +r2 +. + +32 +LENZ, PUCHERT, AND SCHMIDT +Next we compare (r(x) + r(y))2 with λ((Cx ∪ Cy) ∩ B2r(w)) as long as +x ∼ y and x, y ∈ B2r(w). The boundary point w does not belong to +the interior of the discs Cx, Cy. This leads to r(x), r(y) ≤ 2r. Using +this observation and that Cx, Cy are tangent, we obtain +λ((Cx ∪ Cy) ∩ B2r(w)) = λ(Cx ∩ B2r(w)) + λ(Cy ∩ B2r(w)) +≥ C(r(x)2 + r(y)2) +for some constant C > 0 independent of x, y and r (for the last in- +equality we simply estimated the area of the intersection of two discs +with the given parameters). Combining these estimates we infer +Q(fr) ≤ +1 +2r2 +� +x,y∈X +b(x, y)(r(x) + r(y))2 +≤ +1 +Cr2 +� +x,y∈X +b(x, y)λ((Cx ∪ Cy) ∩ B2r(w)) +≤ 2Ω +Cr2λ(B2r(w)) +≤ 8πΩ +C . +Since m is finite, we also have ∥fr∥m → 0, as r → 0+. +Both ob- +servations combined imply that (fr) is bounded in the Hilbert space +H1(G, m). +Using the Banach-Saks theorem we obtain a decreasing +sequence rk → 0 such that +gn = 1 +n +n +� +k=1 +frk +converges in H1(G, m) to some g ∈ H1(G, m). Since convergence in +H1(G, m) implies ℓ2(X, m)-convergence and since frk → 0 in ℓ2(X, m), +we obtain g = 0. By construction we also have gn ≥ 1 on Brn(w), +which leads to +capm({w}) ≤ inf +n∈N +� +Q(gn) + ∥gn∥2 +m +� += 0. +The ’In particular’-part follows from Corollary 5.9 and the observa- +tion that (X, σ) is totally bounded as it is isometric to a bounded and +hence totally bounded subset of R2. +□ +Remark. We do not assume local finiteness in the previous theorem. If +� +x∈X Cx is not dense in R2, then the assumption on the boundedness of +the circle packing can be dropped. In this case, one just uses inversion +at a circle in the complement of � +x∈X Cx to obtain a bounded circle +packing with isomorphic contact graph. For more details see also the +proof of the following corollary. + +RECURRENT AND (STRONGLY) RESOLVABLE GRAPHS +33 +In the following corollary we call a weighted graph planar if the +induced combinatorial graph is planar (for a precise definition of the +latter see e.g. [14, Section 2.1]). +Corollary 6.3. Let G = (X, b) be a locally finite planar graph of +bounded geometry. +Then G is strongly resolvable. +In particular, if +G is transient, then G possesses a non-constant harmonic function of +finite energy. +Proof. According to Claim 4.3 in [15] any locally finite graph is isomor- +phic to the contact graph of a circle packing. We show that the circle +packing can be chosen to be bounded. With this at hand the claim +follows from the previous theorem. +We add one additional point o and one edge from o to a point in +X such that the resulting graph (X′, b′) is still planar. According to +Claim 4.3 in [15] the graph (X′, b′) is isomorphic to a contact graph +of a circle packing Cx, x ∈ X′. In order to make this circle packing +bounded, we use inversion at the circle Co corresponding to the new +vertex o. We denote the inversion map by ψ. Since inversions map +circles to circles, ψ(Cx), x ∈ X, is a circle packing inside the bounded +set Co. By construction its contact graph is the combinatorial graph +underlying (X, b). +□ +Remark. The existence of non-trivial harmonic functions on transient +planar graphs of bounded geometry was one of the main results [2]. +Subsequently, even more explicit description of all harmonic functions +of planar graphs were given via boundaries of sphere packings [1] or +square tilings [5]. For a unified approach we refer to [9]. +Recently the class of canonically compactifiable graphs (see below for +a definition) has gathered some attention. Our previous considerations +allow us to show that locally finite planar graphs of bounded geometry +are never canonically compactifiable. +According to [6] a graph G = (X, b) is called canonically compactifi- +able if D(G) ⊆ ℓ∞(X) (see [18] for different equivalent characterizations +as well). Examples are Zn with n ≥ 3, see [11, Section 6], or graphs +(X, b) for which +� +x,y∈X +1 +b(x, y) < ∞, +see [6, Example 4.6]. Note that the latter condition implies very large +vertex degrees. We note the following. +Theorem 6.4. Infinite canonically compactifiable graphs are not strongly +resolvable. In particular, locally finite infinite planar graphs of bounded +geometry are not canonically compactifiable. +Proof. Let (X, b) be an infinite canonically compactifiable graph and +let m be a finite measure on X. +Canonical compactifiability yields + +34 +LENZ, PUCHERT, AND SCHMIDT +H1(G, m) ⊆ ℓ∞(X). The closed graph theorem implies the existence +of C > 0 such that +∥f∥2 +∞ ≤ C(Q(f) + ∥f∥2 +m) +for all f ∈ H1(G, m). This implies capm(U) ≥ 1/C for any ∅ ̸= U ⊆ X +such that points in any metric boundary have a capacity at least 1/C. +¸It remains to prove that for any intrinsic metric σ with respect to m, +which induces the discrete topology, the space (X, σ) is not complete +(and hence it has at least one boundary point). +According to [18] +(X, b) being canonically compactifiable and σ being an intrisic metric +with respect to a finite measure imply that (X, σ) is totally bounded. +Hence, X +σ is compact. But (X, σ) is not compact as an infinite set +with the discrete topology. This shows ∂σX = X +σ \ X ̸= ∅. +The ’In particular’-part follows from the previous corollary. +□ +References +[1] Omer Angel, Martin T. Barlow, Ori Gurel-Gurevich, and Asaf Nachmias. +Boundaries of planar graphs, via circle packings. Ann. Probab., 44(3):1956– +1984, 2016. +[2] Itai Benjamini and Oded Schramm. Harmonic functions on planar and almost +planar graphs and manifolds, via circle packings. Invent. math., 126:565–587, +1996. +[3] Rupert L. Frank, Daniel Lenz, and Daniel Wingert. Intrinsic metrics for non- +local symmetric Dirichlet forms and applications to spectral theory. J. Funct. +Anal., 266(8):4765–4808, 2014. +[4] Masatoshi Fukushima, Yoichi Oshima, and Masayoshi Takeda. Dirichlet forms +and symmetric Markov processes, volume 19 of de Gruyter Studies in Mathe- +matics. Walter de Gruyter & Co., Berlin, extended edition, 2011. +[5] Agelos Georgakopoulos. The boundary of a square tiling of a graph coincides +with the Poisson boundary. Invent. Math., 203(3):773–821, 2016. +[6] Agelos Georgakopoulos, Sebastian Haeseler, Matthias Keller, Daniel Lenz, and +Rados�law K. Wojciechowski. Graphs of finite measure. J. Math. Pures Appl. +(9), 103(5):1093–1131, 2015. +[7] Batu G¨uneysu, Matthias Keller, and Marcel Schmidt. A Feynman-Kac-Itˆo +formula for magnetic Schr¨odinger operators on graphs. Probab. Theory Related +Fields, 165(1-2):365–399, 2016. +[8] Xueping Huang, Matthias Keller, Jun Masamune, and Rados�law K. Woj- +ciechowski. A note on self-adjoint extensions of the Laplacian on weighted +graphs. J. Funct. Anal., 265(8):1556–1578, 2013. +[9] Tom Hutchcroft and Yuval Peres. Boundaries of planar graphs: a unified ap- +proach. Electron. J. Probab., 22:Paper No. 100, 20, 2017. +[10] Matthias Keller. Intrinsic metrics on graphs: a survey. In Mathematical tech- +nology of networks, volume 128 of Springer Proc. Math. Stat., pages 81–119. +Springer, Cham, 2015. +[11] Matthias Keller, Daniel Lenz, Marcel Schmidt, and Rados�law K. Woj- +ciechowski. Note on uniformly transient graphs. Rev. Mat. Iberoam., 33(3):831– +860, 2017. +[12] Matthias Keller, Daniel Lenz, and Rados�law K. Wojciechowski. Graphs and +discrete Dirichlet spaces, volume 358 of Grundlehren der mathematischen + +RECURRENT AND (STRONGLY) RESOLVABLE GRAPHS +35 +Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, +Cham, [2021] ©2021. +[13] Matthias Keller and Florentin M¨unch. A new discrete Hopf-Rinow theorem. +Discrete Math., 342(9):2751–2757, 2019. +[14] Bojan Mohar and Carsten Thomassen. Graphs on surfaces. Johns Hopkins +Studies in the Mathematical Sciences. Johns Hopkins University Press, Balti- +more, MD, 2001. +[15] Asaf Nachmias. Planar maps, random walks and circle packing, volume 2243 +of Lecture Notes in Mathematics. Springer, Cham, 2020. ´Ecole d’´et´e de prob- +abilit´es de Saint-Flour XLVIII—2018, ´Ecole d’´Et´e de Probabilit´es de Saint- +Flour. [Saint-Flour Probability Summer School]. +[16] Tadashi Nakamura and Maretsugu Yamasaki. Generalized extremal length of +an infinite network. Hiroshima Math. J., 6(1):95–111, 1976. +[17] Simon Puchert. Resolvable graphs and harmonic functions, 2019. +[18] Simon Puchert. Characterizations of canonically compactifiable graphs via in- +trinsic metrics and algebraic properties. Arch. Math. (Basel), 116(5):541–549, +2021. +[19] Marcel Schmidt. Global properties of Dirichlet forms on discrete spaces. Dis- +sertationes Math., 522:43, 2017. +[20] Marcel Schmidt. On the existence and uniqueness of self-adjoint realizations +of discrete (magnetic) Schr¨odinger operators. In Analysis and geometry on +graphs and manifolds, volume 461 of London Math. Soc. Lecture Note Ser., +pages 250–327. Cambridge Univ. Press, Cambridge, 2020. +[21] Marcel Schmidt. Energy forms. PhD thesis, arXiv:1703.04883, Friedrich- +Schiller-Universit¨at Jena 2017. +[22] Paolo M. Soardi. Potential theory on infinite networks, volume 1590 of Lecture +Notes in Mathematics. Springer-Verlag, Berlin, 1994. +[23] Karl-Theodor Sturm. Analysis on local Dirichlet spaces. I. Recurrence, conser- +vativeness and Lp-Liouville properties. J. Reine Angew. Math., 456:173–196, +1994. +[24] Karl-Theodor Sturm. Analysis on local Dirichlet spaces. II. Upper Gaussian +estimates for the fundamental solutions of parabolic equations. Osaka J. Math., +32(2):275–312, 1995. +[25] Karl-Theodor Sturm. Analysis on local Dirichlet spaces. III. The parabolic +Harnack inequality. J. Math. Pures Appl. (9), 75(3):273–297, 1996. +[26] Wolfgang Woess. Random walks on infinite graphs and groups, volume 138 of +Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, +2000. +[27] Maretsugu Yamasaki. Extremum problems on an infinite network. Hiroshima +Math. J., 5(2):223–250, 1975. +[28] Maretsugu Yamasaki. Parabolic and hyperbolic infinite networks. Hiroshima +Math. J., 7(1):135–146, 1977. + +36 +LENZ, PUCHERT, AND SCHMIDT +D. Lenz, Institut f¨ur Mathematik, Friedrich-Schiller-Universit¨at +Jena, 07743 Jena, Germany +Email address: daniel.lenz@uni-jena.de +S. Puchert, Institut f¨ur Mathematik, Friedrich-Schiller-Universit¨at +Jena, 07743 Jena, Germany +Email address: simon.puchert@uni-jena.de +M. Schmidt, Mathematisches Institut, Universit¨at Leipzig, 04109 +Leipzig, Germany +Email address: marcel.schmidt@math.uni-leipzig.de + diff --git a/KdA0T4oBgHgl3EQfCv-W/content/tmp_files/load_file.txt b/KdA0T4oBgHgl3EQfCv-W/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..50bb3455a4ec5f5fd6619bdf7925d127cb14b559 --- /dev/null +++ b/KdA0T4oBgHgl3EQfCv-W/content/tmp_files/load_file.txt @@ -0,0 +1,1169 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf,len=1168 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='01994v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='FA] 5 Jan 2023 RECURRENT AND (STRONGLY) RESOLVABLE GRAPHS DANIEL LENZ, SIMON PUCHERT, AND MARCEL SCHMIDT Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We develop a new approach to recurrence and the existence of non-constant harmonic functions on infinite weighted graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The approach is based on the capacity of subsets of metric boundaries with respect to intrinsic metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The main tool is a connection between polar sets in such boundaries and null sets of paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' This connection relies on suitably diverging functions of finite energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Introduction Potential theory on infinite weighted graphs (sometimes called net- works) studies the induced graph energy functional and derived quanti- ties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' harmonic functions, random walks, resistances, Laplacians,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' As already noted in the pioneering works by Yamasaki [27, 16, 28], so- called null sets of infinite paths play an important role in this theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' For a comprehensive account of Yamasaki’s work (and beyond) we refer to the book [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Another approach to potential theory on weighted graphs is via Dirichlet forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The graph energy induces a Dirichlet form, which in turn leads to the notion of capacity of sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In this language sets of capacity zero, so-called polar sets, are key to understanding properties of the Dirichlet form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In contrast to the situation on graphs, for general Dirichlet forms the concept of a path may be meaningless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Instead, for many applications it turned out fruitful to consider intrinsic metrics as geometric input, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' [23, 24, 25, 3, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In this paper we follow the intrinsic metric line of thinking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Our main observation relates polar sets in certain metric boundaries of graphs to null sets of paths, see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' This sheds a new light on classical results formulated in terms of null sets of paths and allows us to prove new theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In this text we focus on consequences for recurrence and the existence of non-constant harmonic functions of finite energy: We prove a new characterization of recurrence in terms of the ex- istence of intrinsic metrics with finite balls, see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' This in turn leads to an alternative proof of a classical characterization of re- currence due to Yamasaki, see Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='3 and a characterization of recurrence in terms of metric boundaries being polar, see Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='4 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' 1 2 LENZ, PUCHERT, AND SCHMIDT We then turn to the problem of the existence of non-constant har- monic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Again, we tackle this problem by means of capacity on the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Specifically, we introduce the notion of strong resolv- ability, which is essentially a path-free and capacity-based version of the notion of resolvability studied in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We show that strongly resolv- able transient graphs admit non-constant harmonic functions of finite energy, see Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='8 and Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Since strong resolvability is stronger than resolvability (this is a consequence of our main obser- vation mentioned previously), we also prove that locally finite planar graphs of bounded geometry, the main class of examples of resolvable graphs, are even strongly resolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' This allows us to recover one of the main results from [2] that transient locally finite planar graphs of bounded geometry have non-constant harmonic functions of finite energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Note that recently more precise descriptions of the space of harmonic functions of planar graphs were obtained, see [1, 5, 9], which are beyond the scope of our theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Another consequence of planar graphs of bounded geometry being strongly resolvable is that they are never canonically compactifiable, see Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The latter class of graphs was introduced and studied in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='Our results show that two basic issues in the theory of recurrence viz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='characterization of recurrence and existence of non-constant harmonic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='functions can naturally be understood in terms of capacities of (suit- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='able) metric boundaries: Recurrence means that metric boundaries ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='are negligible in the sense of having capacity zero whereas existence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='of non-constant harmonic functions is implied by some richness in the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='structure of such a boundary in the sense of the positive capacity of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='the whole boundary not being concentrated on a single point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Since our methods do not rely on paths but only on intrinsic metrics, they are not limited to locally finite graphs as is sometimes the case in the classical setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Moreover, they can be adapted to more general Dirichlet spaces and even non-linear energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Both directions will be investigated in upcoming works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Parts of this text are based on Simon Puchert’s master’s thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Acknowledgments: Partial support of DFG, in particular, within the Priority programme ’Geometry at infinity’ is gratefully acknowl- edged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Preliminaries In this section we introduce the notation and the objects that are used throughout the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' For a, b ∈ R we let a ∧ b = min{a, b} and a ∨ b = max{a, b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Moreover, a+ = a ∨ 0 and a− = (−a)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We extend this notation pointwise to real-valued functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Graphs and Dirichlet energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Our study of graphs is based on an analytic tool given by the Dirichlet energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' RECURRENT AND (STRONGLY) RESOLVABLE GRAPHS 3 A graph G = (X, b) consists of a nonempty countable set X, whose elements are called nodes or vertices, and a symmetric edge weight function b : X × X → [0, ∞), satisfying the following conditions: The edge weight b vanishes on the diagonal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' b(x, x) = 0 for all x ∈ X, and the weighted vertex degree deg(x) := � y∈X b(x, y) must be finite for all x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' If the weighted vertex degree is bounded, we say that the graph has bounded geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' If the function b takes only values in {0, 1}, the graph (X, b) is called combinatorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Two vertices x, y ∈ X are said to be connected by the edge (x, y) if b(x, y) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In this case, we write x ∼ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Note that since b is symmetric, we have x ∼ y if and only if y ∼ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The set of all (oriented) edges of G is denoted by E := {(x, y) ∈ X × X | x ∼ y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We say that a graph is locally finite if for all x ∈ X the set of its neighbors {y ∈ X | x ∼ y} is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' A locally finite graph is called a bounded valence graph if the cardinality of the set of neighbors of each vertex is bounded by a universal constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' A path in G is a (finite or infinite) sequence (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=') of nodes such that xi ∼ xi+1, for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='. We say that two points x, y ∈ X are connected if there is a finite path (x = x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' , xn = y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' This defines an equivalence relation on the set of vertices and the resulting equivalence classes are called connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' From now on we will generally assume that the graph G is connected without mentioning this explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' This is not a real restriction because all our considerations in this text can be reduced to connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We equip X with the discrete topology and write C(X) for all real- valued functions on X (the continuous functions on X) and Cc(X) for the finitely supported real-valued functions on X (the continuous functions of compact support).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Any function m : X → (0, ∞) induces a Radon measure of full support on all subsets of X via m(A) := � x∈A m(x), A ⊆ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In what follows we do not distinguish between such measures and strictly positive functions and simply call them measures on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Every weighted graph G = (X, b) gives rise to a quadratic form Q: C(X) → [0, ∞] that assigns to any function f : X → R its Dirichlet energy Q(f) := 1 2 � x,y∈X b(x, y)|f(x) − f(y)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' 4 LENZ, PUCHERT, AND SCHMIDT The space of functions of finite energy is D(G) := {f : X → R | Q(f) < ∞}, on which Q acts as a bilinear form by polarization, namely Q(f, g) = 1 2 � x,y∈X b(x, y)(f(x) − f(y))(g(x) − g(y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Here we abuse notation so that Q(f) = Q(f, f) for every f ∈ D(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The form Q has the following fundamental semi-continuity property, which is a direct consequence of Fatou’s lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='1 (Semicontinuity of Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let (X, b) be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let (fn) be a sequence of functions on X converging pointwise to the function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then, Q(f) ≤ lim inf n→∞ Q(fn), where the value ∞ is allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' A map C : R → R is called contraction if |C(s)−C(t)| ≤ |s−t| holds for all s, t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The Dirichlet energy has the important property that it is reduced by contractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Specifically, the following proposition is a direct consequence of the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='2 (Fundamental contraction property).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let (X, b) be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' For all f ∈ D(G) and all contractions C : R → R we have C ◦ f ∈ D(G) and Q(C ◦ f) ≤ Q(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Two useful families of contractions that will be of use later are the following: For a, b ∈ R we define the clamping function C[a,b] : R → R by C[a,b](x) := (x ∧ b) ∨ a and for c ∈ R we define the slicing function Sc : R → R by Sc(x) := (x − c)+ ∧ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' For o ∈ X we define the (pseudo-)norm ∥·∥o : D(G) → [0, ∞) by ∥f∥o := � Q(f) + |f(o)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We denote the closure of the space of functions of compact support Cc(X) with respect to ∥·∥o by D0(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The following well-known lemma shows that ∥·∥o is indeed a norm and that the space D0(G) does not depend on the choice of o ∈ X, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' [19, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Its proof relies on the connectedness of G, which we always assume in this text (see above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let G = (X, b) be a graph and let o ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (a) ∥·∥o is a norm and (D(G), ∥·∥o) is a Banach space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (b) fn → f with respect to ∥·∥o implies fn → f pointwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' RECURRENT AND (STRONGLY) RESOLVABLE GRAPHS 5 (c) For o′ ∈ X the norms ∥·∥o and ∥·∥o′ are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' A measure m on X induces the Hilbert space ℓ2(X, m) = {f ∈ C(X) | � x∈X m(x)|f(x)|2 < ∞} with inner product ⟨f, g⟩m = � x∈X m(x)f(x)g(x) and corresponding norm ∥·∥m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We denote by H1(G, m) := ℓ2(X, m) ∩ D(G) the corresponding first-order Sobolev space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Equipped with the inner product ⟨f, g⟩Q,m := ⟨f, g⟩m + Q(f, g) it is a Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We denote the associated norm by ∥·∥Q,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' For a proof of the completeness of this space we refer the reader to [19, Sec- tion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='3], where the completeness of H1(G, m) is discussed as closedness of the Dirichlet form Q(N) (in the notation used there).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Intrinsic metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In this section we introduce the (pseudo)metrics relevant for our considerations and discuss their properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' A symmetric function σ : X × X → [0, ∞) is called a pseudometric if it satisfies the triangle inequality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' if for all x, y, z ∈ X it satisfies σ(x, y) ≤ σ(x, z) + σ(z, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' For r ≥ 0 and x ∈ X we denote the corresponding ball of radius r around x by Bσ r (x) := {y ∈ X | σ(x, y) ≤ r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' For x ∈ X the distance σU from a nonempty subset U ⊆ X is defined by σU(x) := σ(x, U) := inf y∈U σ(x, y) and the diameter of U with respect to σ is diamσ(U) := sup x,y∈U σ(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' A function f on X is called Lipschitz-function with respect to the pseudometric σ if there exists a C > 0 with |f(x) − f(y)| ≤ Cσ(x, y) for all x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We then also say that f is a C-Lipschitz function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The set of all Lipschitz-functions with respect to σ is denoted by Lipσ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' For a graph G = (X, b) a pseudometric σ is called intrinsic with respect to the measure m if for all x ∈ X it satisfies 1 2 � y∈X b(x, y)σ(x, y)2 ≤ m(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' 6 LENZ, PUCHERT, AND SCHMIDT We write M(G) for the set of pseudometrics that are intrinsic with respect to a finite measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Clearly, a pseudometric σ belongs to M(G) if and only if � x,y∈X b(x, y)σ(x, y)2 < ∞ holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Remark (Background on intrinsic metrics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Intrinsic metrics have long proven to be a useful tool in spectral geometry of manifolds and, more generally, for strongly local Dirichlet spaces, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Sturm’s seminal work [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' For general Dirichlet spaces, including graphs, a system- atic approach was developed in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' A key point in [3] is a Rademacher type theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In the context of graphs this theorem says that a pseudo- metric σ is intrinsic if and only if for all 1-Lipschitz functions f : X → R with respect to σ we have |∇f|2 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Here, for f ∈ C(X) and x ∈ X the quantity |∇f|2(x) := 1 2m(x) � y∈X b(x, y)(f(x) − f(y))2 can be interpreted as the square of the norm of the discrete gradient of f at x (with respect to the measure m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' For graphs with measure m for which the scaled degree deg /m is uniformly bounded the com- binatorial metric is an intrinsic metric (up to a constant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' For graphs with unbounded degree this is not the case anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' For such graphs, intrinsic metrics (rather than the combinatorial metric) have turned out to be the right metrics for various questions, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' the survey [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The present article can also be seen as a point in case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' There are strong ties between functions of finite Dirichlet energy and intrinsic pseudometrics with respect to a finite measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' These will be of relevance for some of our theorems below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='4 (From M(G) to D(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let G = (X, b) be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let σ be an intrinsic pseudometric with respect to the finite measure m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let U be a subset of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then, the following statements hold: (a) Any function f that is C-Lipschitz with respect to σ and constant on U satisfies Q(f) ≤ C2 min{m(X), 2m(X \\ U)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (b) The inequality Q(σU) ≤ min{m(X), 2m(X \\ U)} is valid and, in particular, σU belongs to D(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (c) The inequality Q(f) ≤ C2m(X) holds for any C-Lipschitz function f with respect to σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In particu- lar, any Lipschitz function with respect to σ belongs to D(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' RECURRENT AND (STRONGLY) RESOLVABLE GRAPHS 7 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Clearly, both (b) and (c) are immediate consequences of (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Thus, we only show (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' It suffices to consider the case C = 1 as for general C > 0 the function g = 1 Cf is 1-Lipschitz with Q(g) = 1 C2Q(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' So, let f be a 1-Lipschitz function vanishing on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The bound Q(f) ≤ m(X) follows easily from the estimate Q(f) = 1 2 � x,y b(x, y)(f(x) − f(y))2 ≤ � x∈X � 1 2 � y∈X b(x, y)σ(x, y)2 � and the fact that σ is intrinsic with respect to m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The other estimate can be shown as follows: Using f(x) − f(y) = 0 for all x, y ∈ U and |f(x) − f(y)| ≤ σ(x, y) for all x, y ∈ X we infer Q(f) = 1 2 � x,y∈X b(x, y)(f(x) − f(y))2 = 1 2 � (x,y)∈X2\\U2 b(x, y)(f(x) − f(y))2 ≤ 1 2 � (x,y)∈X2\\U2 b(x, y)σ(x, y)2 = � y∈X\\U � 1 2 � x∈X b(x, y)σ(x, y)2 � + � x∈X\\U � 1 2 � y∈U b(x, y)σ(x, y)2 � ≤ 2m(X \\ U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' □ Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Clearly, the estimates given in the previous lemma trivially continue to hold if m is not a finite measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='5 (From to D(G) to M(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let G = (X, b) be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then, for any function f of finite energy the function σf : X × X → [0, ∞), σf(x, y) := |f(x) − f(y)| is an intrinsic pseudometric with respect to the finite measure mf that is given by mf(x) = 1 2 � y∈X b(x, y)σf(x, y)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The function f is 1-Lipschitz with respect to σf and mf(X) = Q(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' This is already shown in [6, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' □ By the preceding lemma any f of finite energy comes with a pseudo- metric σf that is intrinsic with respect to a finite measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In general, this σf will not be a metric (as values of f in different points need not be distinct).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' However, this can easily be achieved by an arbitrarily small perturbation as the next proposition shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' 8 LENZ, PUCHERT, AND SCHMIDT Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='6 (Small perturbation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let G = (X, b) be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (a) For any ε > 0 there exist sx > 0, x ∈ X, such that any function g : X → R with g(x) ∈ (−sx, sx) for all x ∈ X satisfies Q(g) < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (b) For any f ∈ D(G) and any ε > 0 there exists a function fε ∈ D(G) with fε(x) ̸= fε(y) for all x, y ∈ X with x ̸= y and sup x∈X |f(x) − fε(x)| < ε and |Q(f − fε)| < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (a): We write 1x for the characteristic function of x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then Q(1x) = deg(x) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' For x ∈ X we choose sx > 0 with � x∈X sxQ(1x)1/2 < √ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Choose a sequence (Fn) of finite subsets of X with Fn ⊆ Fn+1 and X = � n Fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then, any g : X → R with g(x) ∈ (−sx, sx) is the point- wise limit of the functions gn := � x∈Fn g(x)1x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The pointwise lower semicontinuity of Q together with Cauchy-Schwarz inequality yield Q(g) ≤ lim inf n→∞ Q(gn) ≤ lim inf n→∞ � x,y∈Fn |g(x)||g(y)|Q(1x)1/2Q(1y)1/2 ≤ �� x∈X sxQ(1x)1/2 �2 < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (b): This follows from (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let ε > 0 be given and chose sx, x ∈ X, according to (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Without loss of generality we can assume sx < ε 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Now let real numbers ux, x ∈ X, be given such that ux − uy is irrational for any x ̸= y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then, for any x ∈ X we can choose an tx ∈ (−sx, sx) such that f(x) − ux − tx is rational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then, fε with fε(x) = f(x) − tx for all x ∈ X satisfies supx |f(x) − fε(x)| ≤ supx sx < ε as well as Q(f − fε) < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Moreover, the values of fε are pairwise different as for x ̸= y we have (f(x)−tx)−(f(y)−ty) = (ux −uy)+(f(x)−ux −tx)−(f(y)−uy −ty) can not vanish (as it is the sum of an irrational number and a rational number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=') □ For us a special class of pseudometrics will be particularly useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' They will be introduced next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Given a symmetric function w: X×X → [0, ∞) and a (possibly infinite) path γ = (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=') in G we define the length of γ with respect to w by Lw(γ) := � i w(xi, xi+1) ∈ [0, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' RECURRENT AND (STRONGLY) RESOLVABLE GRAPHS 9 Since we always assume connectedness, this induces the path pseudo- metric dw on X via dw(x, y) = inf{Lw(γ) | γ a path from x to y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We say that σ is a path pseudometric on X if σ = dw for some symmetric function w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' A symmetric function w: X × X → [0, ∞) is called edge weight if w(x, y) > 0 for all (x, y) ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' A symmetric function w is called adapted with respect to the graph G = (X, b) and the measure m if for all x ∈ X it satisfies 1 2 � y∈X b(x, y)w(x, y)2 ≤ m(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The following lemma summarizes some elementary properties of path pseudometrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='7 (Path pseudometrics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let G = (X, b) be a graph and let w: X ×X → [0, ∞) be a symmetric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then dw is a pseudomet- ric that satisfies dw(x, y) ≤ w(x, y) if x ∼ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Moreover, the following are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (a) If w is a pseudometric, then dw ≥ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In particular, dw(x, y) = w(x, y) for all x, y ∈ X with x ∼ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (b) If w is adapted with respect to the measure m, then dw is intrinsic with respect to the measure m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (c) For σ = dw the equality dσ = σ holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The trivial path (x, y) is one of the paths over which the infimum in the definition of dw is taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' As Lw((x, y)) = w(x, y), the inequality dw(x, y) ≤ w(x, y) is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (a): Given a path γ = (x = x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' , xn = y), an iteration of the triangle inequality w(x1, xk+1) ≤ w(x1, xk) + w(xk, xk+1) yields w(x, y) ≤ n−1 � i=1 w(xi, xi+1) = Lw(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (b): This is an immediate consequence of the inequality dw(x, y) ≤ w(x, y) for all x, y ∈ X with x ∼ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (c): As σ = dw is a pseudometric, (a) gives σ ≤ dσ and, for x, y ∈ X with x ∼ y, even σ(x, y) = dσ(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' For arbitrary x, y ∈ X let a path γ = (x = x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' , xn = y) be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then a short computation involving what we have shown already and the triangle inequality gives Lw(γ) = n−1 � j=1 w(xj, xj+1) ≥ n−1 � j=1 σ(xj, xj+1) = n � j=1 dσ(xj, xj+1) ≥ dσ(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Taking the infimum over all γ we find σ(x, y) ≥ dσ(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' □ We note the following consequence of our considerations: If f is a function of finite energy on the graph (X, b), then σf (defined in 10 LENZ, PUCHERT, AND SCHMIDT Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='5) is an intrinsic pseudometric with respect to mf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Now, we can also consider σf as a symmetric function (adapted to mf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' This induces the path pseudometric df := dσf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The preceding lemma immediately gives the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let G = (X, b) be a graph and let f ∈ D(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then, df = dσf is an intrinsic metric with respect to mf and df(x, y) = |f(x) − f(y)| holds for all x, y ∈ X with b(x, y) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' By the preceding lemma we have df = dσf ≤ σf as well as df(x, y) = σf(x, y) = |f(x)−f(y)| for all x, y ∈ X with b(x, y) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' □ Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let a graph G = (X, b) be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Define �Q on the set of symmetric functions w: X × X → [0, ∞) by �Q(w) := 1 2 � x,y b(x, y)w(x, y)2 ∈ [0, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then, part of our considerations can be understood in terms of �Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' As this may be instructive we give a brief discussion in the present remark: For a symmetric w: X × X → [0, ∞) we define mw : X → [0, ∞], mw(x) = 1 2 � y∈X b(x, y)w(x, y)2 and mw(X) := � x∈X mw(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Finally, for f : X → R define the sym- metric function σf : X × X → [0, ∞) with σf(x, y) := |f(x) − f(y)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then, the following holds: (a) Let w be a symmetric weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then, �Q(w) = mw(X), where the value ∞ is allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' If w is actually a pseudometric, then �Q(dw) = �Q(w) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (b) Let σ be a pseudometric on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then, mσ is a finite measure if and only if σ belongs to M(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' If mσ is a finite measure it is the smallest measure with respect to which σ is an intrinsic metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (c) For f : X → R the equality Q(f) = �Q(σf) holds, where the value ∞ is allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Moreover, f belongs to D(G) if and only if σf belongs to M(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (d) The function f : X → R is 1-Lipschitz with respect to the pseudo- metric σ if and only if σf ≤ σ holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In this case, Q(f) ≤ �Q(σf) is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' As mentioned already we think of the space X underlying the graph (X, b) as equipped with discrete topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Thus, metrics compatible with the discrete topology are of particular relevance for us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The fol- lowing lemma ensures the existence of such metrics in M(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' RECURRENT AND (STRONGLY) RESOLVABLE GRAPHS 11 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let G = (X, b) be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then, there exists a metric in M(G) that induces the discrete topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let N → X, n �→ xn, be an enumeration of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We define f : X → (0, ∞), f(xn) = 1 � 2ndeg(xn) , and σ: X × X → [0, ∞), σ(x, y) = � max{f(x), f(y)} for x ̸= y 0 else .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' It is readily verified that σ is a metric (and even an ultrametric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' By σ(x, y)2 ≤ f(x)2 + f(y)2, the symmetry of b and Fubini’s theorem we find � x∈X b(x, y)σ(x, y)2 ≤ � x,y∈X b(x, y)(f(x)2 + f(y)2) = 2 � x,y∈X b(x, y)f(x)2 = 2 � x∈X deg(x)f(x)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Now, the definition of f gives 2 � x∈X deg(x)f(x)2 ≤ 2 and it follows that σ is an intrinsic metric with respect to a finite measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The metric σ induces the discrete topology as the distance from any point to x ∈ X is bounded from below by f(x) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Boundaries of graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' As outlined in the introduction comple- tions and boundaries of graphs will be most relevant for our consider- ations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Here we introduce the corresponding notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let X be a countable set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let σ be a pseudometric on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The completion of X with respect to σ is defined as the set of equivalence classes of σ-Cauchy sequences in X, where two such sequences (an) and (bn) are considered to be equivalent if lim n→∞ σ(an, bn) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' This set is denoted by X σ and contains a quotient of the vertex set X as the classes of the constant sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' learly, σ can be extended to a pseudometric on X and this extension will - by a slight abuse of notation - also denoted by σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Subsequently, the boundary is defined as ∂σX = X σ \\ (X/ ≃), 12 LENZ, PUCHERT, AND SCHMIDT where x ≃ y if σ(x, y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' A graph is called metrically complete with respect to a pseudometric if the boundary is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Clearly, if σ is a metric then X σ contains a copy of X, this copy is dense, and and our definition of metric completeness agrees with the usual definition (that any Cauchy-sequence converges).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' There are further notions of completeness relevant to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let G = (X, b) be a graph and let w be an edge weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The pseudometric space (X, dw) is called geodesically complete if every infinite path has infinite length with respect to w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' For later purposes we recall the following discrete Hopf-Rinow type theorem that characterizes geodesic completeness, see [8, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='1] and, for further generalizations, see also [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='10 (Hopf-Rinow type theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let G = (X, b) be a lo- cally finite graph and let w be an edge weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then dw is a metric that induces the discrete topology on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Moreover, the following assertions are equivalent: (i) (X, dw) is a complete metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (ii) (X, dw) is geodesically complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (iii) Every distance ball is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (iv) Every bounded and closed set is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Boundaries of graphs can not only arise from metric completions but also from compactifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In fact, they can arise whenever the set X underlying the graph is suitable extended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We finish this section with a short discussion of this aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let X be a countable set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let Y be a topological Hausdorff space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We say that X embeds densely in the topological space Y if Y contains a copy of X, the restriction of the topology of Y on X is the discrete topology, and X is dense in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Clearly, Y must be separable whenever X embeds densely in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Whenever X embeds densely in Y we define the boundary ∂Y X of X in Y by ∂Y X := Y \\ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The complement (in Y ) of any finite subset of X is open in Y (as any finite set is compact and then must be closed due to Hausdorffness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Hence, any such a complement is an open neighborhood of ∂Y X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In particular, any function h with finite support on X can be extended (by zero) to a continuous function on Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Clearly, X embeds densely in X σ whenever σ is a metric on X in- ducing the discrete topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' This is what we have discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then, ∂σX = ∂X σX holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' If X embeds densely in a compact Y , then Y is called a compactifica- tion of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In this case the open neighborhoods of ∂X are exactly given RECURRENT AND (STRONGLY) RESOLVABLE GRAPHS 13 by the complements of finite sets of X (as the complement of any open neighborhood of ∂Y X must be a closed, and hence, compact subset of X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' A particular instance is given by the one-point-compactification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' It is given by the set Y = X ∪{pt}, where pt is an arbitrary additional point, and this set is equipped with topology given by the family of all subsets of Y that are either subsets of X or whose complement is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In this case the boundary of ∂Y X is just pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Capacity of sets in the boundary and infinite paths In this section we introduce the capacity and study the capacity of sets in the boundary with respect to an intrinsic metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let G = (X, b) be a graph and let m: X → (0, ∞) be a measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The capacity of a subset U ⊆ X is defined by capm(U) := inf{∥f∥2 Q,m | f ∈ H1(G, m) with f ≥ 1 on U}, with the convention that capm(U) = ∞ if the set in the above definition is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Using the fundamental contraction property, we can assume 0 ≤ f ≤ 1 in this definition, since (f ∧1)+ satisfies the same constraints as f but reduces the ∥·∥Q,m-norm compared to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Whenever X embeds densely in Y we can extend the capacity to subsets of Y by setting capm(A) := inf{capm(O ∩ X) | O open in Y with A ⊆ O}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Since by assumption every subset A ⊆ X is open in Y (as the topology of Y induces the discrete topology on X) both definitions of capacity on X are compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' This definition can in particular be applied to the completion X σ, whenever the metric σ induces the discrete topology on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The capacity is an outer measure on the power set of Y with m(A) ≤ capm(A) for all A ⊆ X and capm(Y ) ≤ m(X), see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' [4, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='1 and Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Next we discuss how the vanishing of the capacity of subsets of the boundary can be characterized with limits of functions of finite energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='1 (Limes inferior).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let X be a countable set, let σ be a metric on X that induces the discrete topology and let f : X → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' For A ⊆ X σ the limes inferior of f at A with respect to σ is defined by lim inf x→A f(x) := sup{inf{f(x) | x ∈ U∩X} | U open in X σ with A ⊆ U}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Moreover, we define the limes inferior at infinity by lim inf x→∞ f(x) := sup{inf{f(x) | x ∈ X \\ F} | F ⊆ X finite}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' For us the case where lim inf equals ∞ and the case of com- pact X σ is is particularly relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In this context we note the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (a) We have lim infx→A f(x) = ∞ if and only if limn→∞ f(xn) = ∞ for each sequence (xn) in X with σ(xn, A) → 0, n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' 14 LENZ, PUCHERT, AND SCHMIDT (b) We have lim infx→∞ f(x) = ∞ if and only if limn→∞ f(xn) = ∞ for any sequence (xn) converging to pt in the one-point-compactification of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In fact, this easily shows that lim infx→∞ f(x) = ∞ if and only if limn→∞ f(xn) = ∞ for any sequence (xn) converging to some y ∈ ∂Y X, where Y is a compactification of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (c) If σ induces the discrete topology on X and (X, σ) is pre-compact, then every every open neighborhood U of ∂σX in X σ has the form U = X \\ F for some finite F ⊆ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Hence, in this case lim inf x→∞ f(x) = lim inf x→∂σX f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In this sense, the limes inferior at infinity is the limes inferior at the boundary for metric compactifications of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' It turns out that lim infx→∞ governs lim infx→∂σX in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let X be a countable set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let f : X → R be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then, lim inf x→∂σX f(x) ≥ lim inf x→∞ f(x) for any metric σ on X that induces the discrete topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Any finite set F in X is compact in X σ and, hence, closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Thus, for any finite set F in X the set X σ \\F is an open neighborhood of ∂σX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We obtain lim inf x→∂σX f(x) = sup{inf{f(x) | x ∈ U ∩ X} | U open with ∂σX ⊆ U} ≥ sup{inf{f(x) | x ∈ X \\ F ∩ X} | F ⊆ X finite} ≥ lim inf x→∞ f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' □ With the help of the limes inferior we can characterize sets of capacity zero in the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='3 (Characterization of zero capacity sets in the boundary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let G = (X, b) be an infinite graph and σ be a metric on G that induces the discrete topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Further, let A ⊆ ∂σX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The following assertions are equivalent: (i) For one finite measure m on X we have capm(A) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (ii) For all finite measures m on X we have capm(A) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (iii) There exists f ∈ D(G) with lim inf x→A f(x) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (i) ⇒ (iii): The statement capm(A) = 0 implies the existence of sequences of open sets Un ⊇ A and functions fn ≥ 1Un that satisfy lim n→∞ ∥fn∥Q,m = lim n→∞ � Q(fn) + ∥fn∥2 m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' RECURRENT AND (STRONGLY) RESOLVABLE GRAPHS 15 By restricting to a subsequence we can assume without loss of generality that � n∈N ∥fn∥Q,m < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' This implies that the sum f := � n∈N fn converges in the Hilbert space H1(G, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In particular, f ∈ D(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' By the choice of the fn we have f ≥ N on the set �N n=1 Un, which is an open set that contains A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' This proves that lim infx→A f(x) is at least N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Since this is true for all N ∈ N, the supremum has to be infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (iii) ⇒ (ii): Let f ∈ D(G) satisfy (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Without loss of generality we assume f ≥ 0, for otherwise we can replace f by |f|, which also has finite energy due to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We slice f into the parts fn := (f − n)+ ∧ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' First, we observe that 0 ≤ fn ≤ 1 and f = �∞ n=0 fn pointwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' More- over, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='2 yields fn ∈ D(G) and since m is finite and the fn are bounded, we have fn ∈ H1(G, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Because lim infx→A f(x) = ∞, for every n ∈ N0 there is an open set Un ⊇ A such that f ≥ n on Un ∩X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' By construction, for x ∈ X the inequality f(x) ≥ n+1 implies fn(x) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Combining these observations we conclude fn ≥ 1X∩Un+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Altogether, this shows that the functions fn are usable in the defini- tion of the capacity of A and capm(A) ≤ ∥fn∥Q,m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We prove ∥fn∥Q,m → 0, as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' It is readily verified that for n ̸= m and x, y ∈ X the product (fm(x) − fm(y))(fn(x) − fn(y)) is always nonnegative, so that Q(fn, fm) = 1 2 � x,y∈X b(x, y)(fm(x) − fm(y))(fn(x) − fn(y)) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Recall that for N ∈ N we defined C[0,N]f = f+ ∧ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Using Proposi- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='2, the definition of fn and the previous observation, we obtain N−1 � n=0 Q(fn) ≤ N−1 � n=0 Q(fn) + � 0≤m,n≤N−1 m̸=n Q(fm, fn) = Q �N−1 � n=0 fn � = Q(C[0,N]f) ≤ Q(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Since N was arbitrary and Q(f) < ∞, we arrive at limn→∞ Q(fn) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The convergence limn→∞ ∥fn∥m = 0 follows from Lebesgue’s dominated convergence theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' This leads to limn→∞ ∥fn∥Q,m = 0 and thus, capm(A) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (ii) ⇒ (i): This is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' □ 16 LENZ, PUCHERT, AND SCHMIDT Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (a) The lemma shows that having capacity zero does not depend on the choice of the finite measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Indeed, we do not even need to assume that m is strictly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' If we only assume that m(o) > 0 for one o ∈ X, the space H1(X, m) continuously embeds into (D(G), ∥·∥o) and the proof can be carried out in the space (D(G), ∥·∥o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The advantage of working in H1(G, m) is that it is related to intrinsic metrics with respect to m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (b) The inequality used in the proof of the implication (iii) ⇒ (ii) can be extended to a more general form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let f : X → R be a function of finite energy and let C1, C2 : R → R be two monotone increasing 1-Lipschitz functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then Q(C1 ◦ f + C2 ◦ f) ≥ Q(C1 ◦ f) + Q(C2 ◦ f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In the above proof this observation is applied to the monotone increasing contractions Sn, n ∈ N0, with Sn(x) = (x − n)+ ∧ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' One can understand the preceding result also as saying that the sets A ⊆ ∂σX with zero capacity are infinitely far away from any finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' More specifically, the following holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='4 (Capacity zero sets in the boundary have infinite dis- tance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let G = (X, b) be an infinite graph and σ a metric on G that induces the discrete topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Further, let A ⊆ ∂σX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then, ∂σA has zero capacity (with respect to any finite measure) if and only if there exists an intrinsic metric ̺ ∈ M(G) such that for any finite F in X and any r > 0 there exists an open neighborhood U of A with ̺(U ∩ X, F) := inf{̺(z, x) | z ∈ U ∩ X, x ∈ F} ≥ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Assume that A has capacity zero (with respect to any finite measure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' By the previous lemma, there exists an f ∈ D(G) with lim infx→A f(x) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Without loss of generality we can assume f(x) ̸= f(y) for all x, y ∈ X with x ̸= y (else we could add an arbitrary small perturbation by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then, ̺ := σf is an intrinsic metric with respect to a finite measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' By lim infx→A f(x) = ∞ the metric ̺ has the given property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Assume now that there exists an intrinsic metric ̺ with respect to the finite measure m that has the given property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let an arbitrary finite set F be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then, there exists an open neighborhood U of A with ̺(U ∩ X, F) ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Hence, gF := ̺F ∧ 1 satisfies 0 ≤ gF ≤ 1, equals 0 on F and equals 1 on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Hence, capm(A) ≤ Q(gF) + |gF|2 m ≤ Q(̺F) + m(X \\ F) ≤ 3m(X \\ F) holds, where we used Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='4 in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' As this holds for arbitrary F we infer capm(A) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' By the previous lemma this implies that the capacity of A vanishes with respect to any finite measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' □ Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Replacing ̺ by d̺ ≤ ̺ we can even take the intrinsic metric in the preceding corollary to be a path metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' RECURRENT AND (STRONGLY) RESOLVABLE GRAPHS 17 Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='3 and its corollary deal with metric completions of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' However, they can directly be extended to any topological Haus- dorff space Y in which X embeds densely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Indeed, both the definition of lim infx→A and the proofs of the lemma and its corollary carry ver- batim over to this more general situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' This means in particular that these considerations also holds for compactifications of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Next we discuss how the capacity of sets in the metric boundary is related to infinite paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We recall the following standard notion for sets of infinite paths in a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='5 (Null set of paths).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' A set of infinite paths Γ in G = (X, b) is called null if there exists an edge weight w with � x,y∈X b(x, y)w(x, y)2 < ∞ such that Lw(γ) = ∞ for all γ ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let σ be a metric on X that induces the discrete topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' For A ⊆ ∂σX we denote by ΓA,σ the set of infinite paths which have at least one accumulation point with respect to σ lying in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' With the help of our characterization of sets of capacity zero in the boundary we obtain the following relation between sets of capacity zero in the boundary and sets of paths with accumulation point in this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' This is the key observation relating our approach to the classical approach to recurrence by means of null sets of paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' As noted in the introduction, this observation was our motivation to write this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='6 (Capacity and null sets of paths).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let G = (X, b) be an infinite graph and let σ be an intrinsic metric with respect to a finite measure m that induces the discrete topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let A ⊆ ∂σX with capm(A) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then ΓA,σ is null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' According to the previous lemma there is f ∈ D(G) such that lim infx→A f(x) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We consider the function w: X × X → R, w(x, y) = |f(x) − f(y)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Without loss of generality we can assume w(x, y) > 0 whenever (x, y) is an edge (else at each vertex x add a small quantity to f(x) if necessary, see Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then � x,y∈X b(x, y)w(x, y)2 ≤ 2Q(f) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let γ = (x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=') be an infinite path with an accumulation point in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We obtain |f(xn) − f(x1)| ≤ n−1 � k=1 |f(xk) − f(xk+1)| ≤ Lw(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Since lim infx→A f(x) = ∞, the left hand side of this inequality diverges along a suitable subsequence and so we obtain Lw(γ) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Hence, ΓA,σ is null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' □ 18 LENZ, PUCHERT, AND SCHMIDT The converse seems not to hold due to the complicated behavior of paths at metric boundaries of general graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' For trees however we have the following converse for path metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Recall that (X, b) is a tree if it does not have non-trivial cycles (injective paths (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' , xn) with x1 ∼ xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let G = (X, b) be a tree and let σ be a path metric that induces the discrete topology on X and is intrinsic with respect to a finite measure m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' If for A ⊆ ∂σX the set of paths ΓA,σ is null, then capm(A) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let w be an edge weight for ΓA,σ as in the definition of null sets of paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Fix o ∈ X and for x ∈ X let γx be the unique shortest path with respect to the combinatorial distance connecting o and x (uniqueness follows from (X, b) being a tree).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We define f : X → R, f(x) = Lw(γx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Since (X, b) is a tree, for neighbors x, y ∈ X we have |f(x) − f(y)| = w(x, y) showing f ∈ D(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let (xn) be a sequence in X with limit in x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We construct a monotone path γ = (y1, y2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=') (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' the combinatorial distance of y1 and yn+1 is larger or equal than the combinatrial distance of y1 and yn) starting in o such that yn → x and f(xk) = f(ynk) + dw(xk, ynk) ≥ f(ynk) for a suitable subsequence (ynk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The monotonicity of γ and that (X, b) is a tree imply lim inf k→∞ f(xk) ≥ lim inf k→∞ f(ynk) = Lw(γ) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Construction of γ: We consider o as a root for the graph and denote by |x| the combinatorial distance of x to o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We say that y is an ancestor of c if all paths from x to o pass through y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Since (X, b) is a tree, every A ⊆ X has a unique greatest common ancestor, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=', there exists and element y ∈ X with: y is an ancestor of every element of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' For every x ∈ X with |x| > |y| there exists an a ∈ A such that x is not an ancestor of a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We let zn be the greatest common ancestor of {xn, xn+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' This sequence is monotone as zn is an ancestor of {xn+1, xn+2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='} and hence an ancestor of zn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' For every n ∈ N there exists N > n such that the greatest common ancestor of {xn, xN} is zn (otherwise zn would not be a greatest common ancestor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Every path from xn to xN passes through zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Since σ is a path metric, this implies σ(xn, xN) = σ(xn, zn) + σ(zn, xN) RECURRENT AND (STRONGLY) RESOLVABLE GRAPHS 19 and we obtain σ(xn, zn) ≤ sup{σ(xn, xN) | N > n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Hence, (zn) also converges to x but it need not be a path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We make it a path by inserting monotone paths from zn to zn+1 (these exist since yn is an ancestor of yn+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Using that σ is a path metric yields that any such additional point z lying between zn and zn+1 satisfies σ(zn, zm) = σ(zn, z) + σ(z, zm) ≥ σ(zn, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Hence, also the so-constructed monotone path (yn) converges to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We choose nk such that ynk = zk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Using that zk is an ancestor of xk, we obtain f(xk) = f(zk) + Lw(γxk) − Lw(γzk) = f(ynk) + dw(ynk, xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' □ Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In the previous proof we used the following observation uti- lizing that (X, b) is a tree and σ is a path metric: For every x ∈ ∂σX and every sequence (xn) in X converging to x there exists a monotone path (yn) converging to x such that any xn has an element from the path (yn) as an ancestor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Recurrence and intrinsic metrics In this section we use similar technics as in Section 3 to give a new characterization of recurrence in terms of intrinsic metrics and to study the relation of recurrence to the vanishing of the capacity of the bound- ary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Moreover, we provide an alternative proof for a classical charac- terization of recurrence due to Yamasaki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' For general background on recurrence we refer the reader to the textbooks [12, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The word recurrence stems from the stochastic perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In this perspective the graph gives rise to a Markov process modeling a par- ticle jumping between the points of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Recurrence then describes the phenomenon that the particle comes back to any point of X again and again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In the analytic description, which is our concern here, this is encoded by various forms of irrelevance of what is happening far away (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' outside of finite sets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We will see precise versions as we go along and this is the main topic of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='1 (Recurrence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' A graph G = (X, b) is called recurrent if the constant function 1 is contained in D0(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Graphs that are not recurrent are called transient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (a) The definition of recurrence means that there exists a sequence of functions (fn) in Cc(X) with fn → 1 pointwise and Q(fn) = Q(fn −1) → 0, n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' As the fn have finite support this can be seen as an instance of how the behaviour outside of compact sets (in this case the supports of the fn) becomes irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' 20 LENZ, PUCHERT, AND SCHMIDT (b) Recurrence is equivalent to D0(G) = D(G), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=', Cc(X) being dense in D(G) with respect to the norm ∥·∥o, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' [22, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (c) For disconnected graphs transience is a stronger property than not being recurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Since all the graphs in this paper are assumed to be connected, we may well use the above definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' For further background on recurrence we refer the reader to [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Next we connect recurrence, vanishing of the capacity and finiteness of metric balls, to the existence of certain unbounded functions of finite energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='2 (Characterization of recurrence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let G = (X, b) be an infinite graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The following conditions are equivalent: (i) G is recurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (ii) There is a function of finite energy f ∈ D(G) that satisfies lim inf x→∞ f(x) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (iii) There is an intrinsic metric σ ∈ M(G) that induces the discrete topology on X such that distance balls with respect to σ are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (iii)’ There exists a finite measure m and an edge weight w adapted to it such that the distance balls with respect to dw are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (iv) For one (every) finite measure m on X and one (every) compact- ification Y of X the equality capm(∂Y X) = 0 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (v) One (every) finite measure has the following feature: For any ε > there exists a finite set F in X with capm(X \\ F) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (i) ⇒ (iv): Let m be an arbitrary finite measure on X and Y a compactification of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' By (i) there exists a sequence (fn) in Cc(X) with fn → 1 pointwise and Q(fn) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Replacing fn by (fn ∨ 0) ∧ 1 we can assume without loss of generality 0 ≤ fn ≤ 1 for each n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then, 0 ≤ 1 − fn ≤ 1 holds and 1 − fn is 1 outside the finite support of fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Hence, capm(∂Y X) ≤ Q(1 − fn) + |1 − fn|2 m holds for each n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' It suffices to show that both terms on the right hand side converge to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The first term satisfies Q(1 − fn) = Q(fn) → 0, n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The second term satisfies |1 − fn|2 m → 0, n → ∞ by Lebesgue theorem on dominated convergence (as 0 ≤ 1 − fn ≤ 1 holds and 1 − fn converges pointwise to 0 and m is a finite measure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (iv) ⇒ (ii): : This follows by a straightforward adaption of the proof of (i)⇒ (iii) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (ii) ⇒ (iii): Let f be a function satisfying (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='6 we can assume without loss of generality that the values of f are pairwise distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Set σ(x, y) = σf(x, y) = |f(x) − f(y)| for all x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' This yields a pseudo metric that is intrinsic with respect to a finite measure, see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In fact, it is even a metric as the values of f are RECURRENT AND (STRONGLY) RESOLVABLE GRAPHS 21 pairwise distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Its distance balls are given by Bσ r (o) = {x ∈ X | f(o) − r ≤ f(x) ≤ r + f(o)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Since lim infx→∞ f(x) = ∞, they are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In particular, this metric induces the discrete topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (iii) ⇒ (i): Let an intrinsic metric σ with respect to a finite measure m be given according to (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Hence, σ induces the discrete topology and its distance balls are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Now, let F be an arbitrary finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then, σF := σ(·, F) satisfies σF = 0 on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' σF ≥ 1 outside of F1 := {x ∈ X : σ(F, x) < 1} and F1 is finite (as F is a finite set and distance balls with respect to σ are finite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Define gF := (1 − σF)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then gF equals to 1 on F (by the first bullet point) and has finite support contained in F1 (by the second bullet point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Moreover, as Q is compatible with contractions and Q(1) = 0 holds we find from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='4 the estimate Q(gF) ≤ Q(1 − σF) = Q(σF) ≤ 2m(X \\ F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' So, choosing an increasing sequence (Fn) of finite sets with ∪nFn = X we obtain a sequence fn := gFn in Cc(X) converging pointwise to 1 with Q(fn) ≤ 2m(X \\ Fn) → 0, n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' This shows (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The equivalence between (iv) and (v) is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (iii)’ ⇒ (iii): By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='7 the metric dw is intrinsic with respect to the finite measure m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' To show that it induces the discrete topology we note that finiteness of dw-balls implies that for all R > 0 and x ∈ X the set {y ∈ X | y ∼ x with w(x, y) < R} is finite (otherwise the R-ball around x would contain infinitely many points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' This is known as essential local finiteness of w and, according to [13, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='2], implies that dw induces the discrete topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (iii) ⇒ (iii)’: We choose w := σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then, w is adapted to a finite measure m and dσ is then an intrinsic metric with respect to m with σ ≤ dσ by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In particular, balls with respect to dw are contained in the corresponding balls with respect to σ and are, hence, finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' □ Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (a) The equivalence between (i) and (iv) can be seen as a special instance of the recurrence theory developed by the third author in his (unpublished) PhD thesis [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (b) Clearly, a metric with finite distance balls must induce the discrete topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' 22 LENZ, PUCHERT, AND SCHMIDT (c) In the proof of (ii) ⇒ (iii) we have seen that for f of finite energy with lim infx→∞ f(x) = ∞ the intrinsic (pseudo)metric σf has finite distance balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' There is a converse of sorts to this: Let σ be a metric and define for x ∈ X the function fx by fx(y) := σ(y, {x}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then, the distance balls around one x ∈ X are finite if and only if the distance balls around any x ∈ X are finite and this holds if and only if lim infy→∞ fx(y) = ∞ holds for one (all) x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (d) The proof of (iii) ⇒ (i) only uses that the balls of radius 1 are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In fact, the number 1 is irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' It suffices that there is an r > 0 such that all balls of radius r are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' However, if σ is an intrinsic metric all of whose distance balls of radius r are finite then for any sequence Fn of finite sets in X with Fn ⊆ Fn+1 and ∪Fn = X we can define f := �∞ n=1 σ(Fn, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then, f will be well- defined with lim infx→∞ f(x) = ∞ (by finiteness of r-balls).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' With a suitable choice of Fn then f will have finite energy and σ + σf will be an intrinsic metric with respect to a finite measure that has finite distance balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (e) The existence of an intrinsic metric with respect to a (finite) mea- sure m that has finite distance balls has strong consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In particular, as observed in [8], it implies that associated graph Lapla- cians on ℓ2(X, m) (and more general magnetic Schr¨odinger opera- tors [7, 20]) are essentially self-adjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' It is somewhat surprising that recurrence implies essential self-adjointness for a particular fi- nite measure, as in general recurrence is strictly weaker than essen- tial self-adjointness for all finite measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We refer to discussion after Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='15 in the survey [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' This survey contains a version of the previous theorem, which was first obtained in the second author’s master’s thesis [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We are now going to derive some consequence of the preceding the- orem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' As a a first consequence of it we obtain an alternative proof for the (by now) classical recurrence criterion of Yamasaki [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='3 (Yamasaki’s criterion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let G = (X, b) be a locally finite graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then G is recurrent if and only if the set of all infinite paths is null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let G be recurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' According to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='2 there exists a finite measure m and a weight w adapted to m such that the intrinsic path metric dw has finite distance balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' As w is adapted to m we have � x,y b(x, y)w(x, y)2 ≤ 2m(X) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' It suffices to show that the length Lw(γ) of any infinite path is ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We consider two cases: RECURRENT AND (STRONGLY) RESOLVABLE GRAPHS 23 Case 1: The path γ leaves any finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then, the path leaves in particular any ball of finite radius (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' dw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Hence, the path must have infinite length (as the metric is a path metric).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Case 2: The path γ stays within a fixed finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then, it must have infinite length anyway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Suppose that the set of all infinite paths is null and let w be a corresponding edge weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The summability condition on w implies that mw(x) := 1 2 � y∈X b(x, y)w(x, y)2 is a finite measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We consider the path metric dw induced by w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='7 (a) ensures that it is intrinsic with respect to mw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Thus, dw is an intrinsic metric with respect to a finite measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Since all infinite paths have infinite length, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='10 implies that dw has finite distance balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' This yields recurrence by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' □ Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' For proving nullity of the set of all paths on recurrent graphs we did not use local finiteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Another consequence of our characterization of recurrence is van- ishing of the capacity of all boundaries of metric completions in the recurrent case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let G = (X, b) be a recurrent infinite graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' For any finite measure m on X and any metric σ on X that induces the discrete topology we have capm(∂σX) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' This follows immediately from (v) of the previous theorem as ∂σX is contained in X \\ F for any finite F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' □ In some cases the vanishing of the capacity of the boundary is equiv- alent to recurrence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' For this one needs that σ is intrinsic with respect to the finite measure m and some more geometric data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In the following theorem we discuss two situations where this is the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='5 (Capacity criterion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let G = (X, b) be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let σ be a metric that induces the discrete topology and is intrinsic with respect to a finite measure m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then G is recurrent if capm(∂σX) = 0 and one of the following conditions is satisfied: (a) G is locally finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (b) (X, σ) is totally bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (a): According to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='2 it suffices to construct an intrin- sic metric e with respect to a finite measure that has finite distance balls and induces the discrete topology on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The metric e that we 24 LENZ, PUCHERT, AND SCHMIDT construct is a path metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Since G is locally finite, it automatically in- duces the discrete topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' By the discrete Hopf-Rinow theorem, The- orem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='10, the finiteness of distance balls is equivalent to the complete- ness of (X, e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' According to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='3 the assumption capm(∂σX) = 0 yields a function f ∈ D(G) with lim infx→∂σX f(x) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We let e := dσ+σf be the path metric that is induced by the weight σ + σf with σf(x, y) = |f(x)−f(y)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='5 shows that the pseudometric σf is intrinsic with respect to a finite measure and so σ +σf is intrinsic with respect to a finite measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We infer from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='7 that also the induced path metric e is intrinsic with respect to a finite measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' It remains to show the completeness of (X, e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let (xn) be Cauchy with respect to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='7 yields e ≥ σ + σf ≥ σ, so that (xn) must also be a Cauchy sequence with respect to σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Due to completeness it has a limit x ∈ X σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We show that x ∈ X and that (xn) also converges to x with respect to e by considering two cases: Case 1: x ∈ ∂σX: lim inf y→∂σX f(y) = ∞ yields lim inf n→∞ f(xn) = ∞, so that for each m ∈ N e(xm, xn) ≥ σf(xn, xm) = |f(xn) − f(xm)| is unbounded in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In particular, this contradicts the assumption that (xn) is Cauchy with respect to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Case 2: x ∈ X: Since σ induces the discrete topology on X, con- vergence with respect to σ to some point in X yields that (xn) must eventually be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Hence, it also converges with respect to e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (b): By assumption X σ is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Hence, vanishing capacity of ∂σX implies recurrence by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' □ Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (a) Part (a) of this Theorem is a generalization of [8, Theo- rem 3], which only treats certain path metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Note that for finite underlying measures the equality D(Q) = D(Qmax) discussed in this reference is equivalent to recurrence, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' [19, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (b) The condition of (X, σ) being totally bounded means that it can be isometrically embedded into a compact metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Below we consider examples of bounded discrete X ⊆ R2 equipped with the Euclidean metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (c) For general graphs it remains an open question whether or not the previous theorem is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (d) For locally finite G = (X, b) we established the equivalence of the following assertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (i) G is recurrent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (ii) For one/all intrinsic metrics σ with respect to a finite measure m that induce the discrete topology we have capm(∂σX) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (iii) The set of all infinite paths is null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' RECURRENT AND (STRONGLY) RESOLVABLE GRAPHS 25 The implication (iii) ⇒ (ii) can be seen as a sort of converse to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='6 when considering the set A = ∂σX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Resolvable graphs and harmonic functions In this section we turn to the study of transient graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' By the re- sults of the last section, transience is characterized by positivity of the capacity of (suitable) boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Here, we turn to a different aspect based on a characterization of recurrence and transience in terms of su- perharmonic functions (read on for the precise definition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Specifically, a graph is transient if and only if it admits non-constant superhar- monic functions of finite energy (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In general, these super- harmonic functions will not be harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' This stimulates interest in those (transient) graphs which admit non-constant harmonic functions of finite energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The aim of this section is to derive a capacity based sufficient condition for existence of such functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' To this end, we introduce strong resolvability for graphs, which is a somewhat stronger property than resolvability that was introduced in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We prove that (most) transient strongly resolvable graphs admit harmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='1 (Resolvability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' A graph G = (X, b) is called resolvable if there is an edge weight w with � x,y∈X b(x, y)w(x, y)2 < ∞ such that for every point x ∈ ∂dwX the set of paths converging to x with respect to dw is null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In this case, w is called a resolving weight for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The previous definition relied on the concept of path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We now aim at a path-free definition which captures essentially the same concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' This yields the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='2 (Strong resolvability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' A graph G = (X, b) is called strongly resolvable, if there exists an intrinsic metric σ with respect to a finite measure m that induces the discrete topology such that capm({x}) = 0 for all x ∈ ∂σX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In this case, σ is called a resolving metric for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' A strongly resolvable graph is resolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let σ be a resolving metric for G that is intrinsic with respect to the finite measure m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='7, the path metric dσ induced by σ satisfies σ ≤ dσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Hence, there is a continuous map ι: X dσ → X σ that extends the identity on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let x ∈ ∂dσX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We first show that ι(x) ∈ ∂σX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Suppose that this is not the case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' ι(x) ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We choose a sequence (xn) in X with xn → x with respect to dσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Since xn → ι(x) with respect to σ and σ 26 LENZ, PUCHERT, AND SCHMIDT induces the discrete topology, (xn) must be eventually constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Hence, x = ι(x) ∈ X, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Any path converging to x with respect to dσ converges to ι(x) with respect to σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Hence, the set of all such paths is contained in Γ{ι(x)},σ, the set of paths having ι(x) as an accumulation point with respect to σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' It therefore suffices to show that the latter set is null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Since ι(x) ∈ ∂σX, we have capm({ι(x)}) = 0 by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='6 implies that Γ{ι(x)},σ is null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' □ Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (a) Strong resolvability transfers the geometric notion of re- solvability introduced in [2] to a notion of potential theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' This has two advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' With strong resolvability one can also treat non-locally finite graphs, as potential theory does not distinguish between locally finite and non-locally finite graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' This is an advantage of potential theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Indeed, for notions invoking infi- nite paths in general the non-locally finite case poses problems, as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' the discrete Hopf-Rinow theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='7 does not hold on non- locally finite graphs, see the discussion in [8, Appendix A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' More- over, strong resolvability is also available on more general spaces that admit a potential theory, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Riemannian manifolds, fractals or metric graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (b) As discussed after Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='6 we believe that capm({x}) = 0 for all x ∈ ∂σX is strictly stronger than Γ{x},σ being null for all x ∈ ∂σX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Hence, resolvability seems strictly stronger that resolvability (even though we do not have concrete examples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' However, below we shall see that planar graphs, the main examples for resolvable graphs in [2], are also strongly resolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='4 ((Super)Harmonic functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let G = (X, b) be a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' A function f : X → C is called superharmonic if for all x ∈ X it satisfies f(x) ≥ 1 deg(x) � y∈X b(x, y)f(y), where we assume absolute convergence of the sum on the right side of the equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The function f is called harmonic if both f and −f are superharmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We write H(G) for the space of harmonic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' An important property of these functions is that on transient graphs functions of finite energy are uniquely represented as sums of harmonic functions of finite energy and functions in D0(G), see Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='3 in [22] for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='5 (Royden decomposition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let G = (X, b) be a transient graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' For all f ∈ D(G) there exists a unique f0 ∈ D0(G) and a unique harmonic fh ∈ D(G) such that f = f0 + fh RECURRENT AND (STRONGLY) RESOLVABLE GRAPHS 27 and Q(f) = Q(f0) + Q(fh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The function fh is the unique function in D(G) that satisfies Q(fh) = inf{Q(f − g) | g ∈ D0(G)} = inf{Q(f − g) | g ∈ Cc(X)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Moreover, if f is bounded, then f0 and fh are bounded as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Resolvability was introduced to prove the existence of non-constant harmonic functions on transient locally finite resolvable graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' This result of Benjamini and Schramm carries over to strongly resolvable graphs that need not be locally finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Before we prove this we need a result on harmonic functions induced by Lipschitz functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let σ ∈ M(G) which is intrinsic with respect to the finite measure m and suppose now that (X, b) is transient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Using the Royden decom- position we define the map Φ: D(G) ∩ Cb(X σ) → D(G) ∩ H(G), f �→ fh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Since the Royden decomposition preserves boundedness and m is finite, we even obtain that Φ maps to H1(G, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The following is the main observation in this section, which will be used to construct many non- constant hamronic functions of finite energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let (X, b) be a graph and let σ be an intrinsic metric with respect to a finite measure m that induces the discrete topology such that capm(∂σX) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Moreover, let S = {x ∈ ∂σX | capm(U ∩ ∂σX) > 0 all open x ∈ U ⊆ X σ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then ker Φ ⊆ {f ∈ D(G) ∩ Cb(X σ) | f|S = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In particular, if Φ(f) is constant, then f is constant on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let f ∈ D(G) ∩ Cb(X σ) with f|S ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We show fh = Φ(f) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Without loss of generality there exists ε > 0 such that f(x) ≥ ε for some x ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Using Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='5 we choose a sequence (gn) in Cc(X) with Q(fh) = infn≥1 Q(f − gn) = limn→∞ Q(f − gn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' As noted above we have fh ∈ H1(G, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Claim: The sequence (gn) can be chosen such that f − gn → fh in H1(G, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' It suffices to show that (gn) can be chosen such that f − gn → fh in ℓ2(X, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We can assume ∥gn∥∞ ≤ 2 ∥f∥∞, as otherwise we could write ((f − gn) ∧ ∥f∥∞) ∨ (− ∥f∥∞) = f − hn with appropriate hn ∈ Cc(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' These satisfy ∥hn∥∞ ≤ 2 ∥f∥∞ and, using the compatibility of Q with contractions, also Q(fh) ≤ Q(f − hn) ≤ Q(f − gn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' 28 LENZ, PUCHERT, AND SCHMIDT Now assume (gn) is chosen with ∥gn∥∞ ≤ 2 ∥f∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The Royden decomposition shows Q(f0 − gn) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Moreover, our assumption capm(∂σX) > 0 implies transience of the graph, see Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' On transient graphs convergence on D0(G) with respect to Q implies pointwise convergence, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' [11, Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Hence, we obtain f0 − gn → 0 pointwise, which implies f − gn → fh pointwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Since the functions f − gn are uniformly bounded by 3 ∥f∥∞ and since m is a finite measure, Lebesgue’s dominated convergence theorem yields f − gn → fh in ℓ2(X, m), which shows the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let now (gn) be a sequence as in the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Since σ induces the discrete topology, the compactly supported function gn can be contin- uously extended to X σ by letting gn = 0 on ∂σX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then f − gn is continuous on X σ with f − gn = f on ∂σX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' By the continuity of f there exists a relatively open neighborhood Ux ⊆ ∂σX of x in ∂σX with f − gn = f ≥ ε/2 on Ux for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Using that x ∈ S we obtain capm(Ux) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' By the continuity of f − gn there exists an open On ⊆ X σ with Ux ⊆ On and f − gn ≥ ε/4 on On such that 4(f −gn)/ε ≥ 1 on X ∩On.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The way the capacity is defined for subsets of the boundary yields 16 ε2 ∥f − gn∥2 Q,m ≥ capm(X ∩ On) ≥ capm(Ux).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Using f − gn → fh in H1(G, m), we obtain ∥fh∥2 Q,m = lim n→∞ ∥f − gn∥2 Q,m ≥ ε2 16capm(Ux) > 0 and arrive at fh ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' For the ’In particular’-part assume that Φ(f) is constant equal to C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Since the harmonic part of a constant function is just the constant function itself, we obtain Φ(f − C) = Φ(f) − C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Hence, what we previously proved shows f = C on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' □ Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The set S in this lemma is the support of the outer measure capm restricted to subsets of ∂σX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Assume σ ∈ M(G) is intrinsic with respect to the finite measure m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We denote the set of bounded Lipschitz functions with respect to σ by Lipb(X) = Lipb,σ(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' If f ∈ Lipb(X), then Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='4 shows that f ∈ H1(G, m) (the lemma implies f ∈ D(G) and the boundedness of f yields f ∈ ℓ2(X, m)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Moreover, f can be uniquely extended to a Lipschitz function X σ, which we also denote by f with a slight abuse of notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Hence, Lipb(X) ⊆ D(G) ∩ C(X σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' This observation is used in the proof of the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let (X, b) be a graph and let σ be an intrinsic metric with respect to a finite measure m that induces the discrete topology RECURRENT AND (STRONGLY) RESOLVABLE GRAPHS 29 such that capm(∂σX) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Moreover, let S = {x ∈ ∂σX | capm(U ∩ ∂σX) > 0 all open x ∈ U ⊆ X σ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then dim(D(G)∩H(G)) ≥ |S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In particular, if |S| ≥ 2, then the graph admits a non-constant harmonic function of finite energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Without loss of generality we can assume |S| ≥ 2 for oth- erwise the statement is trivial because constant functions belong to D(G) ∩ H(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' , fn ∈ Lipb(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The previous lemma shows that if f1|S, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' , fn|S are linearly independent, then Φ(f1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' , Φ(fn) are linearly independent in D(G)∩H(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' With this at hand the statement follows from dim Lipb(X) ≥ dim Lipb(S) ≥ |S| (use that any bounded Lipschitz function on S can be extended to a bounded Lipschitz func- tion on X σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' □ Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='8 (Existence of non-constant harmonic functions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let G = (X, b) be a strongly resolvable graph and let σ be a resolving metric that is intrinsic with respect to the finite measure m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' If capm(∂σX) > 0, then the space of bounded harmonic functions of finite energy is infinite dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let S be the support of the capacity on the boundary introduced in the previous theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Its complement is given by ∂σX\\S = {x ∈ ∂σX | ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' open x ∈ U ⊆ X σ with capm(U∩∂σX) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Since (X σ, σ) is separable, its topology has a countable basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Hence, the σ-sub-additivity of capm yields capm(∂σX \\ S) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Using the subadditivity of the capacity again shows capm(S) = capm(∂σX) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' By our assumption every point in S has capacity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Hence, S must be uncountable for otherwise σ-subadditivity would imply capm(S) = 0, which contradicts our previous considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' With this at hand the claim follows from the previous theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' □ This theorem is a version of [2, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='1] for strongly resolvable but possibly non-locally finite graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We had to replace the transience assumption of [2] by the stronger capm(∂σX) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' As discussed in The- orem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='5, for some classes of graphs transience implies this condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We mention these situations in the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let G = (X, b) be a transient, strongly resolvable graph and let one of the following conditions be fulfilled: (a) G is locally finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' (b) σ is a resolving metric and (X, σ) is totally bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then the space of bounded harmonic functions of finite energy is infinite dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' 30 LENZ, PUCHERT, AND SCHMIDT Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='5 both conditions imply capm(∂σX) > 0 with respect to a resolving metric σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Hence, the claim follows from the previous corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' □ Constructing harmonic functions from functions on a potential theo- retic boundary (the support of the capacity on the metric boundary) is reminiscent of solving the Dirichlet problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Under suitable additional conditions on the graph and on the function on the boundary this can be made precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Remark (Solving the Dirichlet problem on S for uniformly transient graphs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let (X, b) be a graph with D0(G) ⊆ C0(X), where C0(X) denotes the uniform closure of Cc(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Graphs with this property are called uniformly transient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' As the name suggests uniformly transient graphs are transient, see [11] for this fact and further background on uniform transience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let σ ∈ M(G) be intrinsic with respect to the finite measure m and let S denote the support of the capacity on the boundary discussed above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then for any bounded Lipschitz function ϕ: S → R the Dirichlet problem � h ∈ H(G) ∩ D(G) h ∈ Cb � X σ� with h|S = ϕ has a unique solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Uniqueness: This follows directly from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Existence: The bounded Lipschitz function ϕ: S → R can be ex- tended to a bounded Lipschitz function f : X σ → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Consider the Royden decomposition f = f0 + fh with f0 ∈ D0(G) ⊆ C0(X) and harmonic fh ∈ D(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Any sequence in X converging to a point in ∂σX must eventually leave any finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Hence, f0 can be extended to a continuous function on X σ by letting f0 = 0 on ∂σX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' This shows that also fh = f − f0 has a bounded continuous extension to X σ with fh|∂σX = f|∂σX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' By constructions this yields fh|S = ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Planar and canonically compactifiable graphs In this section we show that circle packings of bounded geometry and hence locally finite planar graphs of bounded geometry are always strongly resolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Moreover, we prove that canonically compactifiable graphs are never strongly resolvable showing that planar graphs of bounded geometry can never be canonically compactifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' First we recall the notion of circle packings and their contact graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' For an extensive background on these topics we refer to the book [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='1 (Circle packing and subordinated graphs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' A circle packing is a set X ̸= ∅ and two maps r: X → (0, ∞) and ϕ: X → R2 such that the collection of closed circles Cx = Br(x)(ϕ(x)), x ∈ X, in R2 satisfies C◦ x ∩ C◦ y = ∅ whenever x ̸= y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' It is called bounded if � x∈X Cx RECURRENT AND (STRONGLY) RESOLVABLE GRAPHS 31 is a bounded set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' An edge weight b on X is called subordinate to the circle packing if b(x, y) > 0 implies Cx ∩ Cy ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In what follows we simply write Cx, x ∈ X, to denote a circle packing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The contact graph or nerve of a circle packing Cx, x ∈ X, is the combinatorial graph on X with x ∼ y if Cx ∩ Cy ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Hence, an edge weight b on X is subordinate to the circle packing if and only if the induced discrete graph is a subgraph of the contact graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The following is our main observation in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let Cx, x ∈ X, be a bounded circle packing and suppose (X, b) is subordinate to the circle packing and has bounded geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then (X, b) is strongly resolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In particular, if (X, b) is transient, then it possesses a non-constant harmonic function of finite energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let Ω = supx∈X deg(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then Ω < ∞ due to (X, b) having bounded geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' As above we let r, ϕ denote the maps inducing the circle packing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We consider the metric σ on X defined by σ(x, y) = |ϕ(x) − ϕ(y)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We first show that σ is an intrinsic metric with respect to a finite measure inducing the discrete topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The assumption Cx ∩ Cy ̸= ∅ for all x ̸= y implies σ(x, y) ≥ r(x) + r(y) > r(x) for all y ̸= x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Hence, σ induces the discrete topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Since b is subordinate to the circle packing, we also have Cx ∩ Cy ̸= ∅ whenver x ∼ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' For x ∼ y this implies σ(x, y) = r(x) + r(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We infer � x,y∈X b(x, y)σ(x, y)2 ≤ 2 � x,y∈X b(x, y)(r(x)2 + r(y)2) ≤ 4Ω � x∈X r(x)2 ≤ 4Ω π λ(A) < ∞, with A = � x∈X Cx and λ the Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' This shows that σ is intrinsic with respect to a finite measure m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Using that ϕ is an isometry we identify (X, σ) with ϕ(X) in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In particular, the boundary with respect to σ is just the Euclidean boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Given w ∈ ∂X we show capm({w}) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' For r > 0 we consider the function fr : X → R, fr(x) = � 2 − |x − w| r � + ∧ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' It satisfies fr = 1 on Br(w) ∩ X and fr = 0 on X \\ B2r(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Moreover, for x ∼ y we have |fr(x) − fr(y)|2 ≤ |x − y|2 r2 = (r(x) + r(y))2 r2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' 32 LENZ, PUCHERT, AND SCHMIDT Next we compare (r(x) + r(y))2 with λ((Cx ∪ Cy) ∩ B2r(w)) as long as x ∼ y and x, y ∈ B2r(w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The boundary point w does not belong to the interior of the discs Cx, Cy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' This leads to r(x), r(y) ≤ 2r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Using this observation and that Cx, Cy are tangent, we obtain λ((Cx ∪ Cy) ∩ B2r(w)) = λ(Cx ∩ B2r(w)) + λ(Cy ∩ B2r(w)) ≥ C(r(x)2 + r(y)2) for some constant C > 0 independent of x, y and r (for the last in- equality we simply estimated the area of the intersection of two discs with the given parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Combining these estimates we infer Q(fr) ≤ 1 2r2 � x,y∈X b(x, y)(r(x) + r(y))2 ≤ 1 Cr2 � x,y∈X b(x, y)λ((Cx ∪ Cy) ∩ B2r(w)) ≤ 2Ω Cr2λ(B2r(w)) ≤ 8πΩ C .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Since m is finite, we also have ∥fr∥m → 0, as r → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Both ob- servations combined imply that (fr) is bounded in the Hilbert space H1(G, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Using the Banach-Saks theorem we obtain a decreasing sequence rk → 0 such that gn = 1 n n � k=1 frk converges in H1(G, m) to some g ∈ H1(G, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Since convergence in H1(G, m) implies ℓ2(X, m)-convergence and since frk → 0 in ℓ2(X, m), we obtain g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' By construction we also have gn ≥ 1 on Brn(w), which leads to capm({w}) ≤ inf n∈N � Q(gn) + ∥gn∥2 m � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The ’In particular’-part follows from Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='9 and the observa- tion that (X, σ) is totally bounded as it is isometric to a bounded and hence totally bounded subset of R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' □ Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We do not assume local finiteness in the previous theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' If � x∈X Cx is not dense in R2, then the assumption on the boundedness of the circle packing can be dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In this case, one just uses inversion at a circle in the complement of � x∈X Cx to obtain a bounded circle packing with isomorphic contact graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' For more details see also the proof of the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' RECURRENT AND (STRONGLY) RESOLVABLE GRAPHS 33 In the following corollary we call a weighted graph planar if the induced combinatorial graph is planar (for a precise definition of the latter see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' [14, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let G = (X, b) be a locally finite planar graph of bounded geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Then G is strongly resolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In particular, if G is transient, then G possesses a non-constant harmonic function of finite energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' According to Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='3 in [15] any locally finite graph is isomor- phic to the contact graph of a circle packing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We show that the circle packing can be chosen to be bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' With this at hand the claim follows from the previous theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We add one additional point o and one edge from o to a point in X such that the resulting graph (X′, b′) is still planar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' According to Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='3 in [15] the graph (X′, b′) is isomorphic to a contact graph of a circle packing Cx, x ∈ X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In order to make this circle packing bounded, we use inversion at the circle Co corresponding to the new vertex o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We denote the inversion map by ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Since inversions map circles to circles, ψ(Cx), x ∈ X, is a circle packing inside the bounded set Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' By construction its contact graph is the combinatorial graph underlying (X, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' □ Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The existence of non-trivial harmonic functions on transient planar graphs of bounded geometry was one of the main results [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Subsequently, even more explicit description of all harmonic functions of planar graphs were given via boundaries of sphere packings [1] or square tilings [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' For a unified approach we refer to [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Recently the class of canonically compactifiable graphs (see below for a definition) has gathered some attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Our previous considerations allow us to show that locally finite planar graphs of bounded geometry are never canonically compactifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' According to [6] a graph G = (X, b) is called canonically compactifi- able if D(G) ⊆ ℓ∞(X) (see [18] for different equivalent characterizations as well).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Examples are Zn with n ≥ 3, see [11, Section 6], or graphs (X, b) for which � x,y∈X 1 b(x, y) < ∞, see [6, Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Note that the latter condition implies very large vertex degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' We note the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Infinite canonically compactifiable graphs are not strongly resolvable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' In particular, locally finite infinite planar graphs of bounded geometry are not canonically compactifiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Let (X, b) be an infinite canonically compactifiable graph and let m be a finite measure on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Canonical compactifiability yields 34 LENZ, PUCHERT, AND SCHMIDT H1(G, m) ⊆ ℓ∞(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The closed graph theorem implies the existence of C > 0 such that ∥f∥2 ∞ ≤ C(Q(f) + ∥f∥2 m) for all f ∈ H1(G, m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' This implies capm(U) ≥ 1/C for any ∅ ̸= U ⊆ X such that points in any metric boundary have a capacity at least 1/C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' ¸It remains to prove that for any intrinsic metric σ with respect to m, which induces the discrete topology, the space (X, σ) is not complete (and hence it has at least one boundary point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' According to [18] (X, b) being canonically compactifiable and σ being an intrisic metric with respect to a finite measure imply that (X, σ) is totally bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Hence, X σ is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' But (X, σ) is not compact as an infinite set with the discrete topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' This shows ∂σX = X σ \\ X ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' The ’In particular’-part follows from the previous corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' □ References [1] Omer Angel, Martin T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Barlow, Ori Gurel-Gurevich, and Asaf Nachmias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Boundaries of planar graphs, via circle packings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' [9] Tom Hutchcroft and Yuval Peres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Boundaries of planar graphs: a unified ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} +page_content=', 22:Paper No.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdA0T4oBgHgl3EQfCv-W/content/2301.01994v1.pdf'} diff --git a/L9E2T4oBgHgl3EQfqQgh/content/tmp_files/2301.04036v1.pdf.txt b/L9E2T4oBgHgl3EQfqQgh/content/tmp_files/2301.04036v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..59ce988313e36ff1a92da78342f77ee134c87b50 --- /dev/null +++ b/L9E2T4oBgHgl3EQfqQgh/content/tmp_files/2301.04036v1.pdf.txt @@ -0,0 +1,967 @@ +Deep Reinforcement Learning for Autonomous Ground Vehicle +Exploration Without A-Priori Maps +Shathushan Sivashangaran and Azim Eskandarian, Senior Member, IEEE +Abstract— Autonomous Ground Vehicles (AGVs) are essential +tools for a wide range of applications stemming from their +ability to operate in hazardous environments with minimal +human operator input. Efficient and effective motion planning +is paramount for successful operation of AGVs. Conventional +motion planning algorithms are dependent on prior knowledge +of environment characteristics and offer limited utility in +information poor, dynamically altering environments such as +areas where emergency hazards like fire and earthquake occur, +and unexplored subterranean environments such as tunnels +and lava tubes on Mars. We propose a Deep Reinforcement +Learning (DRL) framework for intelligent AGV exploration +without a-priori maps utilizing Actor-Critic DRL algorithms to +learn policies in continuous and high-dimensional action spaces, +required for robotics applications. The DRL architecture com- +prises feedforward neural networks for the critic and actor +representations in which the actor network strategizes linear +and angular velocity control actions given current state inputs, +that are evaluated by the critic network which learns and +estimates Q-values to maximize an accumulated reward. Three +off-policy DRL algorithms, DDPG, TD3 and SAC, are trained +and compared in two environments of varying complexity, +and further evaluated in a third with no prior training or +knowledge of map characteristics. The agent is shown to learn +optimal policies at the end of each training period to chart +quick, efficient and collision-free exploration trajectories, and +is extensible, capable of adapting to an unknown environment +with no changes to network architecture or hyperparameters. +I. INTRODUCTION +Autonomous Ground Vehicles (AGVs) are indispensable +tools for mapping uncharted terrain, search & rescue mis- +sions, disaster response, military operations, mining, and +extraterrestrial planetary exploration owing to their ability +to operate in hazardous, unstructured environments reliably +with minimal input from a human operator. Conventional +AGV navigation algorithms are dependent on specific envi- +ronmental configurations [1] which limits their effectiveness +in adapting to dynamically changing environments such as +areas where emergency hazards like fire and earthquake +occur, and unexplored subterranean environments such as +tunnels, caves and lava tubes on Mars. +Recent advancements in Artificial Intelligence (AI), sen- +sors, communication and computer technology facilitate in- +telligent AGVs capable of high autonomy. Simultaneous +Localization And Mapping (SLAM) enables AGVs to simul- +taneously estimate vehicle state utilizing on-board sensors +and construct a model of the environment the sensors per- +ceive [2], [3]. The inclusion of LIDAR-centric SLAM in the +Corresponding author: Shathushan Sivashangaran +The authors are with the Autonomous Systems and Intelligent Ma- +chines Laboratory, Virginia Tech, Blacksburg, VA 24061, USA. +(email: +shathushansiva@vt.edu; eskandarian@vt.edu). +perception pipeline is a key enabler for AGV navigation in +environments that are GPS-denied with no access to a-priori +maps [4]. +Mobile robot trajectories require optimization for shortest +path, minimum energy consumption and training time [5]. +Conventional motion planning algorithms offer limited util- +ity in information poor, dynamically altering environments. +These comprise graph search algorithms such as Dijkstra, +A* and D* [6] that are well-defined and simple to use +but are inefficient in complex, dynamic environments and +have poor robustness to noise interference and errors in +the environment model, random sampling algorithms such +as Probability Graph Method (PGM) and Rapid exploration +Random Tree (RRT) [7] that select random scatter points in +the entire environment space to search for the optimal path +between the starting and end points making them susceptible +to poor real-time performance, sub-optimal solutions and +high computation cost, Artificial Potential Field (APF) [8] +that has low computation cost and is efficient but prone to +local minima traps, and nature inspired algorithms such as +fuzzy logic that is robust, but requires prior knowledge in +the form of user defined knowledge based logic and rules, +and Genetic Algorithm (GA) [9] which is ideal for the global +optimal solution and suitable for complex problems, but has +poor local search ability and slow convergence rate. +Motion planning models that incorporate Artificial Neural +Networks (ANN) and Actor-Critic Reinforcement Learning +(RL) enable robotic systems to learn optimal, end-to-end +policies in continuous and high-dimensional action spaces di- +rectly from characteristics of high-dimensional sensory input +data to intelligently select goal driven actions in dynamically +changing, obstacle filled unstructured terrain in the absence +of prior knowledge and detailed maps [10]–[13]. +On-policy Actor-Critic Deep Reinforcement Learning +(DRL) algorithms such as Trust Region Policy Optimization +(TRPO) [14] and Proximal Policy Optimization (PPO) [15] +are robust to hyperparameter tuning and straightforward to +implement, but are sample inefficient as these require new +training samples for every policy update, which makes learn- +ing an effective policy for complex tasks computationally +exorbitant. Off-policy Actor-Critic DRL algorithms such as +Deep Deterministic Policy Gradient (DDPG) [16], Twin +Delayed Deep Deterministic Policy Gradient (TD3) [17] +and Soft Actor-Critic (SAC) [18] reuse past experience for +learning, thus have good sample efficiency. +Given the potential of DRL for AGV navigation in infor- +mation poor environments, this paper presents and evaluates +a DRL architecture for intelligent AGV navigation, and +arXiv:2301.04036v1 [cs.RO] 10 Jan 2023 + +compares state-of-the-art off-policy DRL algorithms’ ability +to safely navigate and explore obstacle filled terrain without +prior knowledge of environment characteristics. Moreover, +this paper answers research questions related to effective +policy transfer between environments, and sheds light on the +importance, and benefits of simulation training for complex +DRL tasks. These questions are answered through multiple +simulations and analyses of learning and post-training per- +formances in environments of varying complexity. +II. BACKGROUND ON DEEP REINFORCEMENT +LEARNING +RL is a Machine Learning (ML) framework inspired by +trial-and-error animal learning to train agents that interact +with the surrounding environment by promoting or discour- +aging actions utilizing reward feedback signals designed to +gauge effectiveness of executed actions. Deep Learning (DL), +a key ML component, utilizes ANNs to form an abstract, dis- +tinguishable high-level representation from low-level input +features. Deep Reinforcement Learning (DRL) algorithms +combine DL and RL to extract unknown environment fea- +tures from high-dimensional input data utilizing ANNs, and +decide control actions using RL. Figure 1 portrays the DRL +framework. +Fig. 1. +Schematic of deep reinforcement learning framework. +A RL agent observes its state si at each time step t, +and selects an action ai from action space A, conforming +to a learned policy π(ai | si) that maps states to actions. +The expectation of a discounted, accumulated reward Ri = +Σ∞ +k=0γkrwi+k at each state is maximized during learning, +where γ ∈ (0,1] is the discount factor, and rwi is the scalar +reward signal for selecting action ai [19]. +A. Actor-Critic Framework +An actor-critic framework utilizing deep function approx- +imators that combines both value-based and policy-based +RL is the preferred method to learn policies in continuous +and high-dimensional action spaces, required for robotics +applications. This method leverages the joint computing +and decision-making abilities of the actor and critic neural +networks to yield low variance and fast speeds when updating +gradients. Figure 2 illustrates the Actor-Critic framework. +The actor network strategizes an action output selected +from a continuous action space using policy gradient, uti- +lizing the current state as the input. The critic evaluates +Fig. 2. +Schematic of Actor-Critic framework. +the chosen actions and outputs the associated approximate +Q-value for the current state and selected action using an +approximated value function to counter the large variance in +the policy gradients. In off-policy algorithms, sample data +accumulated in a replay buffer is utilized to update and +approximate the value function yielding higher sample effi- +ciency than on-policy algorithms. The two networks compute +the action prediction for the current state at each time step +to generate a temporal-difference error signal. +B. Deep Deterministic Policy Gradient +DDPG is a model-free, off-policy actor-critic RL algo- +rithm that combines ANNs with the actor-critic representa- +tion of standard Deterministic Policy Gradient (DPG) [20] +to successfully implement control sequences in a continuous +action space. The actor, π(s | θ) and critic, Q(s, a | φ) each +comprise fully-linked, two-layer feedforward ANNs with a +Rectified Linear Unit (ReLU) activation function. +The loss L is minimized across all sampled experiences +to update the critic parameters, φ, +L = 1 +M +M +� +i=1 +(yi − Q(si, ai | φ))2 +(1) +Here M is a random mini-batch of experiences, and yi is +the target value function computed as follows, +yi = Ri + γQt(si+1, πt(si+1 | θt) | φt) +(2) +θt and φt are parameters of the target actor πt and target +critic Qt respectively, that have the same structure and +parameterization as π and Q. The agent periodically updates +θt and φt using the latest θ and φ values to improve the +stability of the optimization. +The actor parameters, θ are updated using a sampled +policy gradient ∇θJ to maximize the expected discounted +reward, +∇θJ ≈ 1 +M +M +� +i=1 +GaiGπi +(3) +Here Gai is the gradient of the critic output with respect to +the action selected by the actor network computed as follows, + +Reward +Agent +CNN +Take action +State +Environment +Observe stateEnvironment +reward +Critic +Value +action +state +Function +TD error +Policy, +ActorGai = ∇aQ(si, π(si | θ) | φ) +(4) +Gπi is the gradient of the actor output with respect to its +parameters, +Gπi = ∇θπ(si | θ) +(5) +C. Twin-Delayed Deep Deterministic Policy Gradient +TD3 is designed to improve learned policies by preventing +overestimation of the value function. Two Q-value functions +are learned simultaneously, and the minimum is used for pol- +icy updates. Moreover, the policy is updated less frequently +than the Q-value function to further improve learned policies. +The parameters of the critic, Qk(s, a | φk), where k = 2 +is the number of critics, are updated by minimizing the loss +Lk as follows, +Lk = 1 +M +M +� +i=1 +(yi − Qk(si, ai | φk))2 +(6) +The target value function yi is computed as follows, +yi = Ri + γ min +k (Qtk(si+1, clip(πt(si+1 | θt) + ε) | φtk)) +(7) +Here θt and φtk are parameters of the target actor πt and +target critics Qtk, and ε is noise added to the computed action +to promote exploration. The action is clipped based on the +noise limits. +The actor parameters are updated similar to DDPG using +Equation (3) where Gai is computed as follows and Gπi is +computed as in Equation (5). +Gai = ∇a min +k (Qk(si, π(si | θ) | φ)) +(8) +D. Soft Actor-Critic +SAC, similar to DDPG and TD3, is a model-free, off- +policy actor-critic RL algorithm. In addition to maximizing +the long-term expected reward, SAC maximizes the entropy +of the policy, which is a measure of the policy uncertainty +at a given state. A higher policy entropy promotes explo- +ration, hence the learned policy balances exploitation and +exploration of the environment. +The agent utilizes a stochastic actor that outputs mean +and standard deviation, using which an unbounded action is +randomly selected from a Gaussian distribution. The entropy +of the policy is computed during training for the given +observation using this unbounded probability distribution. +Bounded actions that comply with the action space are +generated from the unbounded action by applying tanh and +scaling operations. +The critic parameters are updated at specific time step +periods by minimizing the loss function in Equation (6), +similar to TD3 for k critics. +The target value function yi is computed as the sum of the +minimum discounted future reward from the critic networks +Ri, and the weighted entropy as follows, +yi = Ri + γ min +k (Qtk(si+1, π(si+1 | θ) | φtk)) +−αlnπ(si+1 | θ) +(9) +Here α is the entropy loss weight. The entropy weight is +updated by minimizing the loss function, Lα where H is the +target entropy as follows, +Lα = 1 +M +M +� +i=1 +(−αlnπ(si | θ) − αH) +(10) +The stochastic actor parameters are updated by minimizing +the objective function Jπ, +Jπ = 1 +M +M +� +i=1 +(− min +k (Qtk(si, π(si | θ) | φtk))+αlnπ(si | θ)) +(11) +III. METHODOLOGY +This section presents the DRL architecture, reward design, +training, and evaluation methodologies for collision-free +AGV exploration in unknown environments. The MATLAB +Robotics System [21] and Reinforcement Learning [22] +Toolboxes, and Simulink are utilized to model the AGV, and +train the DRL agent. +A. Network Architecture +In order to maximize the long-term reward, designed to +encourage quick, efficient, and collision-free exploration of +the environment, the DRL agent makes strategic linear and +angular velocity action decisions for the current time step, +vt and ωt. These decisions are based on LiDAR range mea- +surements r, the AGV’s state s = (x, y, ψ), the previous time +step’s action a = (v, ω), and the corresponding reward value, +R. The proposed DRL architecture for AGV exploration is +shown in Figure 3. +Fig. 3. +Ubiquitous Deep Reinforcement Learning architecture for Au- +tonomous Ground Vehicle exploration. + +LiDAR, r +Agent +Lin. Velocity, Vt-1 +Lin. Velocity, Vt +Ang. Velocity, Ot-1 +Ang. Velocity, Ot +Perception +Control +Reward, R +Tanman +Environment +Observation +ActionB. Reward Function +The reward function is designed to encourage the agent +to explore its environment efficiently, quickly and safely, +without collisions. It computes a scalar reward value as +follows, +R = 0.005r2 + 1.3v2 − 0.5ω2 +(12) +A positive reward is applied to the square of the minimum +measurement obtained by the LiDAR sensor, r to incen- +tivize obstacle avoidance. This reward is highest when the +agent is at a greater distance from obstacles, encouraging +the generation of paths devoid of obstacles. The agent is +additionally rewarded for swift navigation through positive +reinforcement of linear velocity, v. To encourage efficient +exploration, a negative reward is applied to angular velocity, +ω to discourage repeated circular motion in the same vicinity. +High coefficients for r2 and v2 lead to a compromise between +obstacle avoidance ability and exploratory behavior, hence an +optimal balance was determined through experimentation to +prioritize both exploration, and collision avoidance. +C. AGV Model +XTENTH-CAR [23], a proportionally scaled experimental +vehicle platform, designed with similar hardware and soft- +ware architectures as the full-size X-CAR [24] connected au- +tonomous vehicle, is modeled and trained in simulation. The +XTENTH-CAR AGV, shown in Figure 4, has a wheelbase +of 0.32 m and utilizes the Ackermann steering mechanism. +Fig. 4. +XTENTH-CAR Ackermann steered AGV platform. +The AGV’s kinematics are computed using a bicycle +model, portrayed in Figure 5, where the front and rear wheels +are represented by a single wheel located at the center of each +axle. This model is accurate for use at low speeds and offers +a good balance between model accuracy and computation +cost [25] for evaluation of the DRL agent. +The bicycle model is represented by the following equa- +tions, +˙x = v cos(ψ + β) +(13) +˙y = v sin(ψ + β) +(14) +Fig. 5. +Schematic of kinematic bicycle model. +˙ψ = v +lr +sin(β) +(15) +β = tan−1 +� +lr +lf + lr +tan(δ) +� +(16) +Here x and y are position coordinates of the AGV’s +center of mass, ψ is the angle of the AGV’s heading with +respect to the inertial reference frame, β is the angle between +the velocity vector of the AGV’s center of mass and its +longitudinal axis, lf and lr are distances from the center +of mass to the front and rear axles respectively, and velocity, +v and steering angle, δ are control inputs. +D. Environments for Training and Evaluation +The DRL agent is trained in two distinct environments of +varying complexity. The first environment, depicted in Figure +6, is a 25 m x 25 m space with walls that the agent must steer +clear of. The second environment, illustrated in Figure 7, is +a more complex 40 m x 40 m space with walls and various +obstacles, marked in black, that the agent must additionally +avoid. +The AGV, identified with a red symbol on the training +maps, is set to a random starting position at the start of +each training episode to enhance policy learning. This reset +ensures that the agent is not biased towards any particular +initial location. +Fig. 6. +First training environment with DRL agent marked in red at a +randomized initial location. + +SmallRigty +x +B +山 +lfFirst Environment +25 +20 +15 +5 +0 +0 +10 +20 +x (m)Fig. 7. +Second training environment with DRL agent marked in red at a +randomized initial location. +The trained agent is evaluated in a third environment, illus- +trated in Figure 8 to evaluate the robustness, and performance +of the learned policy in a new, unknown environment with +the same network architecture and hyperparameters. +Fig. 8. +Evaluation environment with DRL agent marked in red at a +randomized initial location. +E. Training Conditions and Hyperparameters +A training episode is concluded when the agent encounters +an obstacle or completes the maximum number of steps +permitted in a single episode. Subsequently, the agent is reset +to a randomly determined starting location to initiate the +next episode. The DRL agent is trained to a total of 10,000 +episodes, each with a maximum of 1000 steps in the first +environment, and 20,000 episodes, each with a maximum of +2000 steps in the second, to facilitate rapid iterative learning. +Modified hyperparameters with non-default values are listed +in Table I. +TABLE I +NON-DEFAULT HYPERPARAMETERS +Hyperparameter +Value +Discount Factor (γ) +0.995 +Actor Learn Rate +0.00005 +Critic Learn Rate +0.0005 +Target Smooth Factor +0.001 +Mini Batch Size +128 +Experience Buffer Length +1,000,000 +IV. RESULTS AND DISCUSSION +In this section, we present DRL training results, includ- +ing post-training exploration trajectories and corresponding +average return and steps achieved by the agent each episode +iteration during the training period, utilizing DDPG, TD3 and +SAC algorithms. We further evaluate each trained policy in +a new environment with no prior knowledge of environment +characteristics. +A. Training Results +An Intel i7 11700K CPU and GeForce RTX 3070 Ti GPU +were used for training. Table II summarizes the training times +for each DRL algorithm in the evaluated environments. +TABLE II +TRAINING TIMES +Case +Training Time (Hrs) +First Environment: DDPG +66.0 +First Environment: TD3 +95.3 +First Environment: SAC +156.7 +Second Environment: DDPG +84.8 +Second Environment: TD3 +128.1 +Second Environment: SAC +203.9 +SAC required the longest training time, followed by TD3 +and DDPG which required the least. On average, training in +the second, more complex environment required 31% longer +training time than in the first, over twice the number of +training episodes. DDPG required 28.5%, TD3 34.4% and +SAC 30.1% longer to train in the second environment. +In the first environment, TD3 required 44.4% longer to +train than DDPG, and SAC 137.4% longer than DDPG and +64.4% longer TD3. In the second environment, TD3 required +51.1% longer to train than DDPG, and SAC 140.4% longer +than DDPG and 59.2% longer than TD3. On average, TD3 +required 47.8% longer training time than DDPG, and SAC +138.9% longer than DDPG and 61.8% longer training time +than TD3. +Training times ranged from 2.75 days in the first envi- +ronment for DDPG to 8.5 days in the second environment +for SAC. More optimal policies require longer training +times to accommodate increased episode steps in the first +environment, and more training episodes in the second. +1) First Environment: The order 50 moving average re- +turn and agent steps during training in the first environment +are illustrated in Figures 9 and 10. The training results in +the first environment are summarized in Table III. +DDPG converges first at 170 episodes with an average +return of 318 and 865 average steps. TD3 converges last at +960 episodes with an average return of 435 and 1000 average +steps, and SAC converges at 390 episodes with an average +return of 320 and the maximum 1000 average steps. +DDPG learned the least optimal policy with the lowest +average return and agent steps. TD3 achieves the highest +return, and the maximum 1000 steps, however SAC achieves + +Second Environment +40 +30 +20 +10 +0 +0 +20 +40 +x (m)Third Environment +40 +30 +盲20 +y +10 +0 +0 +20 +40 +x (m)Fig. 9. +Order 50 moving average return during training in the first +environment. +Fig. 10. +Order 50 moving average agent steps during training in the first +environment. +1000 exploration steps more consistently post training con- +vergence. Unlike TD3 which solely maximizes the long-term +expected reward, SAC additionally maximizes the entropy of +the policy to promote exploration. Consequently, TD3 learns +a policy with a higher return, but SAC learns the better policy +for agent exploration. +The trajectories in the first environment for each algorithm +after the first training episode are illustrated in Figure 11. +Fig. 11. +Trajectories in the first environment post first training episode. +All three agents collide having no prior training experi- +ence. SAC covers the most ground after one training episode. +The trajectories in the first environment for each algorithm +post training completion are illustrated in Figure 12. +Each algorithm achieves 1000 episode steps without col- +lision. SAC covers the most ground, and exhibits the most +efficient exploratory behavior which will result in the greatest +energy savings. TD3 is next best, followed by DDPG which +TABLE III +FIRST ENVIRONMENT TRAINING RESULTS +Algorithm +Convergence +Average +Average +Episode +Return +Agent Steps +DDPG +170 +318 +865 +TD3 +960 +435 +1000 +SAC +390 +320 +1000 +Fig. 12. +Trajectories in the first environment post training completion. +is the most inefficient, covering the same region multiple +times. +2) Second Environment: The order 50 moving average +return and agent steps during training in the second envi- +ronment are illustrated in Figures 13 and 14. The training +results in the second environment are summarized in Table +IV. +Fig. 13. +Order 50 moving average return during training in the second +environment. +Training for 20,000 episodes is insufficient for the DRL +algorithms to learn an optimal policy in the second environ- +ment. At the end of the training period, DDPG achieves an +average return of 125 and 530 average steps, TD3 obtains +an average return of 230 and 715 average steps, and SAC +converges to a local maximum at 10,620 episodes with an +average return of 210 and 1050 average steps. Training +was limited to 20,000 episodes to gauge performance in +a reasonable time frame, however, continued training over +75,000 to 100,000 episodes will enable the agents to learn +an optimal policy to traverse the more complex terrain over +an indefinite number of exploration steps. +Training DDPG, TD3 and SAC algorithms in the second + +Average Agent Steps in First Environment +1000 +800 +Steps +600 +Average s +400 +-DDPG +200 +-TD3 +-SAC +0 +0 +2000 +4000 +6000 +8000 +10000 +Episode IterationFirst Episode Trajectory in First Environment +25 +-DDPG +-TD3 +20 +-SAC +15 +10 +5 +0 +0 +10 +20 +x (m)Post-Training Trajectory in First Environment +25 +-DDPG +-TD3 +20 +-SAC +15 +10 +5 +0 +10 +20 +0 +x (m)Average Return in Second Environment +400 +300 +200 +100 +DDPG +-TD3 +-SAC +0.5 +1.5 +1 +2 +0 +Episode Iteration +X104Average Return in First Environment +500 +400 +300 +200 +-DDPG +100 +-TD3 +-SAC +2000 +4000 +6000 +8000 +10000 +0 +Episode IterationFig. 14. +Order 50 moving average agent steps during training in the second +environment. +TABLE IV +SECOND ENVIRONMENT TRAINING RESULTS +Algorithm +Average +Average +Return +Agent Steps +DDPG +125 +530 +TD3 +230 +715 +SAC +210 +1050 +environment for 20,000 episodes required a total of 416.8 +hours, as such, it is infeasible to evaluate the algorithms +for 75,000+ episodes with the existing setup. More powerful +computer hardware is required. Similar to the training results +in the first environment, DDPG learned the least optimal +policy achieving the lowest return and agent steps. TD3 +achieved the highest return, however SAC learned a more +optimal policy achieving the highest agent steps. +The trajectories in the second environment for each algo- +rithm after the first training episode are illustrated in Figure +15. +Fig. 15. +Trajectories in the second environment post first training episode. +DDPG covers the least ground after one training episode. +SAC and TD3 cover a similar distance. All three agents +collide after travelling a short distance with no prior training +experience. +The trajectories in the second environment for each algo- +rithm post training completion are illustrated in Figure 16. +SAC achieves the best performance, learning a trajectory +that covers the most distance. TD3 and DDPG yield similar +performance, with TD3 being a marginal improvement. +Fig. 16. +Trajectories in the second environment post training completion. +B. Trained Policy Evaluation +The six agents, DDPG, TD3 and SAC, trained in two +different environments are evaluated in a third unknown envi- +ronment with no prior training or knowledge of environment +characteristics, to evaluate the extensibility of the ubiquitous +DRL architecture for AGV exploration in information poor +environments. Figure 17 portrays the trajectories for each +agent in the third environment. The evaluation results are +summarized in Table V. +Fig. 17. +Trained DRL agents evaluated in the third environment. +TABLE V +THIRD ENVIRONMENT EVALUATION RESULTS +Agent +Agent Steps +DDPG Env1 +2151 +TD3 Env1 +1533 +SAC Env1 +2971 +DDPG Env2 +2893 +TD3 Env2 +321 +SAC Env2 +2839 +The SAC agents demonstrate the best performance, cover- +ing the most ground, efficiently. The DDPG agent trained in +the first environment covers more ground than either TD3 +agent, however is more inefficient. DDPG trained in the +second environment covers the second highest distance, but +yields the worst exploratory behavior, repeatedly traversing +a circular trajectory in the same vicinity. TD3 agents cover +less ground, and exhibit less efficient exploratory behavior + +Average Agent Steps in Second Environment +1500 +000 +500 +DDPG +TD3 +-SAC +0.5 +1.5 +2 +1 +Episode Iteration +X104First Episode Trajectory in Second Environment +-DDPG +40 +-TD3 +-SAC +30 +20 +10 +0 +0 +20 +40 +X (m)Post-Training Trajectory in Second Environment +-DDPG +40 +-TD3 +-SAC +30 +20 +10 +0 +0 +20 +40 +x (m)Trained Policies Evaluated in Third Environment +40 +30 +DDPG Env1 Agent +TD3 Env1 Agent +10 +-SAC Env1 Agent +-DDPG Env2 Agent +-TD3 Env2 Agent +-SAC Env2 Agent +0 +20 +0 +40 +x (m)than SAC. The DRL agents trained in the first environment +performed better than those trained in the second, as the +characteristics of the evaluated environment are more similar +to the first than the second. The SAC agents are most +robust to differences in environment characteristics with both +achieving near identical performance. +The reward function weights and network hyperparameters +can be further engineered for this application, and the agent +trained over a longer period with more episode steps each +episode iteration to learn an improved policy that explores +the surrounding environment indefinitely. +Bridging the simulation to reality gap to transfer policies +learned in simulation to real-world robotic systems is a +current area of active research. The large number of episodes +required to sufficiently train the agent renders simulation +training an essential component for DRL in robotics applica- +tions to minimize cost and possible physical damage caused +by collisions during training. Substantial computation cost is +required for training, however, post-training implementation +of DRL agents is significantly less expensive, which makes +DRL a powerful tool for real-time AGV motion planning and +control in environments without a-priori maps. +V. CONCLUSIONS +This paper presented an ubiquitous DRL architecture for +intelligent AGV exploration without a-priori maps. Three +actor-critic DRL algorithms, DDPG, TD3 and SAC, were +trained in two environments of varying complexity, and +further evaluated in a third with no prior knowledge of +map characteristics. Simulation results demonstrate the effec- +tiveness of the proposed DRL architecture, reward function +and training conditions for quick, efficient and collision- +free AGV navigation. SAC achieves the best performance, +yielding trajectories that cover the highest distance, and +demonstrate the most efficient exploratory behavior. Learning +requires substantial computation cost, requiring up to 8.5 +days for SAC in the second, complex environment using +an Intel i7 11700K CPU and GeForce RTX 3070 Ti GPU. +Improved policies with higher post-training episode steps +require greater training times. Despite the high training cost, +post-training implementation of DRL agents is significantly +less expensive, which makes DRL a powerful tool for real- +time AGV exploration in information poor, dynamically +altering environments. For future work, the simulation to +reality gap will be bridged to transfer policies learned in +simulation to the physical AGV. +REFERENCES +[1] J. Levinson, J. 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Available: https: +//www.mathworks.com/products/reinforcement-learning.html +[23] S. Sivashangaran and A. Eskandarian, “XTENTH-CAR: A proportion- +ally scaled experimental vehicle platform for connected autonomy and +all-terrain research,” arXiv preprint arXiv:2212.01691, 2022. +[24] G. Mehr, P. Ghorai, C. Zhang, A. Nayak, D. Patel, S. Sivashangaran, +and A. Eskandarian, “X-CAR: An experimental vehicle platform +for connected autonomy research,” IEEE Intelligent Transportation +Systems Magazine, pp. 2–19, 2022. +[25] P. Polack, F. Altch´e, B. d’Andr´ea Novel, and A. de La Fortelle, “The +kinematic bicycle model: A consistent model for planning feasible tra- +jectories for autonomous vehicles?” in 2017 IEEE intelligent vehicles +symposium (IV). +IEEE, 2017, pp. 812–818. + diff --git a/L9E2T4oBgHgl3EQfqQgh/content/tmp_files/load_file.txt b/L9E2T4oBgHgl3EQfqQgh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bf1f2558eb7219cb5362e3d1d9c3360190a2f8e6 --- /dev/null +++ b/L9E2T4oBgHgl3EQfqQgh/content/tmp_files/load_file.txt @@ -0,0 +1,485 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf,len=484 +page_content='Deep Reinforcement Learning for Autonomous Ground Vehicle Exploration Without A-Priori Maps Shathushan Sivashangaran and Azim Eskandarian, Senior Member, IEEE Abstract— Autonomous Ground Vehicles (AGVs) are essential tools for a wide range of applications stemming from their ability to operate in hazardous environments with minimal human operator input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Efficient and effective motion planning is paramount for successful operation of AGVs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Conventional motion planning algorithms are dependent on prior knowledge of environment characteristics and offer limited utility in information poor, dynamically altering environments such as areas where emergency hazards like fire and earthquake occur, and unexplored subterranean environments such as tunnels and lava tubes on Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' We propose a Deep Reinforcement Learning (DRL) framework for intelligent AGV exploration without a-priori maps utilizing Actor-Critic DRL algorithms to learn policies in continuous and high-dimensional action spaces, required for robotics applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The DRL architecture com- prises feedforward neural networks for the critic and actor representations in which the actor network strategizes linear and angular velocity control actions given current state inputs, that are evaluated by the critic network which learns and estimates Q-values to maximize an accumulated reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Three off-policy DRL algorithms, DDPG, TD3 and SAC, are trained and compared in two environments of varying complexity, and further evaluated in a third with no prior training or knowledge of map characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The agent is shown to learn optimal policies at the end of each training period to chart quick, efficient and collision-free exploration trajectories, and is extensible, capable of adapting to an unknown environment with no changes to network architecture or hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' INTRODUCTION Autonomous Ground Vehicles (AGVs) are indispensable tools for mapping uncharted terrain, search & rescue mis- sions, disaster response, military operations, mining, and extraterrestrial planetary exploration owing to their ability to operate in hazardous, unstructured environments reliably with minimal input from a human operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Conventional AGV navigation algorithms are dependent on specific envi- ronmental configurations [1] which limits their effectiveness in adapting to dynamically changing environments such as areas where emergency hazards like fire and earthquake occur, and unexplored subterranean environments such as tunnels, caves and lava tubes on Mars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Recent advancements in Artificial Intelligence (AI), sen- sors, communication and computer technology facilitate in- telligent AGVs capable of high autonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Simultaneous Localization And Mapping (SLAM) enables AGVs to simul- taneously estimate vehicle state utilizing on-board sensors and construct a model of the environment the sensors per- ceive [2], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The inclusion of LIDAR-centric SLAM in the Corresponding author: Shathushan Sivashangaran The authors are with the Autonomous Systems and Intelligent Ma- chines Laboratory, Virginia Tech, Blacksburg, VA 24061, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' (email: shathushansiva@vt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' eskandarian@vt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' perception pipeline is a key enabler for AGV navigation in environments that are GPS-denied with no access to a-priori maps [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Mobile robot trajectories require optimization for shortest path, minimum energy consumption and training time [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Conventional motion planning algorithms offer limited util- ity in information poor, dynamically altering environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' These comprise graph search algorithms such as Dijkstra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' A* and D* [6] that are well-defined and simple to use but are inefficient in complex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' dynamic environments and have poor robustness to noise interference and errors in the environment model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' random sampling algorithms such as Probability Graph Method (PGM) and Rapid exploration Random Tree (RRT) [7] that select random scatter points in the entire environment space to search for the optimal path between the starting and end points making them susceptible to poor real-time performance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' sub-optimal solutions and high computation cost,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Artificial Potential Field (APF) [8] that has low computation cost and is efficient but prone to local minima traps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' and nature inspired algorithms such as fuzzy logic that is robust,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' but requires prior knowledge in the form of user defined knowledge based logic and rules,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' and Genetic Algorithm (GA) [9] which is ideal for the global optimal solution and suitable for complex problems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' but has poor local search ability and slow convergence rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Motion planning models that incorporate Artificial Neural Networks (ANN) and Actor-Critic Reinforcement Learning (RL) enable robotic systems to learn optimal, end-to-end policies in continuous and high-dimensional action spaces di- rectly from characteristics of high-dimensional sensory input data to intelligently select goal driven actions in dynamically changing, obstacle filled unstructured terrain in the absence of prior knowledge and detailed maps [10]–[13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' On-policy Actor-Critic Deep Reinforcement Learning (DRL) algorithms such as Trust Region Policy Optimization (TRPO) [14] and Proximal Policy Optimization (PPO) [15] are robust to hyperparameter tuning and straightforward to implement, but are sample inefficient as these require new training samples for every policy update, which makes learn- ing an effective policy for complex tasks computationally exorbitant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Off-policy Actor-Critic DRL algorithms such as Deep Deterministic Policy Gradient (DDPG) [16], Twin Delayed Deep Deterministic Policy Gradient (TD3) [17] and Soft Actor-Critic (SAC) [18] reuse past experience for learning, thus have good sample efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Given the potential of DRL for AGV navigation in infor- mation poor environments, this paper presents and evaluates a DRL architecture for intelligent AGV navigation, and arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='04036v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='RO] 10 Jan 2023 compares state-of-the-art off-policy DRL algorithms’ ability to safely navigate and explore obstacle filled terrain without prior knowledge of environment characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Moreover, this paper answers research questions related to effective policy transfer between environments, and sheds light on the importance, and benefits of simulation training for complex DRL tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' These questions are answered through multiple simulations and analyses of learning and post-training per- formances in environments of varying complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' BACKGROUND ON DEEP REINFORCEMENT LEARNING RL is a Machine Learning (ML) framework inspired by trial-and-error animal learning to train agents that interact with the surrounding environment by promoting or discour- aging actions utilizing reward feedback signals designed to gauge effectiveness of executed actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Deep Learning (DL), a key ML component, utilizes ANNs to form an abstract, dis- tinguishable high-level representation from low-level input features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Deep Reinforcement Learning (DRL) algorithms combine DL and RL to extract unknown environment fea- tures from high-dimensional input data utilizing ANNs, and decide control actions using RL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Figure 1 portrays the DRL framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Schematic of deep reinforcement learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' A RL agent observes its state si at each time step t, and selects an action ai from action space A, conforming to a learned policy π(ai | si) that maps states to actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The expectation of a discounted, accumulated reward Ri = Σ∞ k=0γkrwi+k at each state is maximized during learning, where γ ∈ (0,1] is the discount factor, and rwi is the scalar reward signal for selecting action ai [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Actor-Critic Framework An actor-critic framework utilizing deep function approx- imators that combines both value-based and policy-based RL is the preferred method to learn policies in continuous and high-dimensional action spaces, required for robotics applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' This method leverages the joint computing and decision-making abilities of the actor and critic neural networks to yield low variance and fast speeds when updating gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Figure 2 illustrates the Actor-Critic framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The actor network strategizes an action output selected from a continuous action space using policy gradient, uti- lizing the current state as the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The critic evaluates Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Schematic of Actor-Critic framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' the chosen actions and outputs the associated approximate Q-value for the current state and selected action using an approximated value function to counter the large variance in the policy gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' In off-policy algorithms, sample data accumulated in a replay buffer is utilized to update and approximate the value function yielding higher sample effi- ciency than on-policy algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The two networks compute the action prediction for the current state at each time step to generate a temporal-difference error signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Deep Deterministic Policy Gradient DDPG is a model-free, off-policy actor-critic RL algo- rithm that combines ANNs with the actor-critic representa- tion of standard Deterministic Policy Gradient (DPG) [20] to successfully implement control sequences in a continuous action space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The actor, π(s | θ) and critic, Q(s, a | φ) each comprise fully-linked, two-layer feedforward ANNs with a Rectified Linear Unit (ReLU) activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The loss L is minimized across all sampled experiences to update the critic parameters, φ, L = 1 M M � i=1 (yi − Q(si, ai | φ))2 (1) Here M is a random mini-batch of experiences, and yi is the target value function computed as follows, yi = Ri + γQt(si+1, πt(si+1 | θt) | φt) (2) θt and φt are parameters of the target actor πt and target critic Qt respectively, that have the same structure and parameterization as π and Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The agent periodically updates θt and φt using the latest θ and φ values to improve the stability of the optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The actor parameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' θ are updated using a sampled policy gradient ∇θJ to maximize the expected discounted reward,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' ∇θJ ≈ 1 M M � i=1 GaiGπi (3) Here Gai is the gradient of the critic output with respect to the action selected by the actor network computed as follows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Reward Agent CNN Take action State Environment Observe stateEnvironment reward Critic Value action state Function TD error Policy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' ActorGai = ∇aQ(si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' π(si | θ) | φ) (4) Gπi is the gradient of the actor output with respect to its parameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Gπi = ∇θπ(si | θ) (5) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Twin-Delayed Deep Deterministic Policy Gradient TD3 is designed to improve learned policies by preventing overestimation of the value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Two Q-value functions are learned simultaneously, and the minimum is used for pol- icy updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Moreover, the policy is updated less frequently than the Q-value function to further improve learned policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The parameters of the critic, Qk(s, a | φk), where k = 2 is the number of critics, are updated by minimizing the loss Lk as follows, Lk = 1 M M � i=1 (yi − Qk(si, ai | φk))2 (6) The target value function yi is computed as follows, yi = Ri + γ min k (Qtk(si+1, clip(πt(si+1 | θt) + ε) | φtk)) (7) Here θt and φtk are parameters of the target actor πt and target critics Qtk, and ε is noise added to the computed action to promote exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The action is clipped based on the noise limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The actor parameters are updated similar to DDPG using Equation (3) where Gai is computed as follows and Gπi is computed as in Equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Gai = ∇a min k (Qk(si, π(si | θ) | φ)) (8) D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Soft Actor-Critic SAC, similar to DDPG and TD3, is a model-free, off- policy actor-critic RL algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' In addition to maximizing the long-term expected reward, SAC maximizes the entropy of the policy, which is a measure of the policy uncertainty at a given state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' A higher policy entropy promotes explo- ration, hence the learned policy balances exploitation and exploration of the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The agent utilizes a stochastic actor that outputs mean and standard deviation, using which an unbounded action is randomly selected from a Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The entropy of the policy is computed during training for the given observation using this unbounded probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Bounded actions that comply with the action space are generated from the unbounded action by applying tanh and scaling operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The critic parameters are updated at specific time step periods by minimizing the loss function in Equation (6), similar to TD3 for k critics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The target value function yi is computed as the sum of the minimum discounted future reward from the critic networks Ri, and the weighted entropy as follows, yi = Ri + γ min k (Qtk(si+1, π(si+1 | θ) | φtk)) −αlnπ(si+1 | θ) (9) Here α is the entropy loss weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The entropy weight is updated by minimizing the loss function, Lα where H is the target entropy as follows, Lα = 1 M M � i=1 (−αlnπ(si | θ) − αH) (10) The stochastic actor parameters are updated by minimizing the objective function Jπ, Jπ = 1 M M � i=1 (− min k (Qtk(si, π(si | θ) | φtk))+αlnπ(si | θ)) (11) III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' METHODOLOGY This section presents the DRL architecture, reward design, training, and evaluation methodologies for collision-free AGV exploration in unknown environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The MATLAB Robotics System [21] and Reinforcement Learning [22] Toolboxes, and Simulink are utilized to model the AGV, and train the DRL agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Network Architecture In order to maximize the long-term reward, designed to encourage quick, efficient, and collision-free exploration of the environment, the DRL agent makes strategic linear and angular velocity action decisions for the current time step, vt and ωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' These decisions are based on LiDAR range mea- surements r, the AGV’s state s = (x, y, ψ), the previous time step’s action a = (v, ω), and the corresponding reward value, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The proposed DRL architecture for AGV exploration is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Ubiquitous Deep Reinforcement Learning architecture for Au- tonomous Ground Vehicle exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' LiDAR, r Agent Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Velocity, Vt-1 Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Velocity, Vt Ang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Velocity, Ot-1 Ang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Velocity, Ot Perception Control Reward, R Tanman Environment Observation ActionB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Reward Function The reward function is designed to encourage the agent to explore its environment efficiently, quickly and safely, without collisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' It computes a scalar reward value as follows, R = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='005r2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='3v2 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='5ω2 (12) A positive reward is applied to the square of the minimum measurement obtained by the LiDAR sensor, r to incen- tivize obstacle avoidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' This reward is highest when the agent is at a greater distance from obstacles, encouraging the generation of paths devoid of obstacles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The agent is additionally rewarded for swift navigation through positive reinforcement of linear velocity, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' To encourage efficient exploration, a negative reward is applied to angular velocity, ω to discourage repeated circular motion in the same vicinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' High coefficients for r2 and v2 lead to a compromise between obstacle avoidance ability and exploratory behavior, hence an optimal balance was determined through experimentation to prioritize both exploration, and collision avoidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' AGV Model XTENTH-CAR [23], a proportionally scaled experimental vehicle platform, designed with similar hardware and soft- ware architectures as the full-size X-CAR [24] connected au- tonomous vehicle, is modeled and trained in simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The XTENTH-CAR AGV, shown in Figure 4, has a wheelbase of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='32 m and utilizes the Ackermann steering mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' XTENTH-CAR Ackermann steered AGV platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The AGV’s kinematics are computed using a bicycle model, portrayed in Figure 5, where the front and rear wheels are represented by a single wheel located at the center of each axle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' This model is accurate for use at low speeds and offers a good balance between model accuracy and computation cost [25] for evaluation of the DRL agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The bicycle model is represented by the following equa- tions, ˙x = v cos(ψ + β) (13) ˙y = v sin(ψ + β) (14) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Schematic of kinematic bicycle model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' ˙ψ = v lr sin(β) (15) β = tan−1 � lr lf + lr tan(δ) � (16) Here x and y are position coordinates of the AGV’s center of mass, ψ is the angle of the AGV’s heading with respect to the inertial reference frame, β is the angle between the velocity vector of the AGV’s center of mass and its longitudinal axis, lf and lr are distances from the center of mass to the front and rear axles respectively, and velocity, v and steering angle, δ are control inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Environments for Training and Evaluation The DRL agent is trained in two distinct environments of varying complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The first environment, depicted in Figure 6, is a 25 m x 25 m space with walls that the agent must steer clear of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The second environment, illustrated in Figure 7, is a more complex 40 m x 40 m space with walls and various obstacles, marked in black, that the agent must additionally avoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The AGV, identified with a red symbol on the training maps, is set to a random starting position at the start of each training episode to enhance policy learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' This reset ensures that the agent is not biased towards any particular initial location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' First training environment with DRL agent marked in red at a randomized initial location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' SmallRigty x B 山 lfFirst Environment 25 20 15 5 0 0 10 20 x (m)Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Second training environment with DRL agent marked in red at a randomized initial location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The trained agent is evaluated in a third environment, illus- trated in Figure 8 to evaluate the robustness, and performance of the learned policy in a new, unknown environment with the same network architecture and hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Evaluation environment with DRL agent marked in red at a randomized initial location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Training Conditions and Hyperparameters A training episode is concluded when the agent encounters an obstacle or completes the maximum number of steps permitted in a single episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Subsequently, the agent is reset to a randomly determined starting location to initiate the next episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The DRL agent is trained to a total of 10,000 episodes, each with a maximum of 1000 steps in the first environment, and 20,000 episodes, each with a maximum of 2000 steps in the second, to facilitate rapid iterative learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Modified hyperparameters with non-default values are listed in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' TABLE I NON-DEFAULT HYPERPARAMETERS Hyperparameter Value Discount Factor (γ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='995 Actor Learn Rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='00005 Critic Learn Rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='0005 Target Smooth Factor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='001 Mini Batch Size 128 Experience Buffer Length 1,000,000 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' RESULTS AND DISCUSSION In this section, we present DRL training results, includ- ing post-training exploration trajectories and corresponding average return and steps achieved by the agent each episode iteration during the training period, utilizing DDPG, TD3 and SAC algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' We further evaluate each trained policy in a new environment with no prior knowledge of environment characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Training Results An Intel i7 11700K CPU and GeForce RTX 3070 Ti GPU were used for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Table II summarizes the training times for each DRL algorithm in the evaluated environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' TABLE II TRAINING TIMES Case Training Time (Hrs) First Environment: DDPG 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='0 First Environment: TD3 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='3 First Environment: SAC 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='7 Second Environment: DDPG 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='8 Second Environment: TD3 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='1 Second Environment: SAC 203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='9 SAC required the longest training time, followed by TD3 and DDPG which required the least.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' On average, training in the second, more complex environment required 31% longer training time than in the first, over twice the number of training episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' DDPG required 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='5%, TD3 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='4% and SAC 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='1% longer to train in the second environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' In the first environment, TD3 required 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='4% longer to train than DDPG, and SAC 137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='4% longer than DDPG and 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='4% longer TD3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' In the second environment, TD3 required 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='1% longer to train than DDPG, and SAC 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='4% longer than DDPG and 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='2% longer than TD3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' On average, TD3 required 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='8% longer training time than DDPG, and SAC 138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='9% longer than DDPG and 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='8% longer training time than TD3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Training times ranged from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='75 days in the first envi- ronment for DDPG to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='5 days in the second environment for SAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' More optimal policies require longer training times to accommodate increased episode steps in the first environment, and more training episodes in the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' 1) First Environment: The order 50 moving average re- turn and agent steps during training in the first environment are illustrated in Figures 9 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The training results in the first environment are summarized in Table III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' DDPG converges first at 170 episodes with an average return of 318 and 865 average steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' TD3 converges last at 960 episodes with an average return of 435 and 1000 average steps, and SAC converges at 390 episodes with an average return of 320 and the maximum 1000 average steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' DDPG learned the least optimal policy with the lowest average return and agent steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' TD3 achieves the highest return, and the maximum 1000 steps, however SAC achieves Second Environment 40 30 20 10 0 0 20 40 x (m)Third Environment 40 30 盲20 y 10 0 0 20 40 x (m)Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Order 50 moving average return during training in the first environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Order 50 moving average agent steps during training in the first environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' 1000 exploration steps more consistently post training con- vergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Unlike TD3 which solely maximizes the long-term expected reward, SAC additionally maximizes the entropy of the policy to promote exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Consequently, TD3 learns a policy with a higher return, but SAC learns the better policy for agent exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The trajectories in the first environment for each algorithm after the first training episode are illustrated in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Trajectories in the first environment post first training episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' All three agents collide having no prior training experi- ence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' SAC covers the most ground after one training episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The trajectories in the first environment for each algorithm post training completion are illustrated in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Each algorithm achieves 1000 episode steps without col- lision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' SAC covers the most ground, and exhibits the most efficient exploratory behavior which will result in the greatest energy savings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' TD3 is next best, followed by DDPG which TABLE III FIRST ENVIRONMENT TRAINING RESULTS Algorithm Convergence Average Average Episode Return Agent Steps DDPG 170 318 865 TD3 960 435 1000 SAC 390 320 1000 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Trajectories in the first environment post training completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' is the most inefficient, covering the same region multiple times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' 2) Second Environment: The order 50 moving average return and agent steps during training in the second envi- ronment are illustrated in Figures 13 and 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The training results in the second environment are summarized in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Order 50 moving average return during training in the second environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Training for 20,000 episodes is insufficient for the DRL algorithms to learn an optimal policy in the second environ- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' At the end of the training period, DDPG achieves an average return of 125 and 530 average steps, TD3 obtains an average return of 230 and 715 average steps, and SAC converges to a local maximum at 10,620 episodes with an average return of 210 and 1050 average steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Training was limited to 20,000 episodes to gauge performance in a reasonable time frame, however, continued training over 75,000 to 100,000 episodes will enable the agents to learn an optimal policy to traverse the more complex terrain over an indefinite number of exploration steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Training DDPG, TD3 and SAC algorithms in the second Average Agent Steps in First Environment 1000 800 Steps 600 Average s 400 DDPG 200 TD3 SAC 0 0 2000 4000 6000 8000 10000 Episode IterationFirst Episode Trajectory in First Environment 25 DDPG TD3 20 SAC 15 10 5 0 0 10 20 x (m)Post-Training Trajectory in First Environment 25 DDPG TD3 20 SAC 15 10 5 0 10 20 0 x (m)Average Return in Second Environment 400 300 200 100 DDPG TD3 SAC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='5 1 2 0 Episode Iteration X104Average Return in First Environment 500 400 300 200 DDPG 100 TD3 SAC 2000 4000 6000 8000 10000 0 Episode IterationFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Order 50 moving average agent steps during training in the second environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' TABLE IV SECOND ENVIRONMENT TRAINING RESULTS Algorithm Average Average Return Agent Steps DDPG 125 530 TD3 230 715 SAC 210 1050 environment for 20,000 episodes required a total of 416.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='8 hours, as such, it is infeasible to evaluate the algorithms for 75,000+ episodes with the existing setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' More powerful computer hardware is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Similar to the training results in the first environment, DDPG learned the least optimal policy achieving the lowest return and agent steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' TD3 achieved the highest return, however SAC learned a more optimal policy achieving the highest agent steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The trajectories in the second environment for each algo- rithm after the first training episode are illustrated in Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Trajectories in the second environment post first training episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' DDPG covers the least ground after one training episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' SAC and TD3 cover a similar distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' All three agents collide after travelling a short distance with no prior training experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The trajectories in the second environment for each algo- rithm post training completion are illustrated in Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' SAC achieves the best performance, learning a trajectory that covers the most distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' TD3 and DDPG yield similar performance, with TD3 being a marginal improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Trajectories in the second environment post training completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Trained Policy Evaluation The six agents, DDPG, TD3 and SAC, trained in two different environments are evaluated in a third unknown envi- ronment with no prior training or knowledge of environment characteristics, to evaluate the extensibility of the ubiquitous DRL architecture for AGV exploration in information poor environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Figure 17 portrays the trajectories for each agent in the third environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The evaluation results are summarized in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Trained DRL agents evaluated in the third environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' TABLE V THIRD ENVIRONMENT EVALUATION RESULTS Agent Agent Steps DDPG Env1 2151 TD3 Env1 1533 SAC Env1 2971 DDPG Env2 2893 TD3 Env2 321 SAC Env2 2839 The SAC agents demonstrate the best performance, cover- ing the most ground, efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The DDPG agent trained in the first environment covers more ground than either TD3 agent, however is more inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' DDPG trained in the second environment covers the second highest distance, but yields the worst exploratory behavior, repeatedly traversing a circular trajectory in the same vicinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' TD3 agents cover less ground, and exhibit less efficient exploratory behavior Average Agent Steps in Second Environment 1500 000 500 DDPG TD3 SAC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content='5 2 1 Episode Iteration X104First Episode Trajectory in Second Environment DDPG 40 TD3 SAC 30 20 10 0 0 20 40 X (m)Post-Training Trajectory in Second Environment DDPG 40 TD3 SAC 30 20 10 0 0 20 40 x (m)Trained Policies Evaluated in Third Environment 40 30 DDPG Env1 Agent TD3 Env1 Agent 10 SAC Env1 Agent DDPG Env2 Agent TD3 Env2 Agent SAC Env2 Agent 0 20 0 40 x (m)than SAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The DRL agents trained in the first environment performed better than those trained in the second, as the characteristics of the evaluated environment are more similar to the first than the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The SAC agents are most robust to differences in environment characteristics with both achieving near identical performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The reward function weights and network hyperparameters can be further engineered for this application, and the agent trained over a longer period with more episode steps each episode iteration to learn an improved policy that explores the surrounding environment indefinitely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Bridging the simulation to reality gap to transfer policies learned in simulation to real-world robotic systems is a current area of active research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' The large number of episodes required to sufficiently train the agent renders simulation training an essential component for DRL in robotics applica- tions to minimize cost and possible physical damage caused by collisions during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Substantial computation cost is required for training, however, post-training implementation of DRL agents is significantly less expensive, which makes DRL a powerful tool for real-time AGV motion planning and control in environments without a-priori maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' CONCLUSIONS This paper presented an ubiquitous DRL architecture for intelligent AGV exploration without a-priori maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Three actor-critic DRL algorithms, DDPG, TD3 and SAC, were trained in two environments of varying complexity, and further evaluated in a third with no prior knowledge of map characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' Simulation results demonstrate the effec- tiveness of the proposed DRL architecture, reward function and training conditions for quick, efficient and collision- free AGV navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' SAC achieves the best performance, yielding trajectories that cover the highest distance, 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in 2017 IEEE intelligent vehicles symposium (IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' IEEE, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} +page_content=' 812–818.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9E2T4oBgHgl3EQfqQgh/content/2301.04036v1.pdf'} diff --git a/MdE0T4oBgHgl3EQf0ALT/vector_store/index.faiss b/MdE0T4oBgHgl3EQf0ALT/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..1bc4667b69df0870b096dde8e8ace654246738ed --- /dev/null +++ b/MdE0T4oBgHgl3EQf0ALT/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:27008ac15577a275fb17191de8635e635a2e1738298c840b32ef6eb763103295 +size 2293805 diff --git a/N9FQT4oBgHgl3EQfWzb0/content/tmp_files/2301.13306v1.pdf.txt b/N9FQT4oBgHgl3EQfWzb0/content/tmp_files/2301.13306v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b95e6cfee22be7d327853d8ced7c53ac4828acdb --- /dev/null +++ b/N9FQT4oBgHgl3EQfWzb0/content/tmp_files/2301.13306v1.pdf.txt @@ -0,0 +1,3678 @@ +arXiv:2301.13306v1 [cs.GT] 30 Jan 2023 +Autobidders with Budget and ROI Constraints: +Efficiency, Regret, and Pacing Dynamics∗ +BRENDAN LUCIER, Microsoft Research, USA +SARATH PATTATHIL, Massachusetts Institute of Technology, USA +ALEKSANDRS SLIVKINS, Microsoft Research, USA +MENGXIAO ZHANG, University of Southern California, USA +We study a game between autobidding algorithms that compete in an online advertising platform. Each au- +tobidder is tasked with maximizing its advertiser’s total value over multiple rounds of a repeated auction, +subject to budget and/or return-on-investment constraints. We propose a gradient-based learning algorithm +that is guaranteed to satisfy all constraints and achieves vanishing individual regret. Our algorithm uses only +bandit feedback and can be used with the first- or second-price auction, as well as with any “intermediate" +auction format. Our main result is that when these autobidders play against each other, the resulting expected +liquid welfare over all rounds is at least half of the expected optimal liquid welfare achieved by any allocation. +This holds whether or not the bidding dynamics converges to an equilibrium and regardless of the correlation +structure between advertiser valuations. +1 +INTRODUCTION +As the rules and algorithms governing online markets increase in complexity and scale, platforms +are increasingly providing ML-powered interfaces to help users interact and navigate efficiently. A +prominent example is the rise of autobidding, a service provided by advertising platforms to help +advertisers automate their campaigns. The advertiser only needs to specify high-level objectives +and constraints. A typical example might be “maximize the number of clicks received, subject to +spending at most $1000 per day, at most $2 per click on average, and no more than $10 for any +one click.” This example encodes three different constraints on the outcome: a budget constraint, +an average return-on-investment (ROI) constraint, and a marginal ROI (or value) constraint. The +autobidder then uses an online learning algorithm to tune a detailed advertising campaign so as +to solve this optimization problem on the advertiser’s behalf. While each individual ad impres- +sion is sold by auction, the actual management of individual bids is delegated to the autobidding +algorithms. +Autobidding is popular with advertisers and all major online ad platforms now provide this +service. This success is owed in part to the effectiveness of online learning methods for bid tun- +ing, which has received substantial attention in the academic literature in recent years. Initial +work focused on the problem of tuning bids subject to an aggregate budget constraint, a.k.a., bud- +get pacing [10, 18]. More recent work concerns ROI (or ROAS, Return on Ad Spend) constraints +[8, 22, 27, 32, 33]. The result is a variety of well-understood learning algorithms that can be used +by an autobidder to achieve vanishing regret in any stationary (or near-stationary) auction envi- +ronment. Correspondingly, autobidder interfaces supporting both budget and ROI constraints are +now ubiquitous. +∗Some of the results have been obtained while S. Pattathil and M. Zhang were research interns at Microsoft Research. +The authors are grateful to Bach Ha for many conversations that informed our perspective, and to Sidharth Satya for +providing research support. +Authors’ addresses: Brendan Lucier, Microsoft Research, Cambridge, MA, USA, brlucier@microsoft.com; Sarath Pattathil, +Massachusetts Institute of Technology, Cambridge, MA, USA, sarathp@mit.edu; Aleksandrs Slivkins, Microsoft Research, +New York, NY, USA, slivkins@microsoft.com; Mengxiao Zhang, University of Southern California, Los Angeles, CA, USA, +mengxiao.zhang@usc.edu. + +2 +Brendan Lucier, Sarath Patathil, Aleksandrs Slivkins, and Mengxiao Zhang +The dominance of autobidding in online advertising platforms gives rise to a new concern. Since +autobidders are now predominantly competing against each other, one must worry what happens +when these algorithmic agents interact. What are the implications for the overall health of the +market as a whole, in terms of aggregate objectives such as efficiency and stability? +These concerns are related to algorithmic collusion, whereby algorithmic agents do not converge +to competitive equilibrium play but rather systematically behave in a way that reduces market +efficiency (either by converging to a less efficient outcome, or by failing to converge entirely). Such +“collusion" could be a non-malicious unintended artifact of the agents’ interactions. Algorithmic +collusion has been observed in simulation studies for a variety of repeated auction scenarios [3, +14, 28]. It is also documented to occur in practice for related settings like algorithmic pricing +on Amazon [36]. Given the potential market-wide impact of such interactions, it is important to +account for unintended emergent behaviors that may arise when autobidders compete. +One solution would be to design autobidding algorithms that always converge to equilibria of +the “bidding game" that they are playing against each other. This would help address our concerns +because equilibrium outcomes in these games are known to be approximately efficient. The state +of the art in guarantees concern liquid welfare,1 and state that a pure Nash equilibrium of the +bidding game with budget and/or ROI constraints attains expected liquid welfare at least half of +the optimum, and this bound is tight [1, 5]. Unfortunately, finding an equilibrium of the bidding +game is PPAD-hard, even for the special case of maximizing value subject to a budget constraint +in a second-price auction [15]. We therefore should not expect to design a learning algorithm that +is guaranteed to jointly converge to an equilibrium when deployed by autobidders. This leaves us +with the challenge of analyzing the dynamics of the learning path taken by autobidders who may +not converge as they adjust bids in competition with one another. +Our Contributions. We present a novel algorithm for online bidding with budget and ROI con- +straints. We note that this by itself is not new; it has recently been shown that dual-based methods +can achieve vanishing regret while (approximately) satisfying such constraints [11, 12, 22, 26]. The +critical new feature of our algorithm is that in addition to achieving vanishing individual regret, it +also yields a state-of-the-art aggregate guarantee for the entire market. That is, when all bids are +controlled by autobidders deploying our algorithm, the resulting expected liquid welfare is at least +half of the optimal. This is the best possible bound even for pure equilibria in second-price auctions +with budgets [1, 5]. However, our aggregate guarantee does not rely on convergence to equilibrium; +rather, we directly analyze the outcomes on the dynamic learning path of the algorithms. +While we view this aggregate guarantee as the main result of the paper, we emphasize that it +is crucial to also guarantee vanishing regret for each individual advertiser. Even if the platform’s +goal when designing an autobidder is to maximize the overall market efficiency, individual guar- +antees are needed to convince the advertisers to make use of this service, rather than place bids +themselves. +Our results hold for a broad class of auction formats, including first-price and second-price +auctions, and allow impression types (which determine, e.g., click rates) to be drawn randomly +in each round and potentially be correlated across agents. Our algorithm is guaranteed to satisfy +all constraints ex post with probability 1 (not just with high probability or with small expected +violation). Further, it only requires bandit feedback from the underlying auction (i.e., only the +outcome for the actual bid submitted, not the counterfactual outcomes for the alternative bids). +The latter is important because even when advertising platforms provide sufficient feedback to +infer the counterfactual allocations of ad impressions (which they do not always), it can be difficult +to accurately model whether a particular ad in the counterfactual slate would be clicked by a user. +1Liquid welfare is a standard notion of welfare under constraints; we discuss it more below. + +Autobidders with Budget and ROI Constraints +3 +Our algorithm updates bids using a variation of stochastic gradient descent (SGD). Essentially, +the autobidder’s goal is to maximize her bids (and therefore the total value won) subject to a +collection of constraints that would be violated if bids were too high. When there is only a bud- +get constraint, a common idea is to use an SGD-based update to learn the constraint-pacing bid: +the bid that would cause this constraint to become tight in expectation. With both budget and +ROI constraints, one could strive to maintain a constraint-pacing bid for each constraint, and +somehow aggregate these per-constraint bids into one. For example, Balseiro et al. [11] employ +a primal-dual framework that, in effect, interpolates and places more weight on constraints that +bind more tightly; Feng et al. [22] apply this directly to the setting of ROI and budget constraints. +Our approach has a similar flavor, but aggregates the per-constraint bids differently: we myopi- +cally choose the per-constraint bid that is largest in a given round, then apply a gradient step to +both per-constraint bids using the observed outcome. This differs from a typical SGD-based ap- +proach in that the smaller per-constraint bid is updated using the outcome from the larger bid. +However, as it turns out, this approach maintains an important invariant: the multiplier for each +constraint encodes the total slack (or violation) of that constraint up to the current round. We will +use this to show that all constraints will be satisfied with certainty in all rounds. This invariant +will also be crucial for establishing our efficiency bound, as it will allow us to track value and pay- +ments in a manner that is agnostic to the details of the (potentially chaotic and non-convergent) +bid dynamics. +We measure market efficiency via liquid welfare, the maximum amount the agents are willing to +pay for the allocations that they receive. This is the appropriate notion of welfare in settings like +ours where the agents (autobidders) do not typically have an objective measured in dollars, but +rather seek to maximize value subject to monetary constraints on how much can be paid for dif- +ferent outcomes.2 Liquid welfare generalizes the standard notion of welfare for agents with quasi- +linear preferences in money (or, equivalently, only a constraint on the maximum willingness-to- +pay in each round), as well as the original introduction of liquid welfare for budget constraints [21]. +Our individual guarantees hold for an adversarial environment, without any assumptions on +the other agents. We attain vanishing regret relative to the constraint-pacing sequence: informally, +a sequence of bids that maximize per-round value under the time-averaged constraints. Note that +vanishing-regret results are impossible against the standard benchmark of best fixed bid (as dis- +cussed further in Section 1.1). To account for the changing environment, our regret bound depends +on the path-length of the constraint-pacing sequence. Specialized to a stochastic environment +(where the other agents’ bids are drawn from a fixed joint distribution), we obtain vanishing re- +gret against the best fixed bid. +Our emphasis is on autobidding algorithms that can satisfy both ROI and budget constraints. +Our regret and liquid welfare bounds extend prior analysis in the budget-pacing setting with a +budget constraint only, where it has been shown that the online bidding algorithm of Balseiro and +Gur [10] achieves both aggregate and individual guarantees [23]. Handling the ROI constraint, +and particularly both constraints jointly, introduces a variety of new technical challenges and +motivates our proposed algorithm, as we discuss in Section 4.1 and Section 4.2. +1.1 +Further Related Work +Our work builds on the recent literature analyzing online algorithms for bidding under constraints. +For online bidding subject to a budget constraint, Borgs et al. [13] analyze bidding dynamics un- +der a multiplicative update rule and establish convergence for first-price auctions. Balseiro and +2When/if the advertisers’ objective is expressible in dollars, utilitarian welfare could also be a reasonable objective. How- +ever, it is subject to strong impossibility results [21], which makes it less suitable for theoretical study. + +4 +Brendan Lucier, Sarath Patathil, Aleksandrs Slivkins, and Mengxiao Zhang +Gur [10] consider a different update rule under second-price auctions and show that it converges +under some additional convexity assumptions and guarantees vanishing individual bidder regret. +Balseiro et al. [11] consider a variation of this approach using online mirror descent (OMD) and +extends the guarantees to repeated truthful auctions without requiring any additional convexity +assumptions. Notably, their individual regret bound applies beyond stochastic settings to adver- +sarial environments, with a loss that grows with the deviation from stochasticity. +For online bidding subject to more general constraints, including the ROI constraints we con- +sider, Bernasconi et al. [12] use convex duality to achieve vanishing regret in stochastic environ- +ments, while satisfying the constraints with high probability. A general dual-based optimization +framework for autobidding subject to ROI and budget constraints was proposed and evaluated in +Gao et al. [24]. Golrezaei et al. [26] consider budget and ROI constraints, and achieve low regret +in a stochastic environment with bounds on expected constraint violations. Feng et al. [22] extend +the OMD approach to achieve vanishing regret while satisfying budget and ROI constraints with +vanishing expected violation. Our algorithm’s learning guarantees are incomparable: it satisfies +the ROI and budget constraints with probability 1, with regret rate 푂(푇 7/8) (whereas 푂(푇 1/2) is +achievable for relaxed constraints [22, 26]). +A related line of work studies pacing equilibria: market equilibria in a single-shot game that +abstracts repeated auctions when bidders have global constraints such as budgets and/or ROI. For +second-price auctions, Conitzer et al. [18] characterized the corresponding equilibria. Aggarwal +et al. [1] show that under a broad class of constraints (including budget and ROI constraints), the +expected liquid welfare obtained at any equilibrium is at least half of the optimal liquid welfare. +Babaioff et al. [5] provide a similar 2-approximation result in settings with “soft constraints” where +agents have a separable and convex disutility for spending money. Both results generalize to the +case of budget constraints in a second-price auction, where finding a pure Nash equilibrium is +known to be PPAD-hard [15]. In contrast to this work, our analysis of liquid welfare does not rely +on convergence to equilibrium, but rather applies directly to bidding dynamics that arise from our +autobidding algorithm. +Conitzer et al. [17] extended pacing equilibria to first-price auctions, showed that the equilib- +rium is essentially unique, and analyzed its properties. Balseiro et al. [9] consider an alternative +equilibrium notion for a broad class of auctions, in which agents are not constrained to pacing +but instead can make their bids arbitrarily contingent on realized impression values. They present +a revenue-equivalence result to bound liquid welfare at any equilibrium subject to a budget con- +straint. It has been widely noted that linear pacing is equivalent to optimal contingent bidding for +truthful auctions [5, 10] but not more generally. We take the linear pacing approach, interpreted +as the autobidder placing a bid that scales linearly with the realized impression value. +Learning theory. Repeated bidding under budget is a special case of bandits with knapsacks +(BwK), a multi-armed bandit problem under global constraints [2, 6, 31], see Chapter 10 in [38] for +a survey. BwK problems in adversarial environments do not admit regret bounds: instead, one is +doomed to approximation ratios, even against a time-invariant benchmark and even in relatively +simple examples [31]. A similar impossibility result is derived in [10] specifically for repeated +budget-constrained bidding in second-price auctions. The literature on BwK does not handle “cov- +ering constraints" such as ROI,3 and is not concerned with aggregate guarantees such as ours. +Stochastic gradient descent is a standard algorithm in online convex optimization [30]. +Convergence of learning algorithms in repeated games is well-studied. When algorithms have +vanishing regret in terms of cumulative payoffs, the average play – time-averaged distribution +over chosen actions – converges to a (coarse) correlated equilibrium [4, 29, 35], and this implies +3With a rare exception of Agrawal and Devanur [2], who handle covering constraints for stochastic environments. + +Autobidders with Budget and ROI Constraints +5 +welfare bounds for various auction formats in the absence of budget or ROI constraints [37]. In +contrast, for repeated auctions with budgets, low individual regret on its own does not imply any +bounded approximation for liquid welfare [23]. Convergence in the last iterate is more challeng- +ing. While strong negative results are known even for two-player zero-sum games [7, 16, 34], a +recent line of work [19, 20, 25, 39] achieves last-iterate convergence under full feedback and sub- +stantial convexity-like assumptions, using two specific regret-minimizing algorithms. To the best +of our understanding, these positive results do not apply to repeated auctions with budget or ROI +constraints. +2 +MODEL AND PRELIMINARIES +We study a repeated auction game played by a collection of 푛 bidding agents. At each time 푡 = +1, 2, . . . ,푇, the seller (or platform) has a single unit of good available to sell. An allocation profile is +a vector 풙 = (푥1, . . . ,푥푛) ∈ 푋 ⊆ [0, 1]푛 where 푥푘 is the quantity of the good allocated to agent 푘. +An allocation sequence is a sequence of allocation profiles (풙1, . . . , 풙푇 ) where 풙풕 = (푥1,푡, . . . ,푥푛,푡) +is the allocation profile at time 푡.4 +For each 푡, the good available for sale at time 푡 has a value 푐푘,푡 ∈ [0, 1] for agent 푘. We can think +of 푐푘,푡 as a click probability for agent 푘 and impression 푡.5 We emphasize that 푐푘,푡 is measured in +clicks, not dollars.6 For each 푡, the value profile c푡 = (푐1,푡, . . . ,푐푛,푡) is drawn from a distribution +퐹 independently across time periods. Note that 퐹 is not necessarily a product distribution, so the +value held by different agents can be arbitrarily correlated within each round. +Auctions. At each time 푡 the good is allocated using an auction that proceeds as follows. Each +agent 푘 submits a bid 훽푘,푡 ≥ 0, which can be interpreted as a bid (in dollars) per unit of value. +All agents submit bids simultaneously. The value profile is then realized, and this determines each +agent’s effective bid 푏푘,푡 = 푐푘,푡훽푘,푡. The auction is defined by an allocation rule 풙 and a payment +rule 풑, where 풙(풃) ∈ [0, 1]푛 is the allocation profile generated under a bid profile 풃, and 푝푘(풃) ≥ 0 +is the payment made by agent 푘. Allocation and payment rules are always weakly monotone in +bids, meaning that 푥푘 (푏푘, b−푘) and 푝푘 (푏푘, b−푘) are weakly increasing in 푏푘 for any b−푘. Given an +implied realization of bids, we will often write 푥푘,푡 and 푝푘,푡 to mean the allocation and payment of +agent 푘 in round 푡. +The auction rules we consider will satisfy the following properties. First, winners are always +selected from among those agents with the highest effective bid, with ties resolved arbitrarily. Sec- +ond, each agent’s payment per unit received lies between the highest and second-highest effective +bids. This class includes both first-price and second-price auctions, as well as any auction whose +payments lie between the first and second price. Some of our results actually apply more generally +to any core auction; we define these in Section 4.1. +Constraints and Objectives. Each agent 푘 has a collection of constraints that their payments and +allocations must satisfy. These are as follows: +(1) Value constraint 푣푘: in each round 푡, agent 푘 cannot pay more than 푣푘 per unit of value +received. That is, 푝푘,푡 ≤ 푣푘푐푘,푡푥푘,푡. +(2) ROI constraint 푤푘: in total over all rounds, agent 푘 cannot pay more than 푤푘 per unit of +value received. That is, � +푡 푝푘,푡 ≤ 푤푘 +� +푡 푐푘,푡푥푘,푡. +4We include a full notation table in Appendix A for convenience. +5One could equivalently interpret this as a conversion rate, expected revenue lift, or any other metric of value that depends +on advertiser and impression type. +6We find it most natural in this context to measure value in non-monetary units and express preferences for payments as +constraints. However, we will show in Section 2.1 that this is mathematically equivalent to measuring value in dollars. + +6 +Brendan Lucier, Sarath Patathil, Aleksandrs Slivkins, and Mengxiao Zhang +(3) Budget constraint 퐵푘: the total payment made by agent 푘 over all rounds cannot exceed 퐵푘. +That is, � +푡 푝푘,푡 ≤ 퐵푘. +The objective of each agent푘 is to maximize the effective value received over all rounds, � +푡 푐푘,푡푥푘,푡, +subject to the constraints being satisfied. +We emphasize that all of these constraints bind ex post, and must be satisfied on every realiza- +tion. We observe that since an agent never pays more than its bid, there always exists an agent +strategy that guarantees all constraints will be satisfied for any realization of values and the bids +of others. In particular, the first constraint is always satisfied if 훽푘,푡 ≤ 푣푘 for all 푡. The second con- +straint is always satisfied if 훽푘,푡 ≤ 푤푘 for all 푡. The third constraint is always satisfied if 훽푘,푡 ≤ 퐵푘/푇 +for all 푡. Bidding the minimum of these amounts in every round would therefore necessarily satisfy +all constraints, though of course this may result in low objective value. +2.1 +Interpretation: Constraints and Pacing +We now take a moment to discuss some alternative interpretations and notation for our con- +strained optimization problem. Since 푣푘 is a constant, the agent’s objective is equivalent to maxi- +mizing � +푡 푣푘푐푘,푡푥푘,푡. If we define 푣푘,푡 = 푣푘푐푘,푡 then we can rewrite the agent’s goal as maximizing +� +푡 푣푘,푡푥푘,푡, where (푣1,푡, . . . , 푣푛,푡) is drawn from a distribution each round. We can think of 푣푘,푡 as +an implied maximum willingness to pay. Define 푣 ≜ max푘 푣푘, so that 푣푘,푡 ≤ 푣 for all 푘 and all 푡. +Next, note that if 푤푘 ≥ 푣푘 then the ROI constraint is implied by the value constraint. We can +therefore assume without loss that 푤푘 ≤ 푣푘. If we define 훾푘 = 푣푘/푤푘 ≥ 1 then the ROI constraint +becomes 훾푘 +� +푡 푝푘,푡 ≤ � +푡 푣푘,푡푥푘,푡, or equivalently (� +푡 푣푘,푡푥푘,푡)/(� +푡 푝푘,푡) ≥ 훾푘. This formulation +motivates the name “ROI constraint,” and we will tend to use 훾푘 (rather than 푤푘) to express its +parameter. Finally, for convenience we write 휌푘 = 퐵푘/푇 for agent 푘’s per-round budget constraint. +Bidding Algorithms. Each agent employs a bidding algorithm to determine its bids. Given the +constraint parameters 푣푘, 퐵푘, and 훾푘 and the history of bids and realized outcomes up to round 푡, +the algorithm determines the bid 훽푘,푡 for round 푡. +We note that since each agent’s payment per unit received is always at least the next-highest +effective bid, it is without loss to restrict attention to bids 훽푘,푡 ≤ 푣푘, since choosing any bid 훽푘,푡 > 푣푘 +is a dominated action for the autobidder.7 With this in mind, we can follow the literature on bid +pacing and define 휇푘,푡 = �푣푘/훽푘,푡 +� − 1, so that 훽푘,푡 = 푣푘/(1 + 휇푘,푡). That is, in each round we +think of the bidding algorithm as choosing a multiplier 휇푘,푡 ≥ 0, where 0 corresponds to bidding +the maximum amount 푣푘, and as 휇푘,푡 grows the resulting bid shrinks. The algorithm’s goal is +to optimize 휇푘,푡 ≥ 0 subject to the ROI and budget constraints, since the value constraint will +necessarily be satisfied. We emphasize that this formulation is without loss of generality, as any +undominated choice of 훽푘,푡 can be expressed in this manner. +2.2 +Aggregate Performance Metric: Liquid Welfare +Liquid welfare is a measure of welfare in settings where agents utilities are described by pay- +ment constraints. Intuitively, an agent’s liquid welfare for an allocation sequence is the agent’s +maximum willingness to pay for the allocation. This generalizes the common notion of welfare in +quasi-linear environments. In our setting with value, budget, and ROI constraints, the definition +of liquid welfare specializes as follows. +7Any bid 훽푘,푡 > 푣푘 is dominated by a bid of 푣푘: either the agent loses the auction or would anyway have won at a bid of +푣푘, in which case a bid of 푣푘 does just as well; or the agent wins at bid 훽푘,푡 but loses at bid 푣푘, in which case she must be +paying more than 푣푘 per unit received in violation of the value constraint. + +Autobidders with Budget and ROI Constraints +7 +Definition 2.1. Given a sequence of value profiles v = (푣푘,푡) ∈ [0, 1]푛푇 and any sequence +of feasible allocations x = (푥푘,푡) ∈ [0, 1]푛푇, the liquid value obtained by agent 푘 is 푊푘(x) = +min +� +퐵푘, +1 +훾푘 +�푇 +푡=1 푣푘,푡푥푘,푡 +� +. The liquid welfare of allocation sequence x ∈ 푋푇 is푊 (x) = �푛 +푘=1푊푘(x). +We emphasize that liquid welfare depends on the allocations, but not on the agents’ payments. +However, it follows immediately from the definitions that as long as all constraints are satisfied, an +agent’s total payment is a lower bound on her liquid welfare. That is, the agent’s actual observed +payment cannot be greater than her maximum willingness to pay. +Observation 2.1. Fix any sequence of value profiles and outcomes (allocations and payments) +such that the value, budget, and ROI constraints are satisfied for agent 푘. Then 푊푘(x) ≥ � +푡 푝푘,푡. +Our objective of interest is the expected liquid welfare obtained by the platform over any random- +ness in the value sequence and the agents’ bidding strategies (and hence the resulting randomness +in the allocation sequence). Since the bid placed in one round can depend on allocations obtained +in the previous rounds, we define a mapping from the entire sequence of푇 value profiles to an allo- +cation sequence. An allocation sequence rule is a function x: [0, 1]푛푇 → 푋푇 , where 푥푘,푡 (v1, . . . , v푇 ) +is the allocation obtained by agent 푘 in round 푡. Then the expected liquid welfare of allocation +sequence rule x is Ev1,...,v푇 ∼퐹 [푊 (x(v1, . . . , v푇 ))]. +3 +WARMUP: ROI CONSTRAINTS AND UNLIMITED BUDGET +In this section we warm up with the special case where 퐵푘 = ∞ for all 푘, meaning that there are no +budget constraints. While bidding algorithms are already known for this problem, we will build +intuition by presenting a new algorithm with properties that will be useful when we adapt it to +our general setting in Section 4. +As we focus on a single agent, we will drop the subscripts 푘 from our notation for the remainder +of this section. In the absence of budget constraints, the agent’s goal is to maximize � +푡 푣푡푥푡 subject +to the constraint � +푡 푣푡푥푡 − 훾 � +푡 푝푡 ≥ 0. +Example 3.1. Consider a repeated second-price auction with 푣푡 = 1 for every 푡 and 훾 = 2. In this +case the agent’s ROI constraint is � +푡 푥푡 − 2 � +푡 푝푡 ≥ 0. Suppose the competing bids are stochastic +and stationary, with the highest bid of a competing agent being either 1/4 or 3/4 each round with +equal probability. +Suppose that the bidding agent chooses 휇푡 = 0 for every 푡, meaning that it bids 푏푡 = 푣푡 = 1. +In this case the agent wins every round and pays either 1/4 or 3/4, for an expected payment of +1/2. The constraint is satisfied in expectation but may be violated on some realizations. Indeed, +in the (unlikely) event that the competing bid is 3/4 every round, the only way to satisfy the ROI +constraint would be to lose every round. On the other hand, if the competing bid is 1/4 on at least +half of the instances then it is optimal to win every round. +As discussed in Section 2, setting 휇푡 = 훾 − 1 in every round (i.e., bidding 푣푡/훾) is guaranteed +to satisfy the ROI constraint. But, as in the example above, the optimal choice of 휇푡 may be lower. +The next lemma shows that ROI is monotone in 휇푡 for our auctions. That is, higher bids result in +lower average return on investment. This lets us think of the agent as trying to reduce 휇 as much +as possible subject to the ROI constraint. +Lemma 3.1. Fix any auction in our class, any agent 푘, and any bids b−푘 of the other agents, and +write 푥푡 (휇) and 푝푡 (휇) for the allocation and payment that results when agent 푘 selects multiplier 휇. +Then 푣푡푥푡 (휇) − 훾푝푡 (휇) is weakly increasing in 휇, for 휇 ∈ [0,훾 − 1]. + +8 +Brendan Lucier, Sarath Patathil, Aleksandrs Slivkins, and Mengxiao Zhang +This motivates us to consider Algorithm 1, which initially takes the safe action 휇1 = 훾 − 1 but +updates 휇푡 online in response to auction feedback. In each round 푡 it places bid 푏푡 = 푣푡/(1 + 휇푡).8 +If the observed allocation 푥푡 and payment 푝푡 in round 푡 are such that 푣푡푥푡 > 훾푝푡 then the ROI +constraint is satisfied with room to spare. This suggests the bid was lower than necessary, so the +algorithm reduces 휇 by an amount proportional to 푣푡푥푡 −훾푝푡. Likewise, if 푣푡푥푡 < 훾푝푡, then the ROI +constraint was violated in round 푡 so the algorithm responds by increasing 휇 proportionally to the +violation. This can be interpreted as stochastic gradient descent (SGD): if 휇푡 is such that the ROI +constraint is satisfied in expectation then the expected update is 0. We will make this connection +with SGD more precise in Section 4.2. +ALGORITHM 1: Bidding Under ROI Constraint +Input: ROI constraint parameter 훾 +Initialization: 휇1 = 훾 − 1 and learning rate 휂 +for 푡 = 1, 2, · · · ,푇 do +Set bid 푏푡 = +푣푡 +1+max{휇푡,0} +Observe allocation 푥푡 ∈ [0, 1] and payment 푝푡 +Update the multiplier 휇푡+1 = 휇푡 − 휂(푣푡푥푡 − 훾푝푡) +Algorithm 1 appears myopic at first glance, always updating its bids in response to the latest +outcome. However, we note that 휇푡 implicitly encodes the status of the aggregate ROI constraint +up to round 푡. Indeed, an immediate implication of the update rule is that 휇푡 is 휇1 minus a term +proportional to � +휏<푡 (푣휏푥휏 − 훾푝휏), which is precisely the aggregate slack (or violation) of the ROI +constraint up to time 푡. A small value of 휇푡 (i.e., a high bid) therefore occurs only if there is sub- +stantial slack in the ROI constraint up to round 푡. +What does this mean for the performance of Algorithm 1? Note that the bidding agent may +receive less than the desired return on investment 훾 in any given round. However, as we now +show, the algorithm is guaranteed, with probability 1, to satisfy the ROI constraint in aggregate +over its 푇 rounds (and indeed, over any prefix of the 푇 rounds). Intuitively, as the ROI constraint +gets closer to being violated in aggregate, 휇푡 gets closer to 휇1 = 훾 − 1, the “safe” choice at which +the ROI constraint will necessarily be satisfied each round. +Lemma 3.2. Fix any (possibly adversarial) mapping from sequences of bids to sequences of alloca- +tions and payments such that 푝푡 ≤ 푏푡푥푡 for every 푡, and suppose 휂 ≤ 1/¯푣. Then for the allocations +and payments resulting from applying Algorithm 1, we have � +푡 푣푡푥푡 ≥ 훾 � +푡 푝푡. +An immediate corollary of Lemma 3.2 is that, for every input sequence, the sequence of multipli- +ers {휇푡 } will satisfy 휇푡 ≤ 훾−1 for every푡.9 As we show in the next section, this will imply high liquid +welfare when multiple agents use (a generalization of) this algorithm. As it turns out, Lemma 3.2 +can also be used to show that Algorithm 1 achieves vanishing regret relative to the best choice +of 휇 in hindsight. We prove this formally in the next section in a more general setting, but for +now let us briefly describe the intuition. Since the ROI constraint is satisfied with probability 1 by +Lemma 3.2, any loss in value must come from bidding too low relative to the optimal fixed strategy +in hindsight, say 휇∗. However, the expected update to 휇푡 turns out to be the gradient of a function +that, on the range [휇∗,훾 − 1], is (a) convex and (b) closely related to the value function. Standard +SGD analysis then allows us to bound the total loss due to rounds where 휇푡 is larger than 휇∗. +8With one small caveat: 휇푡 could be negative in some rounds, in which case we treat it as 0 when setting the bid 푏푡. +9The difference 휇1−휇푡 is proportional to the slack in the ROI constraint up to time 푡, which by Lemma 3.2 is never negative. + +Autobidders with Budget and ROI Constraints +9 +4 +BIDDING UNDER ROI AND BUDGET CONSTRAINTS +We now turn to the general problem with both ROI and budget constraints. We study an extension +of our previous algorithm to this setting, listed as Algorithm 2. The algorithm now keeps track +of two multipliers, 휇푅 and 휇퐵, corresponding to the ROI and budget constraints respectively. At +each round, the algorithm will place a bid using whichever of the multipliers is more conservative; +i.e., whichever results in the lower bid. Each of the multipliers is then updated according to the +realized allocation and payment, including the multiplier that did not correspond to the placed bid. +Multiplier 휇푅 +푘,푡 is updated in the same way as Algorithm 1. The idea behind our update rule for +multiplier 휇퐵 +푘,푡 is similar: each round we compare the observed payment 푝푘,푡 with 휌푘 = 퐵푘/푇, the +target per-round payment according to budget constraint 퐵푘, and we update 휇퐵 +푘,푡 proportional to +the difference where 휂푘,푅,휂푘,퐵 > 0 are the corresponding learning rate. +휇푅 +푘,푡+1 = 휇푅 +푘,푡 − 휂푘,푅(푣푘,푡푥푘,푡 − 훾푘푝푘,푡), +휇퐵 +푘,푡+1 = 휇퐵 +푘,푡 − 휂푘,퐵(휌푘 − 푝푘,푡). +ALGORITHM 2: Bidding Under ROI and Budget Constraints (for agent 푘) +Input: per-round budget constraint 휌푘 > 0 and ROI constraint parameter 훾푘. +Initialization: 휇푅 +푘,1 = 훾푘 − 1, 휇퐵 +푘,1 = 푣 +휌 − 1 and learning rate 휂푘,푅,휂푘,퐵 > 0. +for 푡 = 1, 2, · · · ,푇 do +Calculate 휇푘,푡 = max{휇푅 +푘,푡, 휇퐵 +푘,푡, 0}. +Receive value 푣푘,푡 and set the bid 푏푘,푡 = +푣푘,푡 +1+휇푘,푡 . +Receive the allocation 푥푘,푡 ∈ [0, 1] and the payment 푝푘,푡. +Update the ROI-multiplier 휇푅 +푘,푡+1 = 휇푅 +푘,푡 + 휂푘,푅(훾푘푝푘,푡 − 푣푘,푡푥푘,푡). +Update the budget-multiplier 휇퐵 +푘,푡+1 = 휇퐵 +푘,푡 + 휂푘,퐵 (푝푘,푡 − 휌푘). +We note that in the case where the budget constraint is infinite (i.e., 퐵푘 and hence 휌푘 is +∞), +this update rule yields 휇퐵 +푘,푡 < 0 for every round 푡. Thus, this algorithm reduces to Algorithm 1 +in Section 3. On the other hand, if 훾푘 = 1 and hence there is effectively no ROI constraint, then +we will necessarily have 휇푅 +푘,푡 ≤ 0 for every round 푡 (since 푝푘,푡 ≤ 푏푘,푡푥푘,푡 ≤ 푣푘,푡푥푘,푡). The resulting +algorithm is nearly identical to the one proposed by Balseiro and Gur [10] for bidding subject to a +budget constraint. We can therefore view Algorithm 2 as a generalization of both algorithms. +The key insight behind Algorithm 2 is that the multipliers 휇푅 +푘,푡 and 휇퐵 +푘,푡 encode the cumula- +tive slack in the ROI and budget constraints, respectively, up to time 푡. Similar to our analysis +of Algorithm 1, this implies that Algorithm 2 will necessarily satisfy all of its constraints ex post +with probability 1. Formally, we have the following lemma. +Lemma 4.1. Fix any agent 푘 and any (possibly adversarial) mapping from sequences of bids to +sequences of allocations and payments such that 푝푘,푡 ≤ 푏푘,푡푥푘,푡 for every 푡, and suppose 휂푘,푅 ≤ +1 +푣,휂푘,퐵 ≤ min{ 1 +휌푘 , 1 +푣 }. Then for the allocations and payments resulting from applying Algorithm 2, +we have � +푡 푣푘,푡푥푘,푡 ≥ 훾푘 +� +푡 푝푘,푡 and � +푡 푝푘,푡 ≤ 퐵푘. +Also, similar to Algorithm 1, another implication of this interpretation of 휇푅 +푘,푡 and 휇퐵 +푘,푡 is that +these multipliers are never higher than their “safe” levels. . +Lemma 4.2. Algorithm 2 with 휂푘,푅 ≤ 1 +푣 and 휂푘,퐵 ≤ 1 +푣 guarantees that for all 푡 ∈ [푇], 휇푅 +푘,푡 ≤ 훾푘 − 1 +and 휇퐵 +푘,푡 ≤ +푣 +휌푘 − 1. +The remainder of this section is dedicated to establishing our aggregate liquid welfare and in- +dividual regret guarantees for Algorithm 2. In Section 4.1 we prove that in expectation over the + +10 +Brendan Lucier, Sarath Patathil, Aleksandrs Slivkins, and Mengxiao Zhang +realization of values, Algorithm 2 always obtains at least half of the optimal liquid welfare in hind- +sight. Then in Section 4.2 we establish that the algorithm also satisfies strong individual regret +guarantees, even in non-stochastic settings where the optimal bid sequence has bounded path +length. +4.1 +Liquid Welfare Analysis of Algorithm 2 +In this subsection we prove that when all agents employ Algorithm 2 the resulting expected liquid +welfare is approximately optimal, where the expectation is over the realization of agent values. We +will actually show that the expected liquid welfare is at least half of the optimal ex-ante liquid wel- +fare (defined in Definition 4.1), which is an agent’s willingness to pay for her expected allocation +sequence. Note that this is a stronger benchmark compared to the optimal expected liquid welfare +Ev1,...,v푇 [푊 (x)] due to Jensen’s inequality (See Lemma D.3). +Definition 4.1. For any distribution 퐹 over valuation profiles and any allocation rule y : [0,푣]푛 → +푋, the ex-ante liquid value of agent 푘 is +푊 푘 (y, 퐹) := 푇 × min +� +퐵푘, 1 +훾푘 +Ev∼퐹 [푦푘 (v)푣푘] +� +, +(1) +and the ex-ante liquid welfare is given by 푊 (y, 퐹) = �푛 +푖=1푊 푘(y, 퐹). +Recall that to this point we have focused our attention on single-item auctions. Our liquid wel- +fare bound will actually apply to the following more general class of core auctions. +Definition 4.2. Given any downward-closed set 푋 ⊆ [0, 1]푛 of feasible allocations, an auction +with allocation rule 푥 : [0,푣]푛 → 푋 and payment rule 푝 : [0,푣]푛 → +R푛 +≥0 is a core auction if it is +• Welfare-maximizing: x(v) ∈ arg maxx∈푋 {� +푖 푣푖(x)} +• Individually rational: 푝푖(v) ≤ 푣푖(x(v)) for all 푖 +• Deviation-proof: for all 푆 ⊆ [푛] and y ∈ 푋, � +푖∉푆 푝푖(v) ≥ � +푖 ∈푆(푣푖(푦푖) − 푣푖(푥푖(v))) +Core auctions include first-price and second-price single-item auctions, but also more general +formats like generalized second-price auctions for multiple slots and separable click rates. See [23] +for further discussion. +We are now ready to state the main result of this subsection. +Theorem 4.1. Fix any core auction and any distribution 퐹 over agent value profiles. Suppose all +agents bid by employing Algorithm 2 with max{휂푘,푅,휂푘,퐵} ≤ +푣 +푣+휌푘 +√ +log(푣푛푇) +√ +푇 +, and write x : [0,푣]푛푇 → +푋푇 for the corresponding allocation sequence rule. Then for any allocation rule y : [0,푣]푛 → 푋, +Ev1,···v푇 ∼퐹 [푊 (x(v1, · · · , v푇 ))] ≥ 푊 (y, 퐹) +2 +− O(푛푣 +� +푇 log(푣푛푇)) +(2) +4.1.1 +Proof Intuition. Before going into the details of our proof of Theorem 4.1, let us give a sketch +of the high-level ideas. Fix some agent푘 and consider the liquid welfare obtained by agent푘 over all +푇 rounds. By definition, this liquid welfare is either the agent’s budget 퐵푘 or the sum of ROI-scaled +gained values +1 +훾푘 +� +푡 푣푘,푡푥푘,푡. The former is the easy case: since 퐵푘 is an upper bound on ex-ante +liquid welfare, if agent 푘’s liquid welfare is 퐵푘 then this is at least as good as the benchmark. So +the difficulty lies in handling the latter case. +In order to bound 1 +훾푘 +� +푡 푣푘,푡푥푘,푡, we consider the progression of the bidding multiplier 휇푘,푡 over +rounds 푡 = 1, 2, . . . ,푇. The multiplier may drift up and down over time and may not necessarily +converge. We will distinguish between rounds in which 휇푘,푡 lies above 훾푘 − 1 and rounds in which +휇푘,푡 lies below 훾푘 − 1. + +Autobidders with Budget and ROI Constraints +11 +Whenever 휇푘,푡 < 훾푘 − 1, our intuition is that agent 푘 is not bidding “too low." Indeed, the bid +of agent 푘 must be at least 푣푘,푡/훾푘 on any such round. So even if agent 푘 loses in this round, the +winning bidder(s) must be paying at least 푣푘,푡/훾푘. We can therefore charge any loss in liquid welfare +against the total revenue collected, which (by Observation 2.1) is itself at most the liquid welfare. +On the other hand, in any round 푡 where 휇푘,푡 > 훾푘 − 1, we know that 휇푘,푡 = 휇퐵 +푘,푡. This is because +the ROI multiplier 휇푅 +푘,푡 never lies above 훾푘 − 1 according to Lemma 4.2. Thus, over any contiguous +interval of rounds in which 휇푘,푡 > 훾푘 − 1, it must be the budget multiplier that is determining the +bid of agent 푘. Since any such interval must begin and end close to the threshold 훾푘 −1, the update +rule for 휇퐵 +푘,푡 implies that the total spend over the (say) 푡 rounds of that interval is very close to +푡 ×휌푘. As the optimal ex-ante liquid welfare cannot be more than 휌푘 per round, the obtained liquid +welfare must be comparable to the optimal liquid welfare over this interval. Thus, in every case, +we can relate the obtained liquid welfare to the benchmark. +There are some technical challenges to formalizing this intuition. Our very first step was to +condition on whether agent 푘’s liquid welfare is determined by her budget or by her ROI-scaled +gained value. However, this conditioning introduces correlations between rounds, and in particular +it impacts our assertion that the ex-ante liquid welfare is at most 휌푘 per round. We address this +by explicitly bounding the impact of such correlations and arguing that they are small with high +probability. This introduces the additive error term in the theorem statement. We note that similar +issues arise when analyzing the liquid welfare of the budget-pacing algorithm of Balseiro and +Gur [10]; our solution is a variation of an idea due to [23] for the budgeted case. +Another technical issue that arises is specific to handling budget and ROI constraints simulta- +neously. The intuition above does not carefully account for rounds in which 휇푘,푡 switches from +lying strictly below 훾푘 − 1 to strictly above 훾푘 − 1 or vice-versa. It turns out that these transition +rounds introduce error terms that can accumulate substantially; indeed, when we said above that +the total spend over an interval is very close to 푡 ×휌푘, this approximation can be off by up to 푣 +휌푘. +We handle this by considering separately those rounds in which 휇푘,푡 is very close to the boundary +훾푘 − 1, and directly relate the outcomes to what would occur precisely on the boundary itself. The +resulting error terms are yet another source of the additive error in the theorem statement. +4.1.2 +Proof of Theorem 4.1. We now turn to the formal proof of Theorem 4.1. We prove the the- +orem in three steps. We first show that with high probability, there is no significant correlation +between the progression of our algorithm up to time 푡 and the ex-ante benchmark evaluated in +future rounds. We then condition on this event and bound the liquid welfare obtained on a per- +instance basis. Finally, we take an expectation over realizations to obtain the desired bound on +expected liquid welfare. +Step 1: Bounds on ex-ante Allocate Rules. First, it is without loss of generality to consider only +the allocation rules y which satisfy +Ev∼퐹 +� 1 +훾푘 +푦푘 (v)푣푘 +� +≤ 휌푘. +(3) +This is because for any 푦 that violates this constraint, we can always decrease the allocation for +agent 푘 without affecting 푊 (y, 퐹). +We would actually like to make a stronger claim that Eq. (3) holds for every round 푡 in which +휇푘,푡 > 훾푘 − 1. To this end, we will show that, with high probability, the ex-ante optimal allocation +rule 푦 does not generate significantly different outcomes in rounds where 휇푘,푡 > 훾푘 −1 and rounds +where 휇푘,푡 ≤ 훾푘 − 1. + +12 +Brendan Lucier, Sarath Patathil, Aleksandrs Slivkins, and Mengxiao Zhang +For each agent 푘, define the following quantity: +푅푘 (v) ≜ +� +푡 +� +1{휇푘,푡 ≤ 훾푘 − 1} 1 +훾푘 +푦푘 (v)푣푘,푡 + +1{휇푘,푡 > 훾푘 − 1}휌푘 +� +. +(4) +We can then use the theory of concentration of martingales to establish that the following bound +on 푅푘 (v) holds with probability at least 1 − 1/(푣푛푇)2: +푅푘 (v) ≤ 휌푘 ·푇 + 푣 +� +푇 log(푣푛푇). +(5) +We prove Eq. (5) in Lemma D.1. Now, taking a union bound over all agents푘 ∈ [푛], with probability +at least 1 − 1/(푣푇)2, we have: +푅푘 (v) ≤ 휌푘 ·푇 + 푣 +� +푇 log(푣푛푇), +∀푘 ∈ [푛] +(6) +We say that a value realization is “good” if it satisfies Eq. (6). +Step 2: Liquid Welfare of “Good” Value Realizations. Fix any “good” value profile realization +푣. For any advertiser 푘 whose liquid welfare is 퐵푘, from Eq. (6), we know that: +푊푘 (v) = 퐵푘 ≥ 푅푘 (v) − 푣 +� +푇 log(푣푛푇). +(7) +Now we look at those agents 퐴 ⊆ [푛] for which the liquid welfare is strictly less than 퐵푘: +푊푘(v) = 1 +훾푘 +푇� +푡=1 +푥푘,푡푣푘,푡 < 퐵푘. +As we did in (7), we again wish to bound 푊푘 (v) with respect to 푅푘 (v). To that end, we will +derive a bound on 푊푘(v) that accounts for variation in 휇푘,푡. For notational convenience, let 휂푘 = +max{휂푘,푅,휂푘,퐵}. For each round 푡, Let 푆푡 ⊆ 퐴 denote the agents for whom 휇푘,푡 ≤ 훾푘 −1, and푇푡 ⊆ 푆푡 +for the agents for whom훾푘 −1−휂푘(푣 + 휌푘) < 휇푘,푡 ≤ 훾푘 −1. That is, 푆푡 are the agents bidding “high +enough,” and 푇푡 are the agents in 푆푡 that are “close to” the threshold value 훾푘 − 1 in round 푡. +Lemma 4.3. The following inequality is guaranteed if all agents 푘 ∈ [푛] apply Algorithm 2: +� +푘 ∈퐴 +1 +훾푘 +푇 +� +푡=1 +푥푘,푡푣푘,푡 ≥ +� +푘 ∈퐴 +푇 +� +푡=1 +� +1(푘 ∈ 푆푡) 1 +훾푘 +푥푘,푡푣푘,푡 − +1(푘 ∈ 푇푡)푝푘,푡 + +1(휇푘,푡 > 훾푘 − 1)휌푘 +� +(8) +Proof. Fix some agent 푘 ∈ 퐴. Divide the time interval [1,푇] into intervals (퐼1, 퐼2, . . . ) in the +following manner: each interval 퐼 = [푡1,푡2) is a minimal interval such that 휇푘,푡1 ≤ 훾푘 − 1 and +휇푘,푡2 ≤ 훾푘 − 1. That is, 휇푘,푡 > 훾푘 − 1 for all 푡1 < 푡 < 푡2. Note that according to Lemma 4.2, we know +that 휇푘,푡 = 휇퐵 +푘,푡 when 푡 ∈ (푡1,푡2). +We wish to bound 1 +훾푘 +� +푡 ∈퐼 푥푘,푡푣푘,푡 for each such interval 퐼. Note that if 휇푘,푡1 ≤ 훾푘 −1−휂푘(푣 +휌푘), +then we must have 푡2 = 푡1 + 1 (since 휇푘,푡1+1 ≤ 휇푡1 + 휂푘(푣 + 휌푘) ≤ 훾푘 − 1). Thus, when 휇푘,푡1 ≤ +훾푘 − 1 − 휂푘 (푣 + 휌푘), we have 1 +훾푘 +�푡2−1 +푡=푡1 푥푘,푡푣푘,푡 = +1 +훾푘 푥푘,푡1푣푘,푡1. +On the other hand, if 훾푘 − 1 − 휂푘 (푣 + 휌푘) < 휇푘,푡1 ≤ 훾푘 − 1, we have +훾푘 − 1 < 휇푘,푡2−1 = 휇퐵 +푘,푡2−1 = 휇퐵 +푘,푡1 + 휂푘 +푡2−2 +� +휏=푡1 +(푝푘,휏 − 휌푘) ≤ 훾푘 − 1 + 휂푘 +푡2−2 +� +휏=푡1 +(푝푘,휏 − 휌푘), +which means that �푡2−2 +휏=푡1 (푝푘,휏 − 휌푘) ≥ 0. Since 푝푘,푡 ≤ 푏푘,푡푥푘,푡 < +1 +훾푘 푥푘,푡푣푘,푡 for all 푡1 < 푡 < 푡2, we can +conclude that +1 +훾푘 +푡2−1 +� +푡=푡1 +푥푘,푡푣푘,푡 ≥ 1 +훾푘 +푥푘,푡1푣푘,푡1 + +푡2−1 +� +푡=푡1+1 +푝푘,푡 ≥ 1 +훾푘 +푥푘,푡1푣푘,푡1 − 푝푘,푡1 + (푡2 − 푡1 − 1)휌푘. + +Autobidders with Budget and ROI Constraints +13 +Summing over all time steps, we conclude that +1 +훾푘 +푇� +푡=1 +푥푘,푡푣푘,푡 ≥ +푇� +푡=1 +� +1(휇푘,푡 ≤ 훾푘 − 1) 1 +훾푘 +푥푘,푡푣푘,푡 +− +1(훾푘 − 1 − 휂푘(푣 + 휌푘) < 휇푘,푡 ≤ 훾푘 − 1)푝푘,푡 + +1(휇푘,푡 > 훾푘 − 1)휌푘 +� +. +Summing this inequality over all agents yields Eq. (8). +□ +Our next goal is to relate the terms in the right-hand side of Eq. (8) with the corresponding +terms in 푅푘 (v). Fix some round 푡. We will focus on the first two terms in the expression inside the +summation on the right hand side of Eq. (8). Consider the agents in 푇푡, which are the agents for +whom 훾푘 − 1 − 휂푘(푣 + 휌푘) < 휇푘,푡 ≤ 훾푘 − 1. We have +� +푘 ∈푇푡 +� 1 +훾푘 +푥푘,푡푣푘,푡 − 푝푘,푡 +� += +� +푘 ∈푇푡 +� 1 +훾푘 +푦푘 (v푡)푣푘,푡 − 1 +훾푘 +푣푘,푡 (푦푘 (v푡) − 푥푘,푡) − 푝푘,푡 +� +≥ +� +푘 ∈푇푡 +� 1 +훾푘 +푦푘 (v푡)푣푘,푡 − +푣푘,푡 +1 + 휇푘,푡 +(푦푘 (v푡) − 푥푘,푡) − 휂푘(푣 + 휌푘)2 − 푝푘,푡 +� += +� +푘 ∈푇푡 +1 +훾푘 +푦푘 (v푡)푣푘,푡 − +� +푘 ∈푇푡 +푣푘,푡 +1 + 휇푘,푡 +(푦푘 (v푡) − 푥푘,푡) − +� +푘 ∈푇푡 +푝푘,푡 − |푇푡 |휂푘 (푣 + 휌푘)2 +where the inequality follows from the definition of 푇푡: if 푦푘 (v푡) ≥ 푥푘,푡 we use that 휇푘,푡 ≤ 훾푘 − 1 +and hence +1 +훾푘 ≤ +1 +휇푘,푡+1, whereas if 푦푘 (v푡) < 푥푘,푡 we use that 휇푘,푡 ≥ 훾푘 − 1 − 휂푘 (푣 + 휌푘) and hence +1 +훾푘 ≥ +1 +휇푘,푡+1 − 휂푘(푣 + 휌푘). +On the other hand, for agents in 푆푡 \푇푡, we have +� +푘 ∈푆푡 \푇푡 +1 +훾푘 +푥푘,푡푣푘,푡 ≥ +� +푘 ∈푆푡 \푇푡 +� 1 +훾푘 +푦푘 (v푡)푣푘,푡 − 1 +훾푘 +푣푘,푡 (푦푘 (v푡) − 푥푘,푡)+ +� +≥ +� +푘 ∈푆푡 \푇푡 +� 1 +훾푘 +푦푘 (v푡)푣푘,푡 − +푣푘,푡 +1 + 휇푘,푡 +(푦푘 (v푡) − 푥푘,푡)+ +� += +� +푘 ∈푆푡 \푇푡 +1 +훾푘 +푦푘 (v푡)푣푘,푡 − +� +푘 ∈푆푡 \푇푡 +푣푘,푡 +1 + 휇푘,푡 +(푦푘 (v푡) − 푥푘,푡)+. +Let푈푡 be the set of all agents in푇푡, plus the agents in 푆푡 \푇푡 such that 푥푘,푡 ≤ 푦푘 (v푡). Then, adding +our two inequalities together gives +� +푘 ∈푆푡 +� 1 +훾푘 +푥푘,푡푣푘,푡 +� +− +� +푘 ∈푇푡 +푝푘,푡 ≥ +� +푘 ∈푆푡 +1 +훾푘 +푦푘 (v푡)푣푘,푡 − +� +푘 ∈푈푡 +푣푘,푡 +1 + 휇푘,푡 +(푦푘 (v푡) − 푥푘,푡) − +� +푘 ∈푇푡 +푝푘,푡 − |푇푡 |휂푘 (푣 + 휌푘)2. +We wish to bound the term � +푘 ∈푈푡 +푣푘,푡 +1+휇푘,푡 (푦푘 (v푡) −푥푘,푡) from the inequality above. Note that this +is the exactly the difference in declared value (i.e., bid) for 푦푘 (v푡) and 푥푘,푡 for agents in 푈푡. It is +here where we use the fact that the underlying auction is a core auction. From the definition of a +core auction, this difference in bids is at most the sum of payments of agents not in 푈푡. Therefore, + +14 +Brendan Lucier, Sarath Patathil, Aleksandrs Slivkins, and Mengxiao Zhang +� +푘 ∈푆푡 +� 1 +훾푘 +푥푘,푡푣푘,푡 +� +− +� +푘 ∈푇푡 +푝푘,푡 ≥ +� +푘 ∈푆푡 +1 +훾푘 +푦푘 (v푡)푣푘,푡 − +� +푘∉푈푡 +푝푘,푡 − +� +푘 ∈푇푡 +푝푘,푡 − |푇푡 |휂푘(푣 + 휌푘)2 +≥ +� +푘 ∈푆푡 +1 +훾푘 +푦푘 (v푡)푣푘,푡 − +� +푘 +푝푘,푡 − |푇푡 |휂푘(푣 + 휌푘)2, +(9) +where in the second inequality we used the fact that 푇푡 ⊆ 푈푡, so the two sums of over payments +are over disjoint sets of agents. Summing up Eq. (9) over all rounds and substituting into Eq. (8) +and using the definitions of 푊푘(v) and 푅푘 (v), we conclude that +� +푘 ∈퐴 +푊푘(v) ≥ +� +푘 ∈퐴 +푅푘 (v) − +� +푡 +푛 +� +푘=1 +푝푘,푡 − 휂푘(푣 + 휌푘)2푛푇 . +Summing over all agents 푘 ∈ [푛], we have that for every “good” value realization 푣, +� +푘 +푊푘 (v) ≥ +� +푘 +푅푘 (v) − +� +푡 +푛 +� +푘=1 +푝푘,푡 − 푣푛 +� +푇 log(푣푛푇) − 휂푘(푣 + 휌푘)2푛푇 . +(10) +Step 3: Bounding Expected Liquid Welfare. Recall from Observation 2.1 that the total rev- +enue collected over all rounds will never be greater than the liquid welfare of the allocation. In +other words, � +푡 +�푛 +푘=1 푝푘,푡 ≤ � +푘 푊푘 (v). We can therefore rearrange Eq. (10) to conclude that +2 +� +푘 +푊푘(v) ≥ +� +푘 +푅푘 (v) − 푣푛 +� +푇 log(푣푛푇) − 휂푘(푣 + 휌푘)2푛푇 . +Taking expectations over 푣 and conditioning on the good event, we conclude that our expected +liquid welfare is at least half of the expected optimal liquid welfare with an error term that grows +at a rate of O(푣푛 +� +푇 log(푣푛푇)), as we take 휂푘 ≤ +푣 +푣+휌푘 +� +log(푣푛푇) +푇 +. This completes the proof of Theo- +rem 4.1. +4.2 +Individual Regret Guarantees of Algorithm 2 +In this subsection, we consider the performance of an individual autobidder 푘 on its optimization +problem when applying Algorithm 2. We abstract away the bids of other agents as supplied by +an adversary (which may adapt to the outcomes of the previous rounds). In particular, we do not +assume that the other agents are controlled by any particular algorithm. Our regret bound holds +w.r.t. a non-standard benchmark: the per-round pacing multipliers discussed below. +Since we focus on just agent 푘 throughout this subsection, we will drop the subscript 푘. For each +round 푡, we will write 푥푡 = 푥푡 (휇) and 푝푡 = 푝푡 (휇) for the allocation and payment if bidder 푘 picks +multiplier 휇 in round 푡. Note that these depend on the realized value of bidder 푘 as well as the +bids of the other auction participants. Recall also that 푥푡 and 푝푡 are both weakly non-increasing in +휇. The expectations in this section are taken with respect to the randomness in value profiles of +agent 푘 as well as the bids supplied by the adversary. Further, we define the following quantities: +Definition 4.3. For any round푡 ∈ [푇], define the expected budget expenditure 푍퐵 +푡 (휇), the expected +ROI expenditure 푍푅 +푡 (휇), the expected ROI gain 휌푡 (휇), and the expected gained value 푉푡 (휇) when the +bidder chooses multiplier 휇. Letting (푥)+ = max{푥, 0}, +푍퐵 +푡 (휇) ≜ E [푝푡 (휇)] +and +푍푅 +푡 (휇) ≜ E +� +(훾푝푡 (휇) − 푣푡푥푡 (휇))+� +, +휌푡 (휇) ≜ E +� +(푣푡푥푡 (휇) − 훾푝푡 (휇))+� +and +푉푡 (휇) ≜ E [푣푡푥푡 (휇)] . + +Autobidders with Budget and ROI Constraints +15 +Note that 푍퐵 +푡 (휇) and 푍푅 +푡 (휇) − 휌푡 (휇) are both non-increasing functions when 휇 ≥ 0 and 휇 ∈ +[0,훾 − 1], as is 푉푡 (휇). Formally, we have the following lemma. +Lemma 4.4. For any 푡 ∈ [푇], 푍퐵 +푡 (휇) is monotonically non-increasing for 휇 ≥ 0 and 푍푅 +푡 (휇) − 휌푡 (휇) +is monotonically non-increasing for 휇 ∈ [0,훾 − 1]. +Note also that 푍푅 +푡 (휇)−휌푡 (휇) is precisely the expected value of (훾푝푡 (휇)−푣푡푥푡 (휇)); 푍푅 +푡 (휇) captures +the positive part of this random variable, and 휌푡 (휇) captures the negative part. +We also assume Lipschitzness of allocations and payments with respect to 휇, which implies +that 푍퐵 +푡 , 푍푅 +푡 , 휌푡, and 푉푡 are all Lipschitz as well. This can be interpreted as a requirement that the +allocation and payment functions are sufficiently smooth as a function of an autobidder’s bid.10 +Assumption 4.1. 푍퐵 +푡 (휇), 푍푅 +푡 (휇), 푉푡 (휇) and 휌푡 (휇) are all 휆-Lipschitz for all 푡 ∈ [푇], for some 휆 ≥ 0. +Our regret bound holds with respect to a non-standard benchmark based on per-round pac- +ing multipliers 휇∗ +푡 . Informally, it is a value-optimizing multiplier for a given round 푡 subject to +the time-averaged constraints applied to this round. While not necessarily optimal globally, these +multipliers represent a reasonable goal for an online bidding algorithm in a complex, adversarial +environment (recall that vanishing regret bounds with respect to the standard benchmark of the +best-fixed-multiplier are impossible against an adversary [10]). The formal definition follows: +Definition 4.4 (Pacing multipliers). For a given round 푡 ∈ [푇], +• the budget-pacing multiplier 휇퐵∗ +푡 +is any 휇 ∈ [0, 푣 +휌 − 1] with 푍퐵 +푡 (휇) = 휌, or 0 if no such 휇 exists. +• The ROI-pacing multiplier 휇푅∗ +푡 +is any 휇 ∈ [0,훾 − 1] with 푍푅 +푡 (휇) = 휌푡 (휇), or 0 if no such 휇 +exists. +• The pacing multiplier is 휇∗ +푡 = max{휇퐵∗ +푡 , 휇푅∗ +푡 } ≥ 0. +Thus, our notion of regret is defined as follows: +Reg(푇) ≜ � +푡 ∈[푇 ] 푉푡 (휇∗ +푡 ) − 푉푡 (휇푡). +Our regret bound depends on the amount of drift in the environment, captured (in a fairly weak +way) by the path-lengths of the per-round pacing multipliers: +푃푅 +푇 := � +푡 ∈[푇 ] +���휇푅∗ +푡 +− 휇푅∗ +푡+1 +��� and 푃퐵 +푇 := � +푡 ∈[푇 ] +���휇퐵∗ +푡 +− 휇퐵∗ +푡+1 +��� . +We are now ready to state our main individual guarantee: +Theorem 4.2. Fix any distribution over the values of agent푘. Assume Lipschitzness (Assumption 4.1). +Algorithm 2 with parameters 휂퐵 = √휌/ +� +푇 (푣 + 휌) and 휂푅 = +1 +√ +푇 (훾+1)푣 guarantees that +E [Reg(푇)] ≤ O +� +(푃푅 +푇 + 1) +1 +4 휆 +3 +4 ((훾 + 1)푇) +7 +8 + (푣 + 휌) +7 +4 휌− 5 +4 +� +휆(1 + 푃퐵 +푇 )푇 +3 +4 +� +, +When 휆,훾,푣, 휌 are all constants, we have E[Reg] ≤ O((푃푅 +푇 + 1) +1 +4푇 +7 +8 + (푃퐵 +푇 + 1) +1 +2푇 +3 +4 ). +In a stationary stochastic setting, where the competing bids and the agent’s value are drawn +i.i.d from a fixed distribution, our benchmark boils down to the best feasible multiplier, and the +pathlengths are 푃퐵 +푇 = 푃푅 +푇 = 0. +10We note that this assumption could be satisfied by adding 푂 (휆) noise to the multiplier selected by any given autobidder, +at a loss of welfare proportional to 휆. + +16 +Brendan Lucier, Sarath Patathil, Aleksandrs Slivkins, and Mengxiao Zhang +Corollary 4.1. In the stationary stochastic setting, under the same conditions as stated in Theorem 4.2 +with (휆,훾,푣, 휌) all constants, Algorithm 2 guarantees that +max +휇 ∈Π +� +푡 ∈[푇 ] 푉푡 (휇) − 푉푡 (휇푡) ≤ O(푇 +7 +8 ), +where Π is the set of all constraint-feasible multipliers, +Π = {휇 ≥ 0 : � +푡 ∈[푇 ] E[푍퐵 +푡 (휇)] ≤ 퐵 and � +푡 ∈[푇 ] E[푍푅 +푡 (휇) − 휌푡 (휇)] ≤ 0}. +4.2.1 +Proof intuition. Before proving Theorem 4.2, we first provide the high-level idea of the proof +and highlight some technical difficulties. The high-level idea is to define auxiliary stochastic con- +vex functions 퐻푅 +푡 and 퐻퐵 +푡 that achieve their minima at 휇푅∗ +푡 +and 휇퐵∗ +푡 +respectively, and interpret the +update rules for 휇푅 +푡 and 휇퐵 +푡 as applying stochastic gradient descent (SGD) with respect to these +auxiliary functions. We will then relate the difference in obtained value 푉푡 (휇∗ +푡 ) − 푉푡 (휇푡) by the +total loss in these auxiliary functions. We would then ideally use facts about stochastic gradient +descent to show that the total loss in value, relative to the optimal benchmark, is small. +Unfortunately, there are many technical problems with this approach. The first problem is rel- +atively straightforward. Recall that we have both a budget and an ROI constraint, but only one +multiplier is used in each round; this can be either the budget-multiplier 휇퐵 +푡 or the ROI-multiplier +휇푅 +푡 , whichever is larger. To avoid having to reason about which multiplier is being followed each +round, we will actually bound the sum of regret experienced for both multipliers. We can think +of this as decomposing our experienced regret into the sum of two counterfactual regrets: one +corresponding to the case where we have only the budget constraint and bid according to 휇퐵 +푡 , and +one corresponding to the case where we have only the ROI constraint and bid according to 휇푅 +푡 . +For each of these two cases, we can bound the total loss in value 푉푡 with respect to the differences +|푍퐵 +푡 (휇퐵 +푡 ) −푍퐵 +푡 (휇퐵∗ +푡 )| and |푍푅 +푡 (휇푅 +푡 ) −푍푅 +푡 (휇푅∗ +푡 )|, respectively, which we can then relate to correspond- +ing differences in our auxiliary functions. +Unfortunately, this raises a more fundamental problem with our proposed approach. We would +like to argue that 휇퐵 +푡 evolves according to stochastic gradient descent on our auxiliary function +퐻퐵 +푡 , and similarly for 휇푅 +푡 and 퐻푅 +푡 . However, since we only receive feedback with respect to the +larger multiplier, the smaller multiplier (휇퐵 +푡 or 휇푅 +푡 ) may not be updated according to the gradient +of its corresponding loss function. One thing we do know is that, since the auxiliary functions +are convex, the gradient we use to update the smaller multiplier can only be more negative, in +expectation, than its “correct” gradient. At first this seems like an unacceptable source of error; if +gradients are too negative, then (for example) 휇퐵 +푡 could drift arbitrarily far from 휇퐵∗ +푡 +in the negative +direction. However, we are saved by Lemma 4.1: since we know that the budget and ROI constraints +will necessarily be satisfied at the end of the 푇 rounds, our algorithm will not actually suffer any +loss of value due to bids being too large (and hence, multipliers being too small). We can therefore +think of the evolution of one of the multipliers, say 휇퐵 +푡 , as following a variant of stochastic gradient +descent in which an adversary can, at will, perturb any given update step to be more negative; but +in exchange, we only suffer losses when 휇퐵 +푡 > 휇퐵∗ +푡 . As it turns out, the usual analysis of stochastic +gradient descent extends to this variant, so we can conclude that our total loss is not too large. +There are some additional technical challenges to handle as well. Most notably, the auxiliary loss +function for the ROI-multiplier is not convex in general but only convex when 휇 ∈ [0,훾 − 1]. This +requires us to handle separately the case where the budget-multiplier is greater than훾 −1, and omit +such rounds from our accounting of losses with respect to the ROI constraint. This complicates +our definition of counterfactual regret for ROI, but it turns out that the aggregate loss can still be +bounded with some additional effort. + +Autobidders with Budget and ROI Constraints +17 +4.2.2 +Proof sketch of Theorem 4.2. We begin by formalizing our interpretation of Algorithm 2 as +applying a form of stochastic gradient descent. We construct auxiliary loss functions 퐻퐵 +푡 (휇) = +휌휇 − +∫ 휇 +0 푍퐵 +푡 (휏)푑휏 and 퐻푅 +푡 (휇) = +∫ 휇 +0 휌푡 (휏) − 푍푅 +푡 (휏)푑휏. Based on Lemma 4.2, we have the follow- +ing lemma, which shows that if the ROI multiplier is larger than the budget multiplier, then the +ROI-multiplier is updated by applying a stochastic gradient descent on function 퐻푅 +푡 (휇), and if +the budget-multiplier is larger than the ROI-multiplier, then the budget-multiplier is updated by +applying a stochastic gradient descent on function 퐻퐵 +푡 (휇). +Lemma 4.5. Algorithm 2 guarantees that: +• If 휇푅 +푡 ≥ 휇퐵 +푡 , E [훾푝푡 (휇푡) − 푣푡푥푡 (휇푡)] = 푍푅 +푡 (휇푡) − 휌푡 (휇푡). +• If 휇푅 +푡 < 휇퐵 +푡 , E [푝푡 (휇푡) − 휌] = 푍퐵 +푡 (휇푡) − 휌. +We next establish that the convexity and Lipschitzness of 퐻퐵 +푡 (휇) and 퐻푅 +푡 (휇). +Lemma 4.6. 퐻푅 +푡 (휇) is 훾 ¯푣-Lipschitz and convex when 휇 ∈ [0,훾 − 1] and 퐻퐵 +푡 (휇) is (¯푣 + 휌)-Lipschitz +and convex in 휇 ≥ 0. +The proofs of these two lemmas are deferred to Appendix C.1. +Proof sketch of Theorem 4.2. For conciseness, we omit all the problem dependent constants +in the proof sketch and defer them to the formal proof in Appendix C.1. First, as mentioned in the +previous section, we decompose the overall regret into the sum of the counterfactual regret with +respect to budget-multiplier and ROI-multiplier respectively: +Reg ≤ � +푡 ∈[푇 ] +�� +푉푡 (휇퐵∗ +푡 ) − 푉푡 (휇퐵 +푡 ) +� +1{퐸퐵 +푡 } +� ++ � +푡 ∈[푇 ] +�� +푉푡 (휇푅∗ +푡 ) − 푉푡 (휇푅 +푡 ) +� +1{퐸푅 +푡 } +� +, +(11) +where 퐸퐵 +푡 represents the event that 휇푡퐵 ≥ 휇퐵∗ +푡 +and 퐸푅 +푡 represents the event that 휇푅 +푡 +≥ 휇푅∗ +푡 +and +휇퐵 +푡 ≤ 훾 − 1. In the following, we split the proof into four steps. +Step 1: Upper bounding the difference of 푉푡 by the difference of 푍퐵 +푡 and 푍푅 +푡 − 휌푡. +Using the monotonicity of 푝푡 (휇), we show in Lemma C.1 that for ROI-multiplier and budget- +multiplier, we have the following inequalities, +� +푉푡 (휇푅∗ +푡 ) − 푉푡 (휇푅 +푡 ) +� +1{퐸푅 +푡 } ≤ O +� 1 +훽 +� +푍푅 +푡 (휇푅∗ +푡 ) − 휌푡 (휇푅∗ +푡 ) − 푍푅 +푡 (휇푅 +푡 ) + 휌푡 (휇푅 +푡 ) +� ++ 훽 +� +1{퐸푅 +푡 }, +(12) +� +푉푡 (휇퐵∗ +푡 ) − 푉푡 (휇퐵 +푡 ) +� +1{퐸퐵 +푡 } ≤ O +� +푍퐵 +푡 (휇퐵∗ +푡 ) − 푍퐵 +푡 (휇퐵 +푡 ) +� +1{퐸퐵 +푡 }, +(13) +where 훽 > 0 is any positive number whose choice will be specified later. +Step 2: Upper bounding the difference of 푍푅 +푡 (푍퐵 +푡 ) by the difference of 퐻푅 +푡 (퐻퐵 +푡 ). Next, we +need to relate 푍푅 +푡 (휇푅∗ +푡 ) − 푍푅 +푡 (휇푅 +푡 ) with 퐻푅 +푡 (휇푅∗ +푡 ) − 퐻푅 +푡 (휇푅 +푡 ). Direct calculation shows that +� +퐻푅 +푡 (휇푅 +푡 ) − 퐻푅 +푡 (휇푅∗ +푡 ) +� +1{퐸푅 +푡 } = +∫ 휇푅 +푡 −휇푅∗ +푡 +0 +� +휌푡 (휏 + 휇푅∗ +푡 ) − 푍푅 +푡 (휏 + 휇푅∗ +푡 ) +� +푑휏1{퐸푅 +푡 }. +Note that 푓 푅(푥) = 휌푡 (휏 +휇푅∗ +푡 )−푍푅 +푡 (휏 +휇푅∗ +푡 ) is a non-decreasing function of 푥 when 푥 ∈ [0, 휇푅 +푡 −휇푅∗ +푡 ] +according to Lemma 4.6. Also, we have 푓 (0) ≥ 0. Using a technical lemma Lemma 4.12 in [23] (we +include this lemma in Lemma D.2 for completeness), we can show that +� +푍푅 +푡 (휇푅∗ +푡 ) − 휌푡 (휇푅∗ +푡 ) − 푍푅 +푡 (휇푅 +푡 ) + 휌푡 (휇푅 +푡 ) +� +1{퐸푅 +푡 } ≤ O +�� +퐻푅 +푡 (휇푅 +푡 ) − 퐻푅 +푡 (휇푅∗ +푡 ) +� +1{퐸푅 +푡 }. +(14) + +18 +Brendan Lucier, Sarath Patathil, Aleksandrs Slivkins, and Mengxiao Zhang +Similarly, for budget-constraint multiplier, we also have +� +푍퐵 +푡 (휇퐵∗ +푡 ) − 푍퐵 +푡 (휇퐵 +푡 ) +� +1{퐸퐵 +푡 } ≤ O +�� +퐻퐵 +푡 (휇퐵 +푡 ) − 퐻퐵 +푡 (휇퐵∗ +푡 ) +� +1{퐸퐵 +푡 }. +(15) +Step 3: Upper bounding the regret with respect to 퐻푅 +푡 and 퐻퐵 +푡 . Now we analyze the term +퐻푅 +푡 (휇푅 +푡 ) − 퐻푅 +푡 (휇푅∗ +푡 ) (under the event 휇푅 +푡 ≥ 휇푅∗ +푡 +and 휇퐵 +푡 ≤ 훾 − 1) and 퐻퐵 +푡 (휇푅 +푡 ) − 퐻퐵 +푡 (휇퐵∗ +푡 ) (under the +event 퐸퐵 +푡 ). For the first term, note that +� +퐻푅 +푡 (휇푅 +푡 ) − 퐻푅 +푡 (휇푅∗ +푡 ) +� +1{퐸푅 +푡 } ≤ +� +푔푅,푅 +푡 +, 휇푅 +푡 − 휇푅∗ +푡 +� +1{퐸푅 +푡 }, +where 푔푅,푅 +푡 += ∇퐻푅 +푡 (휇푅 +푡 ). This is because 휇푅∗ +푡 +≤ 휇푅 +푡 ≤ 훾 − 1 and 퐻푅 +푡 (휇) is convex when 휇 ∈ [0,훾 − 1] +according to Lemma 4.6. According to Lemma 4.5, if 휇푅 +푡 ≥ 휇퐵 +푡 , then 휇푅 +푡 is updated by a stochastic +gradient with mean 푔푅,푅 +푡 += ∇퐻푅 +푡 (휇푅 +푡 ). However, note that 휇푅 +푡 may not be updated using its own +stochastic gradient on 퐻푅 +푡 (휇), but may be updated by the gradient 푔푅,퐵 +푡 += ∇퐻푅 +푡 (휇퐵 +푡 ) if 휇퐵 +푡 ≥ 휇푅 +푡 . +However, using the convexity of 퐻푅 +푡 (휇) when 휇 ∈ [0,훾 −1], we have 푔푅,퐵 +푡 +≥ 푔푅,푅 +푡 +as 휇푅 +푡 ≤ 휇퐵 +푡 ≤ 훾 −1. +Therefore, let 푔푅 +푡 be the gradient that updates 휇푅 +푡 and we have +� +퐻푅 +푡 (휇푅 +푡 ) − 퐻푅 +푡 (휇푅∗ +푡 ) +� +1{퐸푅 +푡 } ≤ +� +푔푅 +푡 , 휇푅 +푡 − 휇푅∗ +푡 +� +1{퐸푅 +푡 }. +(16) +Therefore, based on Lemma 4.5 and the classic analysis of online gradient descent, we have: +E +� 푇 +� +푡=1 +� +퐻푅 +푡 (휇푅 +푡 ) − 퐻푅 +푡 (휇푅∗ +푡 ) +� +1{퐸푅 +푡 } +� +≤ +푇� +푡=1 +O +� +|휇푅 +푡 − 휇푅∗ +푡 |2 − |휇푅 +푡+1 − 휇푅∗ +푡+1|2 +휂푅 ++ 휂푅 + +|휇푅∗ +푡 +− 휇푅∗ +푡+1| +휂푅 +� +1{퐸푅 +푡 }, +While generally online gradient descent gives O( +√ +푇) regret, the challenge in bounding the term +in Eq. (28) is that with the condition 1{퐸푅 +푡 }, the term |휇푅 +푡 − 휇푅∗ +푡 |2 − |휇푅 +푡+1 − 휇푅∗ +푡+1|2 can not be tele- +scoped after summation. Therefore, we decompose the total horizon [푇], into 푆 intervals 퐼1 = +[1,푒1], . . . , 퐼푆 = [푤푆,푒푆], where each interval is a maximal sequence of consecutive rounds such +that 휇푅 +푡 ≥ 휇푅∗ +푡 +and 휇퐵 +푡 ≤ 훾 − 1. Then we have +E +� 푇� +푡=1 +� +퐻푅 +푡 (휇푅 +푡 ) − 퐻푅 +푡 (휇푅∗ +푡 ) +� +1{퐸푅 +푡 } +� +≤ +푆 +� +푠=1 +O +� |휇푅푤푠 − 휇푅∗ +푤푠 |2 − |휇푅 +푒푠+1 − 휇푅∗ +푒푠+1|2 +휂푅 +� ++ O +� +푃푅 +푇 +휂푅 ++ 휂푅푇 +� +. +With a more careful analysis on the dynamic of 휇푅 +푡 , we can show that the terms |휇푅 +푤푠 − 휇푅∗ +푤푠 |2 − +|휇푅 +푒푠+1 − 휇푅∗ +푒푠+1|2 indeed telescope after summation over 푠 = 1 to 푆 and we obtain that +E +� 푇� +푡=1 +� +퐻푅 +푡 (휇푅 +푡 ) − 퐻푅 +푡 (휇푅∗ +푡 ) +� +1{퐸푅 +푡 } +� +≤ O +� +1 + 푃푅 +푇 +휂푅 ++ 휂푅푇 +� += O +� +(푃푅 +푇 + 1) +√ +푇 +� +, +(17) +where the final equality is by choosing 휂푅 = Θ( 1 +√ +푇 ). Similarly for budget-multiplier, with 휂퐵 = +Θ( 1 +√ +푇 ), we can also obtain that +E +�� +푡 ∈[푇 ] +� +퐻퐵 +푡 (휇퐵 +푡 ) − 퐻푅 +푡 (휇퐵∗ +푡 ) +� +1{퐸퐵 +푡 } +� +≤ O +� +(1 + 푃퐵 +푇 ) +√ +푇 +� +. +(18) + +Autobidders with Budget and ROI Constraints +19 +Step 4: Combining all the above analysis. Finally, we combine Eq. (11), Eq. (12), Eq. (13), Eq. (14), Eq. (15), Eq. (17), Eq. (18) +and obtain the following +E +�� +푡 ∈[푇 ] +�푉푡 (휇∗ +푡 ) − 푉푡 (휇푡)�� +≤ E +� 푇 +� +푡=1 +O +� 1 +훽 +� +푍푅 +푡 (휇푅∗ +푡 ) − 휌푡 (휇푅∗ +푡 ) − 푍푅 +푡 (휇푅 +푡 ) + 휌푡 (휇푅 +푡 ) +� ++ 훽휆 +� +1{퐸푅 +푡 } +� ++ E +� 푇 +� +푡=1 +O +� +푍퐵 +푡 (휇퐵∗ +푡 ) − 푍퐵 +푡 (휇퐵 +푡 ) +� +1{퐸퐵 +푡 } +� +≤ E +�� +푡 ∈[푇 ]O +� 1 +훽 +� +퐻푅 +푡 (휇푅 +푡 ) − 퐻푅 +푡 (휇푅∗ +푡 ) + 훽휆 +� +1{퐸푅 +푡 } +� ++ E +�� +푡 ∈[푇 ]O +�� +퐻퐵 +푡 (휇퐵 +푡 ) − 퐻퐵 +푡 (휇퐵∗ +푡 ) +� +1{퐸퐵 +푡 } +� +≤ E +� +O +� 1 +훽 +� +푇� +푡 ∈[푇 ](퐻푅 +푡 (휇푅 +푡 ) − 퐻푅 +푡 (휇푅∗ +푡 ))1{퐸푅 +푡 } + 훽휆 +�� ++ E +� +O +� +푇� +푡 ∈[푇 ](퐻퐵 +푡 (휇퐵 +푡 ) − 퐻퐵 +푡 (휇퐵∗ +푡 ))1{퐸퐵 +푡 } +� +≤ O +� 1 +훽 +� +푇 1.5(푃푅 +푇 + 1) + 훽푇 + +� +푇 1.5(1 + 푃퐵 +푇 ) +� +≤ O +� +(푃푅 +푇 + 1) +1 +4푇 +7 +8 + +� +(1 + 푃퐵 +푇 )푇 +3 +4 +� +, +where the third inequality is because of Jensen’s inequality and the last inequality is by picking +the optimal 훽 > 0. This completes the proof of the theorem. +□ +5 +CONCLUSIONS AND OPEN PROBLEMS +We consider the problem of online bidding with both budget and ROI constraints, under a broad +class of auction formats including first-price and second-price auction. We set out to achieve both +aggregate and individual guarantees, as expressed, resp., by liquid welfare and vanishing regret. We +accomplish this with a novel variant of constraint-pacing, achieving (i) the best possible guarantee +in expected liquid welfare, (ii) vanishing individual regret against an adversary, and (iii) satisfying +the budget and ROI constraints with probability 1. The regret bound holds against a non-standard +(albeit reasonable) benchmark, side-stepping impossibility results from prior work. +Our work opens up several directions for future work. First, can we obtain similar results for +other algorithms or classes thereof? Second, while any vanishing regret satisfies our desiderata, +can we improve regret rates while maintaining a similar aggregate guarantee? This is a non- +standard question for the literature on online bidding. Particularly interesting are regret bounds +that go beyond a stationary stochastic environment (since the auction environment is often/typically +not stationary in practice). Another open direction is to analyze other aggregate market metrics +such as platform revenue. For the latter, it would be helpful to have a more complete understanding +of the interaction between autobidders and tunable parameters like reserve prices. + +20 +Brendan Lucier, Sarath Patathil, Aleksandrs Slivkins, and Mengxiao Zhang +REFERENCES +[1] Gagan Aggarwal, Ashwinkumar Badanidiyuru, and Aranyak Mehta. Autobidding with constraints. In International +Conference on Web and Internet Economics, pages 17–30. Springer, 2019. +[2] Shipra Agrawal and Nikhil R. Devanur. 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Latest online revision: Jan 2022. +[39] Chen-Yu Wei, Chung-Wei Lee, Mengxiao Zhang, and Haipeng Luo. Linear last-iterate convergence in constrained +saddle-point optimization. In 9th International Conference on Learning Representations (ICLR), 2021. + +22 +Brendan Lucier, Sarath Patathil, Aleksandrs Slivkins, and Mengxiao Zhang +A +NOTATION SUMMARY +Here we list the notations used throughout the paper for convenience. +Table 1. Summary of Notations +푛 +the total number of agents +푇 +the total number of rounds +x푡 = (푥1,푡, . . . ,푥푛,푡) +allocation profile at round 푡 ∈ [푇] +v푡 = (푣1,푡, . . . , 푣푛,푡) +value profile at round 푡 ∈ [푇] +b푡 = (푏1,푡, . . . ,푏푛,푡) +bid profile at round 푡 ∈ [푇] +p푡 = (푝1,푡, . . . ,푝푛,푡) +payment profile at round 푡 ∈ [푇] +훾푘 +ROI constraint parameter for agent 푘 +퐵푘 = 휌푘 ·푇 +total budget for agent 푘 +푣 +upper bound of the value +푊푘(x) +liquid value of agent 푘 of allocation sequence x (Definition 2.1) +푊 (x) +liquid welfare of allocation sequence x (Definition 2.1) +푊 푘 (x, 퐹) +agent 푘’s ex-ante liquid value of allocation x and value dist. 퐹 +(Definition 4.1) +푊 (x, 퐹) +ex-ante liquid welfare of allocation x and value dist. 퐹 (Definition 4.1) +휇퐵 +푘,푡 +agent 푘’s budget-multiplier at round 푡 +휇푅 +푘,푡 +agent 푘’s ROI-multiplier at round 푡. +휂푘,퐵 +learning rate of agent 푘’s budget-multiplier +휂푘,푅 +learning rate of agent 푘’s ROI-multiplier +푍퐵 +푡 +expected budget expenditure at round 푡 (Definition 4.3) +푍푅 +푡 +expected ROI expenditure at round 푡 (Definition 4.3) +휌푡 +expected ROI gain at round 푡 (Definition 4.3) +푉푡 +expected value gain at round 푡 (Definition 4.3) +휇퐵∗ +푡 +budget-pacing multiplier at round 푡 (Definition 4.4) +휇푅∗ +푡 +ROI-pacing multiplier at round 푡 (Definition 4.4) +휇∗ +푡 +pacing multiplier at round 푡 (Definition 4.4) +푃퐵 +푇 = �푇−1 +푡=1 |휇퐵∗ +푡 −휇퐵∗ +푡+1| +path-length of budget-pacing multiplier +푃푅 +푇 = �푇 −1 +푡=1 |휇푅∗ +푡 −휇푅∗ +푡+1| +path-length of ROI-pacing multiplier + +Autobidders with Budget and ROI Constraints +23 +B +OMITTED DETAILS IN SECTION 3 +In this section, we provide the omitted details and proofs in Section 3. We first prove Lemma 3.1, +which shows that monotonicity of 푣푡푥푡 (휇) − 훾푝푡 (휇). +Proof of Lemma 3.1. Let 푑 = max푗≠푘 푏 푗, and let 휇′ be such that 푣푡/(1 + 휇′) = 푑. Then for all +휇 > 휇′ we have 푥푡 (휇) = 푝푡 (휇) = 0 (which is weakly increasing in 휇), and for all 휇 < 휇′ we have +that 푥푡 (휇) = 1 and 푝푡 (휇) is weakly decreasing in 휇, so 푣푡푥푡 (휇) − 훾푝푡 (휇) is weakly increasing. +It only remains to establish what happens at the threshold 휇 = 휇′, and then only when 휇′ ≤ 훾 −1. +Note however that when 휇 = 휇′, the first and second highest bids are equal, so the payment of +agent 푘 is determined to be 푥푡 (휇)푣푡/(1 + 휇) ≥ 푥푡 (휇)푣푡/훾. This implies 푣푡푥푡 (휇′) − 훾푝푡 (휇′) ≤ 0, and +hence 푣푡푥푡 (휇) − 훾푝푡 (휇) ≤ 0 for all 휇 < 휇′ as well. Since 푣푡푥푡 (휇) − 훾푝푡 (휇) = 0 for all 휇 > 휇′, we +conclude that the difference is monotone in 휇 as claimed. +□ +Next, we show that Algorithm 1 guarantees that we never violate the ROI constriant. +Proof of Lemma 3.2. We prove this using induction. The base case follows trivially (since the +multiplier is initialized to 훾 − 1). Now, suppose this is true for all time up to 푡 − 1, i.e., +푡′ +� +휏=1 +푣휏푥휏 ≥ 훾 +푡′ +� +휏=1 +푝휏, +∀푡 ′ ≤ 푡 − 1 +Now, consider time 푡. From the update rule, we have 휇푡 ≥ 휇0 + 휂(�푡−1 +휏=1 푣휏푥휏 − 훾푝휏). We split the +proof into two parts. +Suppose 휂(�푡−1 +휏=1 푣휏푥휏 − 훾푝휏) < 훾 − 1. This gives us: 푝푡 ≤ 푏푡 ≤ +푣푡 +훾−휂(�푡−1 +휏=1 푣휏푥휏−훾푝휏) . Now, using the +fact that 휂푝푡 ≤ 1 since 휂 < 1/¯푣, we have: 훾푝푡 + �푡−1 +휏=1 푝휏 ≤ 푣푡푥푡 + �푡−1 +휏=1 푣휏푥휏, which gives us the +required claim. +Next, if 휂(�푡−1 +휏=1 푣휏푥휏 − 훾푝휏) > 훾 − 1, we have 푝푡 ≤ 푏푡 ≤ 푣푡 which gives us �푡 +휏=1 푣휏푥휏 − 훾푝휏 > 0. +This completes the proof. +□ +C +OMITTED DETAILS IN SECTION 4 +In this section, we present the omitted proofs in Section 4. First, we prove Lemma 4.1, which shows +that Algorithm 2 never violates either the ROI-constraint or the budget-constraint. +Proof of Lemma 4.1. We first show that the ROI constraint �푇 +푡=1 푣푘,푡푥푘,푡 ≥ 훾푘 +�푇 +푡=1 푝푘,푡 is satis- +fied if the agent is applying Algorithm 2. +The proof is basically the same as the one of Lemma 3.2. We prove this using induction. We +consider a fixed individual bidder and omit the subscript 푘 in the following. The base case still +follows trivially, since the multiplier 휇1 is initialized to max{훾 −1, 푣 +휌 −1}. Now, suppose this is true +for all time up to 푡 − 1, i.e., +푡′ +� +휏=1 +푣휏푥휏 ≥ 훾 +푡′ +� +휏=1 +푝휏, +∀푡 ′ ≤ 푡 − 1 +Now, consider time 푡. From the update rule, we have 휇푅 +푡 ≥ 휇푅 +0 + 휂푅 +�푡−1 +휏=1 (훾푝휏 − 푣휏푥휏). We split the +proof into two parts. +Suppose 휂푅 +�푡−1 +휏=1(푣휏푥휏 − 훾푝휏) < 훾 − 1. This gives us: +푝푡 ≤ 푏푡푥푡 = 푣푡푥푡 +1 + 휇푡 +≤ +푣푡푥푡 +1 + 휇푅 +푡 +≤ +푣푡푥푡 +훾 − 휂푅 +�푡−1 +휏=1(푣휏푥휏 − 훾푝휏) + +24 +Brendan Lucier, Sarath Patathil, Aleksandrs Slivkins, and Mengxiao Zhang +Now, using the fact that 휂푅푝푡 ≤ 휂푅푣푡 ≤ 휂푣 ≤ 1, we have: +훾 +푡� +휏=1 +푝휏 = 훾 +� +푝푡 + +푡−1 +� +휏=1 +푝휏 +� +≤ 휂푅푝푡 +푡−1 +� +휏=1 +(푥휏푣휏 − 훾푝휏) + 푣푡푥푡 + 훾 +푡−1 +� +휏=1 +푝휏 ≤ +푡� +휏=1 +푥휏푣휏, +where the last inequality uses the induction hypothesis. +Now, if 휂푅 +�푡−1 +휏=1(푣휏푥휏 − 훾푝휏) > 훾 − 1, we have �푡−1 +휏=1(푣휏푥휏 − 훾푝휏) ≥ (훾 − 1)푣 ≥ (훾 − 1)푣푡푥푡 and +푝푡 ≤ 푏푡 ≤ 푣푡, which means that +훾 +푡� +휏=1 +푝휏 ≤ 훾 +푡−1 +� +휏=1 +푝휏 + 훾푣푡푥푡 ≤ +푡� +휏=1 +푣휏푥휏. +Combining the above two claims finishes the proof for the ROI constraint. For budget constraint, +similarly, we prove the following inequality using induction. +푡� +휏=1 +푝휏 ≤ 휌푡. +(19) +The base case holds as 푝1 ≤ 푏1 ≤ +푣1 +1+휇퐵 +1 ≤ 휌. Suppose that Eq. (19) holds up to time 푡 − 1. According +to the update rule, we know that 휇퐵 +푡 ≥ 휇퐵 +0 + 휂퐵 +�푡−1 +휏=1(푥휏푝휏 − 휌). We also split the proof into two +parts. Suppose 휂퐵 +�푡−1 +휏=1(푥휏푝휏 − 휌) > 1 − 푣 +휌 . Then we have +푝푡 ≤ 푏푡푥푡 = 푣푡푥푡 +1 + 휇푡 +≤ +푣푡푥푡 +1 + 휇퐵 +푡 +≤ +푣푡푥푡 +휂퐵 +�푡−1 +휏=1(푥휏푝휏 − 휌) + 푣 +휌 +. +Therefore, we have: +푡� +휏=1 +푝휏 = +푡−1 +� +휏=1 +푝휏 + 휌 +푣 +� +푣푡푥푡 + 휂퐵푝푡 +푡−1 +� +휏=1 +(휌 − 푝휏) +� +≤ +푡−1 +� +휏=1 +푝휏 + 휌 + 휂퐵휌 +푡−1 +� +휏=1 +(휌 − 푝휏) = 휌푡. +If 휂퐵 +�푡−1 +휏=1(푥휏푝휏 − 휌) < 1 − 푣 +휌 , then we have 푝푡 ≤ 푏푡 ≤ 푣푡, which gives us +푡� +휏=1 +(푥휏푝휏) ≤ 휌(푡 − 1) + 푣 ≤ 휌푡. +Combining the above two inequalities finishes the proof. +□ +Next, we provide the proof of Lemma 4.2 which guarantees that 휇푅 +푡 +≤ 훾 − 1 under auctions +between first-price and second price, when we update the multipliers using Algorithm 2 +Proof of Lemma 4.2. We prove this by induction. For conciseness, we omit the subscript of the +agent index 푘. Base case trivially holds. Suppose that up to round 푡, 휇푅 +푡 ≤ 훾 − 1 and 휇퐵 +푡 ≤ 푣 +휌 − 1. +First, consider 휇푅 +푡 . At round 푡 + 1, if the bidder does not win an auction, then 휇푅 +푡+1 = 휇푅 +푡 . Otherwise, +we have +휇푅 +푡+1 = 휇푅 +푡 + 휂푅 (훾푝푡 (휇푡) − 푣푡) +≤ 휇푅 +푡 + 휂푅 +� +훾푝푡 (휇푅 +푡 ) − 푣푡 +� +(휇푡 = max{휇퐵 +푡 , 휇푅 +푡 , 0} and 푝푡 (휇) is decreasing in 휇) +≤ 휇푅 +푡 + 휂푅 +� +훾 +푣푡 +1 + 휇푅 +푡 +− 푣푡 +� +(payment does not exceed bid) +≤ 휇푅 +푡 + 휂푅 +(훾 − 1 − 휇푅 +푡 )푣푡 +1 + 휇푅 +푡 +≤ 휇푅 +푡 + 훾 − 1 − 휇푅 +푡 = 훾 − 1, + +Autobidders with Budget and ROI Constraints +25 +where the last inequality uses the fact that 휂푅 ≤ 1 +¯푣. This proves the result for 휇푅 +푡 . +Consider the budget-multiplier 휇퐵 +푡 . At round 푡 +1, similarly, if the bidder does not win an auction, +then 휇퐵 +푡+1 = 휇푅 +푡 − 휂퐵휌 ≤ 푣 +휌 − 1. Otherwise, we have +휇퐵 +푡+1 = 휇퐵 +푡 + 휂퐵 (푝푡 (휇푡) − 휌) +≤ 휇퐵 +푡 + 휂퐵 +� +푣푡 +1 + 휇퐵 +푡 +− 휌 +� +(푝푡 (휇푡) ≤ +푣푡 +1+휇푡 ≤ +푣푡 +1+휇퐵 +푡 ) +≤ 1 + 휇퐵 +푡 + +휂퐵푣 +1 + 휇퐵 +푡 +− 휌휂퐵 − 1 ≤ 푣 +휌 + 휂퐵푣 +푣/휌 − 휌휂퐵 − 1 ≤ 푣 +휌 − 1, +where the third inequality is because 휂퐵푣 ≤ 1 and ℎ(푥) = 푥 + 1 +푥 is increasing for 푥 ≥ 1. +□ +C.1 +Omited Details in Section 4.2 +In this section, we provide the proofs for the individual regret guarantee for Algorithm 2. +First, we prove Lemma 4.4, which shows the monotonicity of 푍퐵 +푡 (휇) and 푍푅 +푡 (휇) − 휌푡 (휇). +Proof of Lemma 4.4. For any 푡 ∈ [푇], as 푝푡 is non-decreasing in the bid, which means that +푝푡 (휇) is non-increasing in 휇, 푍퐵 +푡 (휇) is non-increasing for 휇 ≥ 0. Next, consider 휌푡 (휇) − 푍푅 +푡 (휇) = +E[푣푡푥푡 (휇) −훾푝푡 (휇)]. Let 푑푡 denote the competing bid. Also note that the bid 푏푡 = 푣푡/(1+ 휇) > 푣푡/훾. +We split the proof into three cases: +(1) If 푑푡 < 푣푡/훾. We have 푥푡 (휇) = 1 (since 푏푡 > 푑푡) and 푝푡 (휇) is decreasing in 휇. +(2) If 푣푡/훾 ≤ 푑푡 ≤ 푣푡. Suppose the competing bid푑푡 = 푣푡/(1+휇′) for some constant 휇′ ∈ [1,훾 −1]. +In this case, 푥푡 (휇) = 1 for 휇 ∈ [0, 휇′ − 1), and we have 푣푡 − 훾푝푡 (휇) < 0 and increasing. +At 휇 = 휇′, we have 푏푡 = 푑푡, and therefore 푥푡 (휇) ≤ 1. Furthermore, we have 푝푡 (휇) = +푥푡 (휇)푣푡/(1+휇′). Therefore 푣푡푥푡 (휇) −훾푝푡 (휇) = 푥푡 (휇) (푣푡 − 훾푣푡/(1 + 휇′)) ≤ 0, and also greater +than 푣푡푥푡 (휇) − 훾푝푡 (휇), 휇 < 휇′ (since 푝푡 (휇) decreases with 휇 and 푥(휇′) ≤ 1). This shows that +there will be an increase in 푣푡푥푡 (휇) − 훾푝푡 (휇) when we move from 휇 < 휇′ to 휇 = 휇′. +Finally, for 휇 ∈ (휇′,훾 − 1], 푣푡푥푡 (휇) −훾푝푡 (휇) will identically equal to zero. Therefore, we have +푣푡푥푡 (휇) − 훾푝푡 (휇) is increasing in this case. +(3) If 푑푡 > 푣푡. In this case, since 푏푡 < 푑푡, 푥푡 (휇) and 푝푡 (휇) will both be zero. +Combining these three cases, and taking expectations gives the desired result. +□ +Next, we prove the following lemma, which bounds the expected difference in value by the +expected difference in budget and ROI payoff. +Lemma C.1. For any 0 ≤ 휇1 ≤ 휇2 ≤ 훾 − 1, and any 훽 > 0, we have +푉푡 (휇1) − 푉푡 (휇2) ≤ 훾 +훽 +� +푍푅 +푡 (휇1) − 휌푡 (휇1) − 푍푅 +푡 (휇2) + 휌푡 (휇2) +� ++ 훽휆, +where 휆 > 0 is the Lipschitz constant defined in Assumption 4.1. In addition, for any 0 ≤ 휇1 ≤ 휇2 ≤ +푣 +휌 − 1, we have +푉푡 (휇1) − 푉푡 (휇2) ≤ 푣 +휌 +� +푍퐵 +푡 (휇1) − 푍퐵 +푡 (휇2) +� +, +Proof. For any 훽 > 0, as 푉푡 (휇) is 휆-Lipschitz based on Assumption 4.1, we have +푉푡 (휇1) − 푉푡 (휇2) = 푉푡 (휇1) − 푉푡 (min{휇2,훾 − 1 − 훽}) + 푉푡 (min{휇2,훾 − 1 − 훽}) − 푉푡 (휇2) +≤ 푉푡 (휇1) − 푉푡 (min{휇2,훾 − 1 − 훽}) + 훽휆. +(20) + +26 +Brendan Lucier, Sarath Patathil, Aleksandrs Slivkins, and Mengxiao Zhang +Now we show that +푉푡 (휇1) − 푉푡 (min{휇2,훾 − 1 − 훽}) ≤ 훾 +훽 +� +푍푅 +푡 (휇1) − 휌푅 +푡 (휇1) − 푍푅 +푡 (min{휇2,훾 − 1 − 훽}) + 휌푡 (min{휇2,훾 − 1 − 훽}) +� +. +For any 휇 ∈ [휇1,훾 − 1 − 훽], +∇[푍푅 +푡 (휇) − 휌푡 (휇)] = ∇E [((훾푝푡 (휇) − 푣푡)푥푡 (휇))] += E [훾∇푝푡 (휇)푥푡 (휇)}] + ∇E [훾푝푡 (휇)푥푡 (휇)}] − ∇E [푣푡푥푡 (휇)] +≤ ∇E +� +훾 푣푡 +1 + 휇 푥푡 (휇) +� +− ∇푉푡(휇) +(∇푝푡 (휇) ≤ 0, ∇푥푡 (휇) ≤ 0 and 푝푡 (휇) ≤ +푣푡 +1+휇 ) += 훾 − 휇 − 1 +1 + 휇 +∇푉푡 (휇). +In addition, note that 푍푅 +푡 (휇1) − 휌푡 (휇1) − 푍푅 +푡 (휇) + 휌푡 (휇) = +∫ 휇1 +휇 +∇(푍푅 +푡 (휏) − 휌푡 (휏))푑휏 and 푉푡 (휇1) − +푉푡 (휇) = +∫ 휇1 +휇 +∇푉푡 (휏)푑휏. Therefore, we have +푉푡 (휇1) − 푉푡 (min{휇2,훾 − 1 − 훽}) += +∫ min{휇2,훾−1−훽 } +휇1 +−∇푉푡 (휏)푑휏 +≤ +1 + min{휇2,훾 − 1 − 훽} +훾 − min{휇2,훾 − 1 − 훽} − 1 +∫ min{휇2,훾−1−훽 } +휇1 +−∇(푍푅 +푡 (휏) − 휌푡 (휇))푑휏 += +1 + min{휇2,훾 − 1 − 훽} +훾 − min{휇2,훾 − 1 − 훽} − 1 +� +푍푅 +푡 (휇1) − 푍푅 +푡 (min{휇2,훾 − 1 − 훽} − 휌푡 (휇1) + 휌푡 (min{휇2,훾 − 1 − 훽}) +� +≤ 훾 +훽 +� +푍푅 +푡 (휇1) − 휌푡 (휇1) − 푍푅 +푡 (휇2) + 휌푡 (휇2) +� +. +(푍푅 +푡 (휇) − 휌푡 (휇) is non-increasing in 휇 ∈ [0,훾 − 1]) +Plugging the above into Eq. (20) gives +푉푡 (휇1) − 푉푡 (휇2) ≤ 훾 +훽 +� +푍푅 +푡 (휇1) − 휌푡 (휇1) − 푍푅 +푡 (휇2) + 휌푡 (휇2) +� ++ 훽휆, +(21) +which finishes the proof of the first inequality. For the second inequality, note that for any 휇 ∈ +[0, 푣 +휌 − 1] +∇푍퐵 +푡 (휇) = ∇E [푝푡 (휇)푥푡 (휇)] += E [∇푝푡 (휇)푥푡 (휇)] + E [푝푡 (휇)∇푥푡 (휇)] +≤ ∇E +� 푣푡 +1 + 휇 푥푡 (휇) +� +(∇푝푡 (휇) ≤ 0, ∇푥푡 (휇) ≤ 0 and 푝푡 (휇) ≤ +푣푡 +1+휇 ) += +1 +1 + 휇 ∇푉푡 (휇) +≤ 휌 +푣 ∇푉푡 (휇). +Therefore, we have +푉푡 (휇1) − 푉푡 (휇2) = +∫ 휇2 +휇1 +−∇푉푡 (휏)푑휏 ≤ 푣 +휌 +∫ 휇2 +휇1 +−∇푍퐵 +푡 (휏)푑휏 = 푣 +휌 +� +푍퐵 +푡 (휇1) − 푍푅 +푡 (휇2) +� +≤ 푣 +휌 +� +푍퐵 +푡 (휇1) − 푍퐵 +푡 (휇2) +� +. +which completes the proof. +□ + +Autobidders with Budget and ROI Constraints +27 +Next, we prove Lemma 4.5, which shows that Algorithm 2 can be viewed as applying SGD on +the auxiliary loss function 퐻푅 +푡 and 퐻퐵 +푡 respectively. +Proof of Lemma 4.5. For ROI-multiplier, direct calculation shows that +E [(훾푝푡 (휇푡) − 푣푡)푥푡 (휇푡)] += E +� +(훾푝푡 (휇푡) − 푣푡푥푡 (휇푡))+� +− E +� +(푣푡푥푡 (휇푡) − 훾푝푡 (휇푡))+� += 푍푅 +푡 (휇푡) − 휌푡 (휇푡). +For budget-multiplier, direct calculation shows that E [푥푡푝푡 (휇푡) − 휌] = 푍퐵 +푡 (휇푡) − 휌. Combining +the above two equations completes the proof. +□ +Next we prove Lemma 4.6, which shows the Lipshitzness and convexity of auxiliary functions +퐻푅 +푡 and 퐻퐵 +푡 . +Proof of Lemma 4.6. The results for 퐻퐵 +푡 (휇) is proven in Lemma D.2 in [23]. For the function +퐻푅 +푡 (휇) we have +∇퐻푅 +푡 (휇) = 휌푡 (휇) − 푍푅 +푡 (휇) = E [푣푡푥푡 (휇) − 훾푝푡 (휇)] , +which we show is increasing over 휇 ∈ [0,훾 − 1] according to Lemma 4.4. In addition, we have for +all 휇 ∈ [0,훾 − 1], +|∇퐻푅 +푡 (휇)| ≤ max{훾푝푡 (휇), 푣푡 } ≤ 훾 ¯푣. +which completes the proof. +□ +Now we provide the proof of our main result Theorem 4.2, which shows the individual regret +of Algorithm 2. +Proof of Theorem 4.2. First, we decompose the overall regret into the regret with respect to +budget-multiplier and ROI-multiplier as follows: +Reg = +푇 +� +푡=1 +�푉푡 (휇∗ +푡 ) − 푉푡 (휇푡)� +≤ +푇 +� +푡=1 +�푉푡 (휇∗ +푡 ) − 푉푡 (휇푡)� 1{휇푡 ≥ 휇∗ +푡 } +(푉푡 (휇) is decreasing in 휇) +≤ +푇 +� +푡=1 +�� +푉푡 (휇∗ +푡 ) − 푉푡 (휇퐵 +푡 ) +� +1{휇퐵 +푡 ≥ max{휇푅 +푡 , 휇∗ +푡 } +� ++ +푇 +� +푡=1 +�� +푉푡 (휇∗ +푡 ) − 푉푡 (휇푅 +푡 ) +� +1{휇푅 +푡 ≥ max{휇퐵 +푡 , 휇∗ +푡 }} +� +≤ +푇 +� +푡=1 +�� +푉푡 (휇∗ +푡 ) − 푉푡 (휇퐵 +푡 ) +� +1{휇퐵 +푡 ≥ 휇∗ +푡 } +� ++ +푇� +푡=1 +�� +푉푡 (휇∗ +푡 ) − 푉푡 (휇푅 +푡 ) +� +1{휇푅 +푡 ≥ 휇∗ +푡 , 휇퐵 +푡 ≤ 훾 − 1} +� +≤ +푇 +� +푡=1 +�� +푉푡 (휇퐵∗ +푡 ) − 푉푡 (휇퐵 +푡 ) +� +1{휇퐵 +푡 ≥ 휇퐵∗ +푡 } +� ++ +푇 +� +푡=1 +�� +푉푡 (휇푅∗ +푡 ) − 푉푡 (휇푅 +푡 ) +� +1{휇푅 +푡 ≥ 휇푅∗ +푡 , 휇퐵 +푡 ≤ 훾 − 1} +� +, +(22) +where the third inequality is because 휇푅 +푡 ≤ 훾 −1 according to Lemma 4.2, meaning that 휇퐵 +푡 ≤ 휇푅 +푡 ≤ +훾 − 1 in the second term; the fourth inequality is because 푉푡 (휇) is non-increasing in 휇. We split the +rest of the proof into four steps. +Step 1: Upper bounding the difference of 푉푡 by the difference of 푍퐵 +푡 and 푍푅 +푡 . + +28 +Brendan Lucier, Sarath Patathil, Aleksandrs Slivkins, and Mengxiao Zhang +According to Lemma C.1, for ROI-multiplier and budget-multiplier, we have +� +푉푡 (휇푅∗ +푡 ) − 푉푡 (휇푅 +푡 ) +� +1{휇푅 +푡 ≥ 휇푅∗ +푡 , 휇퐵 +푡 ≤ 훾 − 1} +≤ +�훾 +훽 +� +푍푅 +푡 (휇푅∗ +푡 ) − 휌푡 (휇푅∗ +푡 ) − 푍푅 +푡 (휇푅 +푡 ) + 휌푡 (휇푅 +푡 ) +� ++ 훽휆 +� +1{휇푅 +푡 ≥ 휇푅∗ +푡 , 휇퐵 +푡 ≤ 훾 − 1}, +(23) +� +푉푡 (휇퐵∗ +푡 ) − 푉푡 (휇퐵 +푡 ) +� +1{휇퐵 +푡 ≥ 휇퐵∗ +푡 } ≤ 푣 +휌 +� +푍퐵 +푡 (휇퐵∗ +푡 ) − 푍퐵 +푡 (휇퐵 +푡 ) +� +1{휇퐵 +푡 ≥ 휇퐵∗ +푡 }. +(24) +Step 2: Upper bounding the difference of 푍푅 +푡 (푍퐵 +푡 ) by the difference of 퐻푅 +푡 (퐻퐵 +푡 ) +Next, we need to relate 푍푅 +푡 (휇푅∗ +푡 ) −푍푅 +푡 (휇푅 +푡 ) with 퐻푅 +푡 (휇푅∗ +푡 ) −퐻푅 +푡 (휇푅 +푡 ). Direct calculation shows that +� +퐻푅 +푡 (휇푅 +푡 ) − 퐻푅 +푡 (휇푅∗ +푡 ) +� +1{휇푅 +푡 ≥ 휇푅∗ +푡 , 휇퐵 +푡 ≤ 훾 − 1} += +∫ 휇푅 +푡 +휇푅∗ +푡 +� +휌푡 (휏) − 푍푅 +푡 (휏) +� +푑휏1{휇푅 +푡 ≥ 휇푅∗ +푡 , 휇퐵 +푡 ≤ 훾 − 1} += +∫ 휇푅 +푡 −휇푅∗ +푡 +0 +� +휌푡 (휏 + 휇푅∗ +푡 ) − 푍푅 +푡 (휏 + 휇푅∗ +푡 ) +� +푑휏1{휇푅 +푡 ≥ 휇푅∗ +푡 , 휇퐵 +푡 ≤ 훾 − 1}. +Note that 푔푅(푥) = 휌푡 (푥 +휇푅∗ +푡 )−푍푅 +푡 (푥 +휇푅∗ +푡 ) is a non-decreasing function of 푥 when 푥 ∈ [0, 휇푅 +푡 −휇푅∗ +푡 ] +according to Lemma 4.6. Also, we have 푔푅(0) ≥ 0. Therefore, let 푓 푅(푥) = 푔푅(푥) − 푔푅(0) and +according to Lemma 4.13 in [23] (we include this lemma in Lemma D.2 for completeness), we +know that +�� +휌푡 (휇푅 +푡 ) − 푍푅 +푡 (휇푅 +푡 ) +� +− +� +휌푡 (휇푅∗ +푡 ) − 푍푅 +푡 (휇푅∗ +푡 ) +�� +1{휇푅 +푡 ≥ 휇푅∗ +푡 , 휇퐵 +푡 ≤ 훾 − 1} +≤ +� +4휆 �퐻푅 +푡 (휇푅 +푡 ) − 퐻푅 +푡 (휇푅∗ +푡 ) − (휇푅 +푡 − 휇푅∗ +푡 )(휌푡 (휇푅∗ +푡 ) − 푍푅 +푡 (휇푅∗ +푡 ))�1{휇푅 +푡 ≥ 휇푅∗ +푡 , 휇퐵 +푡 ≤ 훾 − 1}. +When 휇푅 +푡 ≥ 휇푅∗ +푡 , we know that (휇푅 +푡 − 휇푅∗ +푡 )(휌푡 (휇푅∗ +푡 ) − 푍푅 +푡 (휇푅∗ +푡 )) is non-negative and we can obtain +that +� +푍푅 +푡 (휇푅∗ +푡 ) − 휌푡 (휇푅∗ +푡 ) − 푍푅 +푡 (휇푅 +푡 ) + 휌푡 (휇푅 +푡 ) +� +1{휇푅 +푡 ≥ 휇푅∗ +푡 , 휇퐵 +푡 ≤ 훾 − 1} +≤ +� +4휆(퐻푅 +푡 (휇푅 +푡 ) − 퐻푅 +푡 (휇푅∗ +푡 ))1{휇푅 +푡 ≥ 휇푅∗ +푡 , 휇퐵 +푡 ≤ 훾 − 1}. +(25) +For budget-constraint multiplier 휇퐵 +푡 , similarly we define 푓 퐵(푥) = 휌 − 푍퐵 +푡 (휇). Note that 푓 퐵(푥) is +also a non-decreasing function and 휆-Lipschitz. Applying Lemma D.2 on 푓 퐵(푥) − 푓 퐵(0), we also +have +� +푍퐵 +푡 (휇퐵∗ +푡 ) − 푍퐵 +푡 (휇퐵 +푡 ) +� +1{휇퐵 +푡 ≥ 휇퐵∗ +푡 } ≤ +� +2휆(퐻퐵 +푡 (휇퐵 +푡 ) − 퐻퐵 +푡 (휇퐵∗ +푡 ))1{휇퐵 +푡 ≥ 휇퐵∗ +푡 }. +(26) +Step 3-1: Upper bounding the regret with respect to 퐻푅 +푡 . +Now we analyze the term 퐻푅 +푡 (휇푅 +푡 ) − 퐻푅 +푡 (휇푅∗ +푡 ) (under the event 휇푅 +푡 ≥ 휇푅∗ +푡 +and 휇퐵 +푡 ≤ 훾 − 1) and +퐻퐵 +푡 (휇푅 +푡 ) − 퐻퐵 +푡 (휇퐵∗ +푡 ) (under the event 휇퐵 +푡 ≥ 휇퐵∗ +푡 ). For the first term, note that +� +퐻푅 +푡 (휇푅 +푡 ) − 퐻푅 +푡 (휇푅∗ +푡 ) +� +1{휇푅 +푡 ≥ 휇푅∗ +푡 , 휇퐵 +푡 ≤ 훾 − 1} ≤ +� +푔푅,푅 +푡 +, 휇푅 +푡 − 휇푅∗ +푡 +� +1{휇푅 +푡 ≥ 휇푅∗ +푡 , 휇퐵 +푡 ≤ 훾 − 1}, +where 푔푅,푅 +푡 += ∇퐻푅 +푡 (휇푅 +푡 ). This is because 휇푅∗ +푡 +≤ 휇푅 +푡 ≤ 훾 − 1 and 퐻푅 +푡 (휇) is convex when 휇 ∈ [0,훾 − 1]. +According to Lemma 4.5, if 휇푅 +푡 ≥ 휇퐵 +푡 , then 휇푅 +푡 is updated by a stochastic gradient with mean 푔푅,푅 +푡 += +∇퐻푅 +푡 (휇푅 +푡 ). However, note that 휇푅 +푡 may not be updated using its own stochastic gradient on 퐻푅 +푡 (휇), +but may be updated by the gradient 푔푅,퐵 +푡 += ∇퐻푅 +푡 (휇퐵 +푡 ) if 휇퐵 +푡 ≥ 휇푅 +푡 . However, using the convexity of +퐻푅 +푡 (휇) when 휇 ∈ [0,훾 − 1], we have 푔푅,퐵 +푡 +≥ 푔푅,푅 +푡 +as 휇푅 +푡 ≤ 휇퐵 +푡 ≤ 훾 − 1. + +Autobidders with Budget and ROI Constraints +29 +Let 푔푅 +푡 be the gradient that updates 휇푅 +푡 . Now we have +� +퐻푅 +푡 (휇푅 +푡 ) − 퐻푅 +푡 (휇푅∗ +푡 ) +� +1{휇푅 +푡 ≥ 휇푅∗ +푡 , 휇퐵 +푡 ≤ 훾 − 1} +≤ +� +푔푅,푅 +푡 +, 휇푅 +푡 − 휇푅∗ +푡 +� +1{휇푅 +푡 ≥ 휇푅∗ +푡 , 휇퐵 +푡 ≤ 훾 − 1} +≤ +� +푔푅 +푡 , 휇푅 +푡 − 휇푅∗ +푡 +� +1{휇푅 +푡 ≥ 휇푅∗ +푡 , 휇퐵 +푡 ≤ 훾 − 1}. +(27) +When 휇퐵 +푡 ≤ 휇푅 +푡 , this inequality directly holds. If 휇퐵 +푡 ≥ 휇푅 +푡 , then 푔푅 +푡 > 푔푅,푅 +푡 +and 휇푅 +푡 ≥ 휇푅∗ +푡 . Let �푔푟 +푡 +denote the empirical gradient of 휇푅 +푡 at round 푡 with E[�푔푅 +푡 ] = 푔푅 +푡 . Then, from the analysis of online +gradient descent we have: +E +� 푇 +� +푡=1 +� +퐻푅 +푡 (휇푅 +푡 ) − 퐻푅 +푡 (휇푅∗ +푡 ) +� +1{휇푅 +푡 ≥ 휇푅∗ +푡 , 휇퐵 +푡 ≤ 훾 − 1} +� +≤ E +�� +�푔푅 +푡 , 휇푅 +푡 − 휇푅∗ +푡 +� +1{휇푅 +푡 ≥ 휇푅∗ +푡 , 휇퐵 +푡 ≤ 훾 − 1} +� +≤ +푇 +� +푡=1 +� +|휇푅 +푡 − 휇푅∗ +푡 |2 − |휇푅 +푡+1 − 휇푅∗ +푡+1|2 +2휂푅 ++ 휂푅 +2 (훾 + 1)2푣2 + +(훾 − 1)|휇푅∗ +푡 +− 휇푅∗ +푡+1| +휂푅 +� +1{휇푅 +푡 ≥ 휇푅∗ +푡 , 휇퐵 +푡 ≤ 훾 − 1}, +(28) +where the last inequality is because: +|휇푅 +푡+1 − 휇푅∗ +푡 |2 ≤ |휇푅 +푡 − 휂푅�푔푅 +푡 − 휇푅∗ +푡 |2 += |휇푅 +푡 − 휇푅∗ +푡 |2 − 2휂푅 +� +�푔푅 +푡 , 휇푅 +푡 − 휇푅∗ +푡 +� ++ 휂2 +푅|�푔푅 +푡 |2, +|휇푅 +푡+1 − 휇푅∗ +푡 |2 = |휇푅 +푡+1 − 휇푅∗ +푡+1|2 + 2 +� +휇푅 +푡+1 − 휇푅∗ +푡+1, 휇푅∗ +푡 +− 휇푅∗ +푡+1 +� ++ |휇푅∗ +푡 +− 휇푅∗ +푡+1|2 +≥ |휇푅 +푡+1 − 휇푅∗ +푡+1|2 − 2|휇푅 +푡+1 − 휇푅∗ +푡+1| · |휇푅∗ +푡 +− 휇푅∗ +푡+1| +≥ |휇푅 +푡+1 − 휇푅∗ +푡+1|2 − 2(훾 − 1)|휇푅∗ +푡 +− 휇푅∗ +푡+1| +and |�푔푅 +푡 | ≤ (훾 +1)¯푣. Next, we decompose the total horizon [푇], into 푆 intervals 퐼1 = [1,푒1], . . . , 퐼푆 = +[푤푆,푒푆], where each interval is a maximal sequence of consecutive rounds such that 휇푅 +푡 ≥ 휇푅∗ +푡 +and +휇퐵 +푡 ≤ 훾 − 1. Then we have +E +� 푇� +푡=1 +� +퐻푅 +푡 (휇푅 +푡 ) − 퐻푅 +푡 (휇푅∗ +푡 ) +� +1{휇푅 +푡 ≥ 휇푅∗ +푡 , 휇퐵 +푡 ≤ 훾 − 1} +� +≤ +푆 +� +푠=1 +� +|휇푅 +푤푠 − 휇푅∗ +푤푠 |2 − |휇푅 +푒푠+1 − 휇푅∗ +푒푠+1|2 +2휂푅 ++ 휂푅 +2 (훾 + 1)2푣2푇 + +�푇−1 +푡=1 (훾 − 1)|휇푅∗ +푡 +− 휇푅∗ +푡+1|2 +휂푅 +� +. +(29) +For 푠 ≥ 2, consider the most recent round 휎푠 before round 푤푠 such that 휇퐵 +휎푠 ≤ 훾 − 1. As 휇푅 +푡 ≤ 훾 − 1, +we know that 휇퐵 +푡 ≥ 휇푅 +푡 when 푡 ∈ [휎푠 + 1,푤푠 − 1]. In addition, according to the update rule of 휇푅 +푡 , +we know that when 푡 ∈ [휎푠 + 1,푤푠 − 1], +휇푅 +푡+1 = 휇푅 +푡 + 휂푅 (훾푝푡 (휇푡) − 푣푡) 푥푡 ≤ 휇푅 +푡 + 휂푅 +� +훾푣푡 +1 + (훾 − 1) +� +푥푡 ≤ 휇푅 +푡 . + +30 +Brendan Lucier, Sarath Patathil, Aleksandrs Slivkins, and Mengxiao Zhang +Next, consider the round 휎푠. If 휎푠 belongs to some interval 퐼푖, according to the definition of 휎푠, 휎푠 +must be the end of 퐼푠−1 (i.e. 휎푠 = 푒푠−1). In this case, we have +|휇푅 +푤푠 − 휇푅∗ +푤푠 |2 ≤ |휇푅 +휎푠+1 − 휇푅∗ +푤푠 |2 +(휇푅∗ +푤푠 ≤ 휇푅 +푤푠 ≤ 휇푅 +휎푠 +1) += |휇푅 +휎푠 +1 − 휇푅∗ +휎푠+1|2 + 2(휇푅 +휎푠+1 − 휇푅∗ +휎푠 +1)(휇푅∗ +휎푠 +1 − 휇푅∗ +푤푠 ) + |휇푅∗ +휎푠 +1 − 휇푅∗ +푤푠 |2 +≤ |휇푅 +푒푠−1+1 − 휇푅∗ +푒푠−1+1|2 + 3(훾 − 1) �� +� +� +푡 ∈[푒푠−1+1,푤푠−1] +|휇푅∗ +푡 +− 휇푅∗ +푡+1|�� +� +. +Otherwise, 휎푠 is outside the interval and 휇푅 +휎푠 < 휇푅∗ +휎푠 . From the update of 휇푅 +푡 , we know that +휇푅 +휎푠+1 ≤ 휇푅 +휎푠 + 휂푅(훾 + 1)푣 < 휇푅∗ +휎푠 + 휂푅(훾 + 1)푣. +Therefore we know that +|휇푅 +푤푠 − 휇푅∗ +푤푠 |2 +≤ |휇푅 +휎푠 +1 − 휇푅∗ +푤푠 |2 +≤ |휇푅∗ +휎푠 + 휂푅(훾 + 1)푣 − 휇푅∗ +푤푠 |2 +≤ |휇푅∗ +휎푠 − 휇푅∗ +푤푠 |2 + 2휂푅(훾 + 1)푣 · +� +휏 ∈[휎푠,푤푠−1] +|휇푅∗ +휏 − 휇푅∗ +휏+1| + 휂2 +푅(훾 + 1)2푣2 +≤ (훾 − 1)|휇푅∗ +휎푠 − 휇푅∗ +푤푠 | + 2휂푅(훾 + 1)푣 · +� +휏 ∈[휎푠,푤푠−1] +|휇푅∗ +휏 − 휇푅∗ +휏+1| + 휂2 +푅(훾 + 1)2푣2 +Combining the above two cases and noticing that in the second case, 휎푠 does not belong to an +interval 퐼푖, we have for any 휂푅 ≤ min{1, 1 +푣 }, +푆 +� +푠=1 +� +|휇푅 +푤푠 − 휇푅∗ +푤푠 |2 − |휇푅 +푒푠 +1 − 휇푅∗ +푒푠+1|2 +휂푅 +� +≤ O +� +1 + (훾 + 1) �푇−1 +푡=1 |휇푅∗ +푡 +− 휇푅∗ +푡+1| +휂푅 ++ 휂푅(훾 + 1)2푣2푇 +� +. +(30) +Combining Eq. (30) with Eq. (29), along with the definition of 푃푅 +푇 , we have +E +� 푇� +푡=1 +� +퐻푅 +푡 (휇푅 +푡 ) − 퐻푅 +푡 (휇푅∗ +푡 ) +� +1{휇푅 +푡 ≥ 휇푅∗ +푡 , 휇퐵 +푡 ≤ 훾 − 1} +� +≤ O +� +1 + (훾 + 1)푃푅 +푇 +휂푅 ++ 휂푅(훾 + 1)2푣2푇 +� +. +Choosing 휂푅 = +1 +√ +푇 (훾+1)푣, we know that +E +� 푇� +푡=1 +� +퐻푅 +푡 (휇푅 +푡 ) − 퐻푅 +푡 (휇푅∗ +푡 ) +� +1{휇푅 +푡 ≥ 휇푅∗ +푡 , 휇퐵 +푡 ≤ 훾 − 1} +� +≤ O +� +(훾 + 1)1.5푣(푃푅 +푇 + 1) +√ +푇 +� +. +(31) +Step 3-2: Upper bounding the regret with respect to 퐻퐵 +푡 . +Next, we consider 퐻퐵 +푡 (휇퐵 +푡 ) − 퐻퐵 +푡 (휇퐵∗ +푡 ) under the condition that 휇퐵 +푡 ≥ 휇퐵∗ +푡 . Let 푔퐵,퐵 +푡 += ∇퐻퐵 +푡 (휇퐵 +푡 ) +and 푔퐵,푅 +푡 += ∇퐻퐵 +푡 (휇푅 +푡 ). Similar to Step 3-1, because of the convexity of 퐻퐵 +푡 (휇) when 휇 ∈ [0, +∞), +we know that if 휇퐵 +푡 ≤ 휇푅 +푡 , we have 푔퐵,푅 +푡 +≥ 푔퐵,퐵 +푡 +. Therefore, we have the following inequality similar +to Eq. (27): +� +퐻퐵 +푡 (휇퐵 +푡 ) − 퐻퐵 +푡 (휇퐵∗ +푡 ) +� +1{휇퐵 +푡 ≥ 휇퐵∗ +푡 } ≤ +� +푔퐵 +푡 , 휇퐵 +푡 − 휇퐵∗ +푡 +� +1{휇퐵 +푡 ≥ 휇퐵∗ +푡 }, +(32) + +Autobidders with Budget and ROI Constraints +31 +where 푔퐵 +푡 = ∇퐻퐵 +푡 (휇푡). Then, similar to Eq. (28), using the fact that 휇퐵∗ +푡 +≤ 푣 +휌 for all 푡 ∈ [푇], we have +E +� 푇� +푡=1 +� +퐻퐵 +푡 (휇퐵 +푡 ) − 퐻퐵 +푡 (휇퐵∗ +푡 ) +� +1{휇퐵 +푡 ≥ 휇퐵∗ +푡 } +� +≤ +푇� +푡=1 +� +|휇퐵 +푡 − 휇퐵∗ +푡 |2 − |휇퐵 +푡+1 − 휇퐵∗ +푡+1|2 +2휂퐵 ++ 휂퐵 +2 (휌 + 푣)2 + 푣|휇퐵∗ +푡 +− 휇퐵∗ +푡+1|2 +휌휂퐵 +� +1{휇퐵 +푡 ≥ 휇퐵∗ +푡 }, +(33) +where (휌 + ¯푣) is a universal upper bound of the empirical gradient for 휇퐵 +푡 . Next, we similarly +decompose the total horizon [푇], into 푆푏 intervals 퐼1 = [1,푒′ +1], . . . , 퐼 ′ +푆푏 = [푤 ′ +푆푏,푒′ +푆푏], where each +interval 퐼 contains a maximal sequence of consecutive rounds such that 휇퐵 +푡 ≥ 휇퐵∗ +푡 . Then we have +E +� 푇� +푡=1 +� +퐻퐵 +푡 (휇퐵 +푡 ) − 퐻푅 +푡 (휇퐵∗ +푡 ) +� +1{휇퐵 +푡 ≥ 휇퐵∗ +푡 } +� +≤ +푆푏 +� +푠=1 +� |휇퐵 +푤′푠 − 휇퐵∗ +푤′푠 |2 − |휇퐵 +푒′푠+1 − 휇퐵∗ +푒′푠+1|2 +2휂퐵 +� ++ 휂퐵 +2 (휌 + 푣)2푇 + 푣 �푇−1 +푡=1 |휇퐵∗ +푡 +− 휇퐵∗ +푡+1|2 +휌휂퐵 +, +(34) +For 푠 ≥ 2, Note that we have 휇퐵 +푤′푠−1 ≤ 휇퐵∗ +푤′푠−1 and according to the update of 휇퐵 +푡 , we also have +휇퐵 +푤′푠 ≤ 휇퐵 +푤′푠−1 + 휂퐵(푣 + 휌). Combining the fact that 휇퐵 +푤′푠 ≥ 휇퐵∗ +푤′푠 , we have +|휇퐵 +푤′푠 − 휇퐵∗ +푤′푠 |2 ≤ |휇퐵 +푤′푠−1 + 휂퐵(푣 + 휌) − 휇퐵∗ +푤′푠 |2 ≤ |휇퐵∗ +푤′푠−1 + 휂퐵(푣 + 휌) − 휇퐵∗ +푤′푠 |2 +≤ +� 푣 +휌 + 2휂퐵(푣 + 휌) +� +|휇퐵∗ +푤′푠−1 − 휇퐵∗ +푤′푠 | + 휂2 +퐵(푣 + 휌)2. +Combining the above with Eq. (34) and choosing 휂퐵 = +√휌 +√ +푇 (푣+휌) , we have +E +� 푇� +푡=1 +� +퐻퐵 +푡 (휇퐵 +푡 ) − 퐻푅 +푡 (휇퐵∗ +푡 ) +� +1{휇퐵 +푡 ≥ 휇퐵∗ +푡 } +� +≤ O +� 푣 +휌휂퐵 +(1 + 푃퐵 +푇 ) + 휂퐵(푣 + 휌)2푇 +� +≤ O +� +휌−0.5(푣 + 휌)1.5(1 + 푃퐵 +푇 ) +√ +푇 +� +(35) +Step 4: Combining all the above analysis + +32 +Brendan Lucier, Sarath Patathil, Aleksandrs Slivkins, and Mengxiao Zhang +Finally, we are ready to prove the main results. Combining Eq. (22), Eq. (23), Eq. (24), Eq. (25), Eq. (26), Eq. (31), Eq. (35), +we have +E +� 푇� +푡=1 +�푉푡 (휇∗ +푡 ) − 푉푡 (휇푡)� +� +≤ E +� 푇� +푡=1 +�훾 +훽 +� +푍푅 +푡 (휇푅∗ +푡 ) − 휌푡 (휇푅∗ +푡 ) − 푍푅 +푡 (휇푅 +푡 ) + 휌푡 (휇푅 +푡 ) +� ++ 훽휆 +� +1{휇푅 +푡 ≥ 휇푅∗ +푡 , 휇퐵 +푡 ≤ 훾 − 1} +� ++ E +� 푇� +푡=1 +푣 + 휌 +휌 +� +푍퐵 +푡 (휇퐵∗ +푡 ) − 푍퐵 +푡 (휇퐵 +푡 ) +� +1{휇퐵 +푡 ≥ 휇퐵∗ +푡 } +� +(Eq. (23), Eq. (24)) +≤ E +� 푇� +푡=1 +�훾 +훽 +� +4휆(퐻푅 +푡 (휇푅 +푡 ) − 퐻푅 +푡 (휇푅∗ +푡 )) + 훽휆 +� +1{휇푅 +푡 ≥ 휇푅∗ +푡 , 휇퐵 +푡 ≤ 훾 − 1} +� ++ E +� 푇� +푡=1 +푣 + 휌 +휌 +� +2휆(퐻퐵 +푡 (휇퐵 +푡 ) − 퐻퐵 +푡 (휇퐵∗ +푡 ))1{휇퐵 +푡 ≥ 휇퐵∗ +푡 } +� +(Eq. (25), Eq. (26)) +≤ E + +��� +� +훾 +훽 +� +� +� +4푇 +푇� +푡=1 +휆(퐻푅 +푡 (휇푅 +푡 ) − 퐻푅 +푡 (휇푅∗ +푡 )) + 훽휆 +��� +� +1{휇푅 +푡 ≥ 휇푅∗ +푡 , 휇퐵 +푡 ≤ 훾 − 1} + ++ E + +푣 + 휌 +휌 +� +� +� +2푇휆 +푇� +푡=1 +(퐻퐵 +푡 (휇퐵 +푡 ) − 퐻퐵 +푡 (휇퐵∗ +푡 ))1{휇퐵 +푡 ≥ 휇퐵∗ +푡 } + +(Jensen’s inequality) +≤ O +�훾 +훽 +� +휆푇 1.5(훾 + 1)1.5푣(푃푅 +푇 + 1) + 훽휆푇 + (푣 + 휌)1.75 +휌1.25 +� +푇 1.5휆(1 + 푃퐵 +푇 ) +� +(Eq. (31), Eq. (35)) +≤ O +� +(푃푅 +푇 + 1) +1 +4 휆 +3 +4 ((훾 + 1)푇) +7 +8 + (푣 + 휌) +7 +4 휌− 5 +4 +� +휆(1 + 푃퐵 +푇 )푇 +3 +4 +� +where we obtain the last inequality is by picking the optimal 훽 > 0. This completes the proof of +the theorem. +□ +Finally, we provide the proof of Corollary 4.1, which shows that Algorithm 2 ensures a �O(푇 7/8) +individual regret bound in the stationary stochastic setting. +Proof of Corollary 4.1. In the stationary stochastic setting, define 푍퐵(휇) ≜ 푍퐵 +푡 (휇), 푍푅(휇) ≜ +푍푅 +푡 (휇), 휌(휇) ≜ 휌푡 (휇), and 푉 (휇) ≜ 푉푡 (휇) for all 푡 ∈ [푇]. Also, we have for all 푡 ∈ [푇], 휇푅∗ +푡 += 휇푅∗ and +휇퐵∗ +푡 += 휇퐵∗. Here 휇푅∗ is any 휇 ∈ [0,훾 − 1] such that E[푍푅(휇) − 휌(휇)] = 0, or 0 if no such 휇 exists; +휇퐵∗ is any 휇 ∈ [0, 푣 +휌 − 1] such that E[푍퐵(휇) − 휌] = 0, or 0 if no such 휇 exists. +Now consider any 휇 ∈ Π. As E[푍퐵(휇)] ≤ 퐵 +푇 = 휌 and E[푍푅(휇) − 휌(휇)] ≤ 0, according to the +monotonicity of 푍퐵(휇) and 푍푅(휇) − 휌(휇) proven in Lemma 4.4, there exists 휇퐵∗ ∈ [0, 푣 +휌 − 1] and +휇푅∗ ∈ [0,훾 − 1] such that 휇 ≥ 휇퐵∗ and 휇 ≥ 휇푅∗. Therefore, according to Theorem 4.2 and the +monotonicity of 푉 (휇), we know that +푇� +푡=1 +(푉 (휇) − 푉 (휇푡)) ≤ +푇� +푡=1 +(푉 (max{휇퐵∗ +푡 , 휇푅∗ +푡 }) − 푉 (휇푡)) ≤ O(푇 +7 +8 ), +where the last inequality is because Theorem 4.2 and 푃푅 +푇 = 푃퐵 +푇 = 0. +□ + +Autobidders with Budget and ROI Constraints +33 +D +AUXILIARY LEMMAS +In this section, we show the auxiliary lemmas that are useful in our analysis. +Lemma D.1 (Lemma 3.7 of [23]). Let 푌1, · · · ,푌푇 be random variables and F0 ⊆ · · · ⊆ F푇 be a +filtration such that: +(1) 0 ≤ 푌푡 ≤ 푣 with probability 1 for some parameter 푣 ≥ 0 for all 푡. +(2) E[푌푡] ≤ 휌 for some parameter 휌 for all 푡. +(3) For all 푡, 푌푡 is F푡-measurable but is independent of F푡−1-measurable. Then: +P( +푇� +푡=1 +푋푡푌푡 + (1 − 푋푡)휌 ≥ 휌 ·푇 + 휃) ≤ exp +�−2휃2 +푇푣2 +� +. +Lemma D.2 (Lemma 4.12 of [23]). Let 푓 : R → R be an increasing 휆−Lipschitz function such that +푓 (0) = 0. Let 푅 = +∫ 푥 +0 푓 (푦)푑푦 for some 푥 ∈ R. Then |푓 (푥)| ≤ +√ +2휆푅. +Lemma D.3 (Lemma B.1 of [23]). Let �y : [0,푣]푛푇 → 푋푇 be an allocation sequence rule that takes +in the entire sequence v1, v2, · · · , v푇 and allocates �푦푘,푡 (v1, v2, · · · , v푇 ) units to agent 푘 at time 푡. Then +there exists a (single-round) allocation rule 푦 : [0,푣]푛 → 푋 such that +� +푊 (�y, 퐹) := +푛 +� +푘=1 +min +� +퐵푘, 1 +훾푘 +Ev1,v2,···,v푇 ∼퐹 +� 푇� +푡=1 +�푦푘,푡 (v1, v2, · · · , v푇 )푣푘,푡 +�� += +푛 +� +푘=1 +푇 · min +� +휌푘, 1 +훾푘 +Ev∼퐹 [푦푘 (v)푣푘] +� += 푊 (y, 퐹). + diff --git a/N9FQT4oBgHgl3EQfWzb0/content/tmp_files/load_file.txt b/N9FQT4oBgHgl3EQfWzb0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ac3537ae450531f92226d560e6e74f95d3c09e51 --- /dev/null +++ b/N9FQT4oBgHgl3EQfWzb0/content/tmp_files/load_file.txt @@ -0,0 +1,1284 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf,len=1283 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='13306v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='GT] 30 Jan 2023 Autobidders with Budget and ROI Constraints: Efficiency, Regret, and Pacing Dynamics∗ BRENDAN LUCIER, Microsoft Research, USA SARATH PATTATHIL, Massachusetts Institute of Technology, USA ALEKSANDRS SLIVKINS, Microsoft Research, USA MENGXIAO ZHANG, University of Southern California, USA We study a game between autobidding algorithms that compete in an online advertising platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Each au- tobidder is tasked with maximizing its advertiser’s total value over multiple rounds of a repeated auction, subject to budget and/or return-on-investment constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We propose a gradient-based learning algorithm that is guaranteed to satisfy all constraints and achieves vanishing individual regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Our algorithm uses only bandit feedback and can be used with the first- or second-price auction, as well as with any “intermediate" auction format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Our main result is that when these autobidders play against each other, the resulting expected liquid welfare over all rounds is at least half of the expected optimal liquid welfare achieved by any allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' This holds whether or not the bidding dynamics converges to an equilibrium and regardless of the correlation structure between advertiser valuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 1 INTRODUCTION As the rules and algorithms governing online markets increase in complexity and scale, platforms are increasingly providing ML-powered interfaces to help users interact and navigate efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' A prominent example is the rise of autobidding, a service provided by advertising platforms to help advertisers automate their campaigns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The advertiser only needs to specify high-level objectives and constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' A typical example might be “maximize the number of clicks received, subject to spending at most $1000 per day, at most $2 per click on average, and no more than $10 for any one click.” This example encodes three different constraints on the outcome: a budget constraint, an average return-on-investment (ROI) constraint, and a marginal ROI (or value) constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The autobidder then uses an online learning algorithm to tune a detailed advertising campaign so as to solve this optimization problem on the advertiser’s behalf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' While each individual ad impres- sion is sold by auction, the actual management of individual bids is delegated to the autobidding algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Autobidding is popular with advertisers and all major online ad platforms now provide this service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' This success is owed in part to the effectiveness of online learning methods for bid tun- ing, which has received substantial attention in the academic literature in recent years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Initial work focused on the problem of tuning bids subject to an aggregate budget constraint, a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=', bud- get pacing [10, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' More recent work concerns ROI (or ROAS, Return on Ad Spend) constraints [8, 22, 27, 32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The result is a variety of well-understood learning algorithms that can be used by an autobidder to achieve vanishing regret in any stationary (or near-stationary) auction envi- ronment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Correspondingly, autobidder interfaces supporting both budget and ROI constraints are now ubiquitous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' ∗Some of the results have been obtained while S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Pattathil and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Zhang were research interns at Microsoft Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The authors are grateful to Bach Ha for many conversations that informed our perspective, and to Sidharth Satya for providing research support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Authors’ addresses: Brendan Lucier, Microsoft Research, Cambridge, MA, USA, brlucier@microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Sarath Pattathil, Massachusetts Institute of Technology, Cambridge, MA, USA, sarathp@mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Aleksandrs Slivkins, Microsoft Research, New York, NY, USA, slivkins@microsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Mengxiao Zhang, University of Southern California, Los Angeles, CA, USA, mengxiao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='zhang@usc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 2 Brendan Lucier, Sarath Patathil, Aleksandrs Slivkins, and Mengxiao Zhang The dominance of autobidding in online advertising platforms gives rise to a new concern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Since autobidders are now predominantly competing against each other, one must worry what happens when these algorithmic agents interact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' What are the implications for the overall health of the market as a whole, in terms of aggregate objectives such as efficiency and stability?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' These concerns are related to algorithmic collusion, whereby algorithmic agents do not converge to competitive equilibrium play but rather systematically behave in a way that reduces market efficiency (either by converging to a less efficient outcome, or by failing to converge entirely).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Such “collusion" could be a non-malicious unintended artifact of the agents’ interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Algorithmic collusion has been observed in simulation studies for a variety of repeated auction scenarios [3, 14, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' It is also documented to occur in practice for related settings like algorithmic pricing on Amazon [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Given the potential market-wide impact of such interactions, it is important to account for unintended emergent behaviors that may arise when autobidders compete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' One solution would be to design autobidding algorithms that always converge to equilibria of the “bidding game" that they are playing against each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' This would help address our concerns because equilibrium outcomes in these games are known to be approximately efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The state of the art in guarantees concern liquid welfare,1 and state that a pure Nash equilibrium of the bidding game with budget and/or ROI constraints attains expected liquid welfare at least half of the optimum, and this bound is tight [1, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Unfortunately, finding an equilibrium of the bidding game is PPAD-hard, even for the special case of maximizing value subject to a budget constraint in a second-price auction [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We therefore should not expect to design a learning algorithm that is guaranteed to jointly converge to an equilibrium when deployed by autobidders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' This leaves us with the challenge of analyzing the dynamics of the learning path taken by autobidders who may not converge as they adjust bids in competition with one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Our Contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We present a novel algorithm for online bidding with budget and ROI con- straints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We note that this by itself is not new;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' it has recently been shown that dual-based methods can achieve vanishing regret while (approximately) satisfying such constraints [11, 12, 22, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The critical new feature of our algorithm is that in addition to achieving vanishing individual regret, it also yields a state-of-the-art aggregate guarantee for the entire market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' That is, when all bids are controlled by autobidders deploying our algorithm, the resulting expected liquid welfare is at least half of the optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' This is the best possible bound even for pure equilibria in second-price auctions with budgets [1, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' However, our aggregate guarantee does not rely on convergence to equilibrium;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' rather, we directly analyze the outcomes on the dynamic learning path of the algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' While we view this aggregate guarantee as the main result of the paper, we emphasize that it is crucial to also guarantee vanishing regret for each individual advertiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Even if the platform’s goal when designing an autobidder is to maximize the overall market efficiency, individual guar- antees are needed to convince the advertisers to make use of this service, rather than place bids themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Our results hold for a broad class of auction formats, including first-price and second-price auctions, and allow impression types (which determine, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=', click rates) to be drawn randomly in each round and potentially be correlated across agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Our algorithm is guaranteed to satisfy all constraints ex post with probability 1 (not just with high probability or with small expected violation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Further, it only requires bandit feedback from the underlying auction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=', only the outcome for the actual bid submitted, not the counterfactual outcomes for the alternative bids).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The latter is important because even when advertising platforms provide sufficient feedback to infer the counterfactual allocations of ad impressions (which they do not always), it can be difficult to accurately model whether a particular ad in the counterfactual slate would be clicked by a user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 1Liquid welfare is a standard notion of welfare under constraints;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' we discuss it more below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Autobidders with Budget and ROI Constraints 3 Our algorithm updates bids using a variation of stochastic gradient descent (SGD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Essentially, the autobidder’s goal is to maximize her bids (and therefore the total value won) subject to a collection of constraints that would be violated if bids were too high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' When there is only a bud- get constraint, a common idea is to use an SGD-based update to learn the constraint-pacing bid: the bid that would cause this constraint to become tight in expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' With both budget and ROI constraints, one could strive to maintain a constraint-pacing bid for each constraint, and somehow aggregate these per-constraint bids into one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' For example, Balseiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' [11] employ a primal-dual framework that, in effect, interpolates and places more weight on constraints that bind more tightly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' [22] apply this directly to the setting of ROI and budget constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Our approach has a similar flavor, but aggregates the per-constraint bids differently: we myopi- cally choose the per-constraint bid that is largest in a given round, then apply a gradient step to both per-constraint bids using the observed outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' This differs from a typical SGD-based ap- proach in that the smaller per-constraint bid is updated using the outcome from the larger bid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' However, as it turns out, this approach maintains an important invariant: the multiplier for each constraint encodes the total slack (or violation) of that constraint up to the current round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We will use this to show that all constraints will be satisfied with certainty in all rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' This invariant will also be crucial for establishing our efficiency bound, as it will allow us to track value and pay- ments in a manner that is agnostic to the details of the (potentially chaotic and non-convergent) bid dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We measure market efficiency via liquid welfare, the maximum amount the agents are willing to pay for the allocations that they receive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' This is the appropriate notion of welfare in settings like ours where the agents (autobidders) do not typically have an objective measured in dollars, but rather seek to maximize value subject to monetary constraints on how much can be paid for dif- ferent outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2 Liquid welfare generalizes the standard notion of welfare for agents with quasi- linear preferences in money (or, equivalently, only a constraint on the maximum willingness-to- pay in each round), as well as the original introduction of liquid welfare for budget constraints [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Our individual guarantees hold for an adversarial environment, without any assumptions on the other agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We attain vanishing regret relative to the constraint-pacing sequence: informally, a sequence of bids that maximize per-round value under the time-averaged constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Note that vanishing-regret results are impossible against the standard benchmark of best fixed bid (as dis- cussed further in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' To account for the changing environment, our regret bound depends on the path-length of the constraint-pacing sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Specialized to a stochastic environment (where the other agents’ bids are drawn from a fixed joint distribution), we obtain vanishing re- gret against the best fixed bid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Our emphasis is on autobidding algorithms that can satisfy both ROI and budget constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Our regret and liquid welfare bounds extend prior analysis in the budget-pacing setting with a budget constraint only, where it has been shown that the online bidding algorithm of Balseiro and Gur [10] achieves both aggregate and individual guarantees [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Handling the ROI constraint, and particularly both constraints jointly, introduces a variety of new technical challenges and motivates our proposed algorithm, as we discuss in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1 and Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1 Further Related Work Our work builds on the recent literature analyzing online algorithms for bidding under constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' For online bidding subject to a budget constraint, Borgs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' [13] analyze bidding dynamics un- der a multiplicative update rule and establish convergence for first-price auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Balseiro and 2When/if the advertisers’ objective is expressible in dollars, utilitarian welfare could also be a reasonable objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' How- ever, it is subject to strong impossibility results [21], which makes it less suitable for theoretical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 4 Brendan Lucier, Sarath Patathil, Aleksandrs Slivkins, and Mengxiao Zhang Gur [10] consider a different update rule under second-price auctions and show that it converges under some additional convexity assumptions and guarantees vanishing individual bidder regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Balseiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' [11] consider a variation of this approach using online mirror descent (OMD) and extends the guarantees to repeated truthful auctions without requiring any additional convexity assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Notably, their individual regret bound applies beyond stochastic settings to adver- sarial environments, with a loss that grows with the deviation from stochasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' For online bidding subject to more general constraints, including the ROI constraints we con- sider, Bernasconi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' [12] use convex duality to achieve vanishing regret in stochastic environ- ments, while satisfying the constraints with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' A general dual-based optimization framework for autobidding subject to ROI and budget constraints was proposed and evaluated in Gao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Golrezaei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' [26] consider budget and ROI constraints, and achieve low regret in a stochastic environment with bounds on expected constraint violations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' [22] extend the OMD approach to achieve vanishing regret while satisfying budget and ROI constraints with vanishing expected violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Our algorithm’s learning guarantees are incomparable: it satisfies the ROI and budget constraints with probability 1, with regret rate 푂(푇 7/8) (whereas 푂(푇 1/2) is achievable for relaxed constraints [22, 26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' A related line of work studies pacing equilibria: market equilibria in a single-shot game that abstracts repeated auctions when bidders have global constraints such as budgets and/or ROI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' For second-price auctions, Conitzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' [18] characterized the corresponding equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Aggarwal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' [1] show that under a broad class of constraints (including budget and ROI constraints), the expected liquid welfare obtained at any equilibrium is at least half of the optimal liquid welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Babaioff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' [5] provide a similar 2-approximation result in settings with “soft constraints” where agents have a separable and convex disutility for spending money.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Both results generalize to the case of budget constraints in a second-price auction, where finding a pure Nash equilibrium is known to be PPAD-hard [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' In contrast to this work, our analysis of liquid welfare does not rely on convergence to equilibrium, but rather applies directly to bidding dynamics that arise from our autobidding algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Conitzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' [17] extended pacing equilibria to first-price auctions, showed that the equilib- rium is essentially unique, and analyzed its properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Balseiro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' [9] consider an alternative equilibrium notion for a broad class of auctions, in which agents are not constrained to pacing but instead can make their bids arbitrarily contingent on realized impression values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' They present a revenue-equivalence result to bound liquid welfare at any equilibrium subject to a budget con- straint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' It has been widely noted that linear pacing is equivalent to optimal contingent bidding for truthful auctions [5, 10] but not more generally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We take the linear pacing approach, interpreted as the autobidder placing a bid that scales linearly with the realized impression value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Learning theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Repeated bidding under budget is a special case of bandits with knapsacks (BwK), a multi-armed bandit problem under global constraints [2, 6, 31], see Chapter 10 in [38] for a survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' BwK problems in adversarial environments do not admit regret bounds: instead, one is doomed to approximation ratios, even against a time-invariant benchmark and even in relatively simple examples [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' A similar impossibility result is derived in [10] specifically for repeated budget-constrained bidding in second-price auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The literature on BwK does not handle “cov- ering constraints" such as ROI,3 and is not concerned with aggregate guarantees such as ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Stochastic gradient descent is a standard algorithm in online convex optimization [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Convergence of learning algorithms in repeated games is well-studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' When algorithms have vanishing regret in terms of cumulative payoffs, the average play – time-averaged distribution over chosen actions – converges to a (coarse) correlated equilibrium [4, 29, 35], and this implies 3With a rare exception of Agrawal and Devanur [2], who handle covering constraints for stochastic environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Autobidders with Budget and ROI Constraints 5 welfare bounds for various auction formats in the absence of budget or ROI constraints [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' In contrast, for repeated auctions with budgets, low individual regret on its own does not imply any bounded approximation for liquid welfare [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Convergence in the last iterate is more challeng- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' While strong negative results are known even for two-player zero-sum games [7, 16, 34], a recent line of work [19, 20, 25, 39] achieves last-iterate convergence under full feedback and sub- stantial convexity-like assumptions, using two specific regret-minimizing algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' To the best of our understanding, these positive results do not apply to repeated auctions with budget or ROI constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 2 MODEL AND PRELIMINARIES We study a repeated auction game played by a collection of 푛 bidding agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' At each time 푡 = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' ,푇, the seller (or platform) has a single unit of good available to sell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' An allocation profile is a vector 풙 = (푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' ,푥푛) ∈ 푋 ⊆ [0, 1]푛 where 푥푘 is the quantity of the good allocated to agent 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' An allocation sequence is a sequence of allocation profiles (풙1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' , 풙푇 ) where 풙풕 = (푥1,푡, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' ,푥푛,푡) is the allocation profile at time 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='4 For each 푡, the good available for sale at time 푡 has a value 푐푘,푡 ∈ [0, 1] for agent 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We can think of 푐푘,푡 as a click probability for agent 푘 and impression 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='5 We emphasize that 푐푘,푡 is measured in clicks, not dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='6 For each 푡, the value profile c푡 = (푐1,푡, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' ,푐푛,푡) is drawn from a distribution 퐹 independently across time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Note that 퐹 is not necessarily a product distribution, so the value held by different agents can be arbitrarily correlated within each round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' At each time 푡 the good is allocated using an auction that proceeds as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Each agent 푘 submits a bid 훽푘,푡 ≥ 0, which can be interpreted as a bid (in dollars) per unit of value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' All agents submit bids simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The value profile is then realized, and this determines each agent’s effective bid 푏푘,푡 = 푐푘,푡훽푘,푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The auction is defined by an allocation rule 풙 and a payment rule 풑, where 풙(풃) ∈ [0, 1]푛 is the allocation profile generated under a bid profile 풃, and 푝푘(풃) ≥ 0 is the payment made by agent 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Allocation and payment rules are always weakly monotone in bids, meaning that 푥푘 (푏푘, b−푘) and 푝푘 (푏푘, b−푘) are weakly increasing in 푏푘 for any b−푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Given an implied realization of bids, we will often write 푥푘,푡 and 푝푘,푡 to mean the allocation and payment of agent 푘 in round 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The auction rules we consider will satisfy the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' First, winners are always selected from among those agents with the highest effective bid, with ties resolved arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Sec- ond, each agent’s payment per unit received lies between the highest and second-highest effective bids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' This class includes both first-price and second-price auctions, as well as any auction whose payments lie between the first and second price.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Some of our results actually apply more generally to any core auction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' we define these in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Constraints and Objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Each agent 푘 has a collection of constraints that their payments and allocations must satisfy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' These are as follows: (1) Value constraint 푣푘: in each round 푡, agent 푘 cannot pay more than 푣푘 per unit of value received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' That is, 푝푘,푡 ≤ 푣푘푐푘,푡푥푘,푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (2) ROI constraint 푤푘: in total over all rounds, agent 푘 cannot pay more than 푤푘 per unit of value received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' That is, � 푡 푝푘,푡 ≤ 푤푘 � 푡 푐푘,푡푥푘,푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 4We include a full notation table in Appendix A for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 5One could equivalently interpret this as a conversion rate, expected revenue lift, or any other metric of value that depends on advertiser and impression type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 6We find it most natural in this context to measure value in non-monetary units and express preferences for payments as constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' However, we will show in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1 that this is mathematically equivalent to measuring value in dollars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 6 Brendan Lucier, Sarath Patathil, Aleksandrs Slivkins, and Mengxiao Zhang (3) Budget constraint 퐵푘: the total payment made by agent 푘 over all rounds cannot exceed 퐵푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' That is, � 푡 푝푘,푡 ≤ 퐵푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The objective of each agent푘 is to maximize the effective value received over all rounds, � 푡 푐푘,푡푥푘,푡, subject to the constraints being satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We emphasize that all of these constraints bind ex post, and must be satisfied on every realiza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We observe that since an agent never pays more than its bid, there always exists an agent strategy that guarantees all constraints will be satisfied for any realization of values and the bids of others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' In particular, the first constraint is always satisfied if 훽푘,푡 ≤ 푣푘 for all 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The second con- straint is always satisfied if 훽푘,푡 ≤ 푤푘 for all 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The third constraint is always satisfied if 훽푘,푡 ≤ 퐵푘/푇 for all 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Bidding the minimum of these amounts in every round would therefore necessarily satisfy all constraints, though of course this may result in low objective value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1 Interpretation: Constraints and Pacing We now take a moment to discuss some alternative interpretations and notation for our con- strained optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Since 푣푘 is a constant, the agent’s objective is equivalent to maxi- mizing � 푡 푣푘푐푘,푡푥푘,푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' If we define 푣푘,푡 = 푣푘푐푘,푡 then we can rewrite the agent’s goal as maximizing � 푡 푣푘,푡푥푘,푡, where (푣1,푡, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' , 푣푛,푡) is drawn from a distribution each round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We can think of 푣푘,푡 as an implied maximum willingness to pay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Define 푣 ≜ max푘 푣푘, so that 푣푘,푡 ≤ 푣 for all 푘 and all 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Next, note that if 푤푘 ≥ 푣푘 then the ROI constraint is implied by the value constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We can therefore assume without loss that 푤푘 ≤ 푣푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' If we define 훾푘 = 푣푘/푤푘 ≥ 1 then the ROI constraint becomes 훾푘 � 푡 푝푘,푡 ≤ � 푡 푣푘,푡푥푘,푡, or equivalently (� 푡 푣푘,푡푥푘,푡)/(� 푡 푝푘,푡) ≥ 훾푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' This formulation motivates the name “ROI constraint,” and we will tend to use 훾푘 (rather than 푤푘) to express its parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Finally, for convenience we write 휌푘 = 퐵푘/푇 for agent 푘’s per-round budget constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Bidding Algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Each agent employs a bidding algorithm to determine its bids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Given the constraint parameters 푣푘, 퐵푘, and 훾푘 and the history of bids and realized outcomes up to round 푡, the algorithm determines the bid 훽푘,푡 for round 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We note that since each agent’s payment per unit received is always at least the next-highest effective bid, it is without loss to restrict attention to bids 훽푘,푡 ≤ 푣푘, since choosing any bid 훽푘,푡 > 푣푘 is a dominated action for the autobidder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='7 With this in mind, we can follow the literature on bid pacing and define 휇푘,푡 = �푣푘/훽푘,푡 � − 1, so that 훽푘,푡 = 푣푘/(1 + 휇푘,푡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' That is, in each round we think of the bidding algorithm as choosing a multiplier 휇푘,푡 ≥ 0, where 0 corresponds to bidding the maximum amount 푣푘, and as 휇푘,푡 grows the resulting bid shrinks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The algorithm’s goal is to optimize 휇푘,푡 ≥ 0 subject to the ROI and budget constraints, since the value constraint will necessarily be satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We emphasize that this formulation is without loss of generality, as any undominated choice of 훽푘,푡 can be expressed in this manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2 Aggregate Performance Metric: Liquid Welfare Liquid welfare is a measure of welfare in settings where agents utilities are described by pay- ment constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Intuitively, an agent’s liquid welfare for an allocation sequence is the agent’s maximum willingness to pay for the allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' This generalizes the common notion of welfare in quasi-linear environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' In our setting with value, budget, and ROI constraints, the definition of liquid welfare specializes as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 7Any bid 훽푘,푡 > 푣푘 is dominated by a bid of 푣푘: either the agent loses the auction or would anyway have won at a bid of 푣푘, in which case a bid of 푣푘 does just as well;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' or the agent wins at bid 훽푘,푡 but loses at bid 푣푘, in which case she must be paying more than 푣푘 per unit received in violation of the value constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Autobidders with Budget and ROI Constraints 7 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Given a sequence of value profiles v = (푣푘,푡) ∈ [0, 1]푛푇 and any sequence of feasible allocations x = (푥푘,푡) ∈ [0, 1]푛푇, the liquid value obtained by agent 푘 is 푊푘(x) = min � 퐵푘, 1 훾푘 �푇 푡=1 푣푘,푡푥푘,푡 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The liquid welfare of allocation sequence x ∈ 푋푇 is푊 (x) = �푛 푘=1푊푘(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We emphasize that liquid welfare depends on the allocations, but not on the agents’ payments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' However, it follows immediately from the definitions that as long as all constraints are satisfied, an agent’s total payment is a lower bound on her liquid welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' That is, the agent’s actual observed payment cannot be greater than her maximum willingness to pay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Fix any sequence of value profiles and outcomes (allocations and payments) such that the value, budget, and ROI constraints are satisfied for agent 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Then 푊푘(x) ≥ � 푡 푝푘,푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Our objective of interest is the expected liquid welfare obtained by the platform over any random- ness in the value sequence and the agents’ bidding strategies (and hence the resulting randomness in the allocation sequence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Since the bid placed in one round can depend on allocations obtained in the previous rounds, we define a mapping from the entire sequence of푇 value profiles to an allo- cation sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' An allocation sequence rule is a function x: [0, 1]푛푇 → 푋푇 , where 푥푘,푡 (v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' , v푇 ) is the allocation obtained by agent 푘 in round 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Then the expected liquid welfare of allocation sequence rule x is Ev1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=',v푇 ∼퐹 [푊 (x(v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' , v푇 ))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 3 WARMUP: ROI CONSTRAINTS AND UNLIMITED BUDGET In this section we warm up with the special case where 퐵푘 = ∞ for all 푘, meaning that there are no budget constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' While bidding algorithms are already known for this problem, we will build intuition by presenting a new algorithm with properties that will be useful when we adapt it to our general setting in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' As we focus on a single agent, we will drop the subscripts 푘 from our notation for the remainder of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' In the absence of budget constraints, the agent’s goal is to maximize � 푡 푣푡푥푡 subject to the constraint � 푡 푣푡푥푡 − 훾 � 푡 푝푡 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Consider a repeated second-price auction with 푣푡 = 1 for every 푡 and 훾 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' In this case the agent’s ROI constraint is � 푡 푥푡 − 2 � 푡 푝푡 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Suppose the competing bids are stochastic and stationary, with the highest bid of a competing agent being either 1/4 or 3/4 each round with equal probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Suppose that the bidding agent chooses 휇푡 = 0 for every 푡, meaning that it bids 푏푡 = 푣푡 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' In this case the agent wins every round and pays either 1/4 or 3/4, for an expected payment of 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The constraint is satisfied in expectation but may be violated on some realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Indeed, in the (unlikely) event that the competing bid is 3/4 every round, the only way to satisfy the ROI constraint would be to lose every round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' On the other hand, if the competing bid is 1/4 on at least half of the instances then it is optimal to win every round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' As discussed in Section 2, setting 휇푡 = 훾 − 1 in every round (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=', bidding 푣푡/훾) is guaranteed to satisfy the ROI constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' But, as in the example above, the optimal choice of 휇푡 may be lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The next lemma shows that ROI is monotone in 휇푡 for our auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' That is, higher bids result in lower average return on investment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' This lets us think of the agent as trying to reduce 휇 as much as possible subject to the ROI constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Fix any auction in our class, any agent 푘, and any bids b−푘 of the other agents, and write 푥푡 (휇) and 푝푡 (휇) for the allocation and payment that results when agent 푘 selects multiplier 휇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Then 푣푡푥푡 (휇) − 훾푝푡 (휇) is weakly increasing in 휇, for 휇 ∈ [0,훾 − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 8 Brendan Lucier, Sarath Patathil, Aleksandrs Slivkins, and Mengxiao Zhang This motivates us to consider Algorithm 1, which initially takes the safe action 휇1 = 훾 − 1 but updates 휇푡 online in response to auction feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' In each round 푡 it places bid 푏푡 = 푣푡/(1 + 휇푡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='8 If the observed allocation 푥푡 and payment 푝푡 in round 푡 are such that 푣푡푥푡 > 훾푝푡 then the ROI constraint is satisfied with room to spare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' This suggests the bid was lower than necessary, so the algorithm reduces 휇 by an amount proportional to 푣푡푥푡 −훾푝푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Likewise, if 푣푡푥푡 < 훾푝푡, then the ROI constraint was violated in round 푡 so the algorithm responds by increasing 휇 proportionally to the violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' This can be interpreted as stochastic gradient descent (SGD): if 휇푡 is such that the ROI constraint is satisfied in expectation then the expected update is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We will make this connection with SGD more precise in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' ALGORITHM 1: Bidding Under ROI Constraint Input: ROI constraint parameter 훾 Initialization: 휇1 = 훾 − 1 and learning rate 휂 for 푡 = 1, 2, · · · ,푇 do Set bid 푏푡 = 푣푡 1+max{휇푡,0} Observe allocation 푥푡 ∈ [0, 1] and payment 푝푡 Update the multiplier 휇푡+1 = 휇푡 − 휂(푣푡푥푡 − 훾푝푡) Algorithm 1 appears myopic at first glance, always updating its bids in response to the latest outcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' However, we note that 휇푡 implicitly encodes the status of the aggregate ROI constraint up to round 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Indeed, an immediate implication of the update rule is that 휇푡 is 휇1 minus a term proportional to � 휏<푡 (푣휏푥휏 − 훾푝휏), which is precisely the aggregate slack (or violation) of the ROI constraint up to time 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' A small value of 휇푡 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=', a high bid) therefore occurs only if there is sub- stantial slack in the ROI constraint up to round 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' What does this mean for the performance of Algorithm 1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Note that the bidding agent may receive less than the desired return on investment 훾 in any given round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' However, as we now show, the algorithm is guaranteed, with probability 1, to satisfy the ROI constraint in aggregate over its 푇 rounds (and indeed, over any prefix of the 푇 rounds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Intuitively, as the ROI constraint gets closer to being violated in aggregate, 휇푡 gets closer to 휇1 = 훾 − 1, the “safe” choice at which the ROI constraint will necessarily be satisfied each round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Fix any (possibly adversarial) mapping from sequences of bids to sequences of alloca- tions and payments such that 푝푡 ≤ 푏푡푥푡 for every 푡, and suppose 휂 ≤ 1/¯푣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Then for the allocations and payments resulting from applying Algorithm 1, we have � 푡 푣푡푥푡 ≥ 훾 � 푡 푝푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' An immediate corollary of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2 is that, for every input sequence, the sequence of multipli- ers {휇푡 } will satisfy 휇푡 ≤ 훾−1 for every푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='9 As we show in the next section, this will imply high liquid welfare when multiple agents use (a generalization of) this algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' As it turns out, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2 can also be used to show that Algorithm 1 achieves vanishing regret relative to the best choice of 휇 in hindsight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We prove this formally in the next section in a more general setting, but for now let us briefly describe the intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Since the ROI constraint is satisfied with probability 1 by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2, any loss in value must come from bidding too low relative to the optimal fixed strategy in hindsight, say 휇∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' However, the expected update to 휇푡 turns out to be the gradient of a function that, on the range [휇∗,훾 − 1], is (a) convex and (b) closely related to the value function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Standard SGD analysis then allows us to bound the total loss due to rounds where 휇푡 is larger than 휇∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 8With one small caveat: 휇푡 could be negative in some rounds, in which case we treat it as 0 when setting the bid 푏푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 9The difference 휇1−휇푡 is proportional to the slack in the ROI constraint up to time 푡, which by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2 is never negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Autobidders with Budget and ROI Constraints 9 4 BIDDING UNDER ROI AND BUDGET CONSTRAINTS We now turn to the general problem with both ROI and budget constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We study an extension of our previous algorithm to this setting, listed as Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The algorithm now keeps track of two multipliers, 휇푅 and 휇퐵, corresponding to the ROI and budget constraints respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' At each round, the algorithm will place a bid using whichever of the multipliers is more conservative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=', whichever results in the lower bid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Each of the multipliers is then updated according to the realized allocation and payment, including the multiplier that did not correspond to the placed bid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Multiplier 휇푅 푘,푡 is updated in the same way as Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The idea behind our update rule for multiplier 휇퐵 푘,푡 is similar: each round we compare the observed payment 푝푘,푡 with 휌푘 = 퐵푘/푇, the target per-round payment according to budget constraint 퐵푘, and we update 휇퐵 푘,푡 proportional to the difference where 휂푘,푅,휂푘,퐵 > 0 are the corresponding learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 휇푅 푘,푡+1 = 휇푅 푘,푡 − 휂푘,푅(푣푘,푡푥푘,푡 − 훾푘푝푘,푡), 휇퐵 푘,푡+1 = 휇퐵 푘,푡 − 휂푘,퐵(휌푘 − 푝푘,푡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' ALGORITHM 2: Bidding Under ROI and Budget Constraints (for agent 푘) Input: per-round budget constraint 휌푘 > 0 and ROI constraint parameter 훾푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Initialization: 휇푅 푘,1 = 훾푘 − 1, 휇퐵 푘,1 = 푣 휌 − 1 and learning rate 휂푘,푅,휂푘,퐵 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' for 푡 = 1, 2, · · · ,푇 do Calculate 휇푘,푡 = max{휇푅 푘,푡, 휇퐵 푘,푡, 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Receive value 푣푘,푡 and set the bid 푏푘,푡 = 푣푘,푡 1+휇푘,푡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Receive the allocation 푥푘,푡 ∈ [0, 1] and the payment 푝푘,푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Update the ROI-multiplier 휇푅 푘,푡+1 = 휇푅 푘,푡 + 휂푘,푅(훾푘푝푘,푡 − 푣푘,푡푥푘,푡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Update the budget-multiplier 휇퐵 푘,푡+1 = 휇퐵 푘,푡 + 휂푘,퐵 (푝푘,푡 − 휌푘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We note that in the case where the budget constraint is infinite (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=', 퐵푘 and hence 휌푘 is +∞), this update rule yields 휇퐵 푘,푡 < 0 for every round 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Thus, this algorithm reduces to Algorithm 1 in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' On the other hand, if 훾푘 = 1 and hence there is effectively no ROI constraint, then we will necessarily have 휇푅 푘,푡 ≤ 0 for every round 푡 (since 푝푘,푡 ≤ 푏푘,푡푥푘,푡 ≤ 푣푘,푡푥푘,푡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The resulting algorithm is nearly identical to the one proposed by Balseiro and Gur [10] for bidding subject to a budget constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We can therefore view Algorithm 2 as a generalization of both algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The key insight behind Algorithm 2 is that the multipliers 휇푅 푘,푡 and 휇퐵 푘,푡 encode the cumula- tive slack in the ROI and budget constraints, respectively, up to time 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Similar to our analysis of Algorithm 1, this implies that Algorithm 2 will necessarily satisfy all of its constraints ex post with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Formally, we have the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Fix any agent 푘 and any (possibly adversarial) mapping from sequences of bids to sequences of allocations and payments such that 푝푘,푡 ≤ 푏푘,푡푥푘,푡 for every 푡, and suppose 휂푘,푅 ≤ 1 푣,휂푘,퐵 ≤ min{ 1 휌푘 , 1 푣 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Then for the allocations and payments resulting from applying Algorithm 2, we have � 푡 푣푘,푡푥푘,푡 ≥ 훾푘 � 푡 푝푘,푡 and � 푡 푝푘,푡 ≤ 퐵푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Also, similar to Algorithm 1, another implication of this interpretation of 휇푅 푘,푡 and 휇퐵 푘,푡 is that these multipliers are never higher than their “safe” levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Algorithm 2 with 휂푘,푅 ≤ 1 푣 and 휂푘,퐵 ≤ 1 푣 guarantees that for all 푡 ∈ [푇], 휇푅 푘,푡 ≤ 훾푘 − 1 and 휇퐵 푘,푡 ≤ 푣 휌푘 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The remainder of this section is dedicated to establishing our aggregate liquid welfare and in- dividual regret guarantees for Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1 we prove that in expectation over the 10 Brendan Lucier, Sarath Patathil, Aleksandrs Slivkins, and Mengxiao Zhang realization of values, Algorithm 2 always obtains at least half of the optimal liquid welfare in hind- sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Then in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2 we establish that the algorithm also satisfies strong individual regret guarantees, even in non-stochastic settings where the optimal bid sequence has bounded path length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1 Liquid Welfare Analysis of Algorithm 2 In this subsection we prove that when all agents employ Algorithm 2 the resulting expected liquid welfare is approximately optimal, where the expectation is over the realization of agent values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We will actually show that the expected liquid welfare is at least half of the optimal ex-ante liquid wel- fare (defined in Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1), which is an agent’s willingness to pay for her expected allocation sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Note that this is a stronger benchmark compared to the optimal expected liquid welfare Ev1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=',v푇 [푊 (x)] due to Jensen’s inequality (See Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' For any distribution 퐹 over valuation profiles and any allocation rule y : [0,푣]푛 → 푋, the ex-ante liquid value of agent 푘 is 푊 푘 (y, 퐹) := 푇 × min � 퐵푘, 1 훾푘 Ev∼퐹 [푦푘 (v)푣푘] � , (1) and the ex-ante liquid welfare is given by 푊 (y, 퐹) = �푛 푖=1푊 푘(y, 퐹).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Recall that to this point we have focused our attention on single-item auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Our liquid wel- fare bound will actually apply to the following more general class of core auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Given any downward-closed set 푋 ⊆ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 1]푛 of feasible allocations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' an auction with allocation rule 푥 : [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푣]푛 → 푋 and payment rule 푝 : [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푣]푛 → R푛 ≥0 is a core auction if it is Welfare-maximizing: x(v) ∈ arg maxx∈푋 {� 푖 푣푖(x)} Individually rational: 푝푖(v) ≤ 푣푖(x(v)) for all 푖 Deviation-proof: for all 푆 ⊆ [푛] and y ∈ 푋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' � 푖∉푆 푝푖(v) ≥ � 푖 ∈푆(푣푖(푦푖) − 푣푖(푥푖(v))) Core auctions include first-price and second-price single-item auctions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' but also more general formats like generalized second-price auctions for multiple slots and separable click rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' See [23] for further discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We are now ready to state the main result of this subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Fix any core auction and any distribution 퐹 over agent value profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Suppose all agents bid by employing Algorithm 2 with max{휂푘,푅,휂푘,퐵} ≤ 푣 푣+휌푘 √ log(푣푛푇) √ 푇 , and write x : [0,푣]푛푇 → 푋푇 for the corresponding allocation sequence rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Then for any allocation rule y : [0,푣]푛 → 푋, Ev1,···v푇 ∼퐹 [푊 (x(v1, · · · , v푇 ))] ≥ 푊 (y, 퐹) 2 − O(푛푣 � 푇 log(푣푛푇)) (2) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1 Proof Intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Before going into the details of our proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1, let us give a sketch of the high-level ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Fix some agent푘 and consider the liquid welfare obtained by agent푘 over all 푇 rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' By definition, this liquid welfare is either the agent’s budget 퐵푘 or the sum of ROI-scaled gained values 1 훾푘 � 푡 푣푘,푡푥푘,푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The former is the easy case: since 퐵푘 is an upper bound on ex-ante liquid welfare, if agent 푘’s liquid welfare is 퐵푘 then this is at least as good as the benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' So the difficulty lies in handling the latter case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' In order to bound 1 훾푘 � 푡 푣푘,푡푥푘,푡, we consider the progression of the bidding multiplier 휇푘,푡 over rounds 푡 = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' ,푇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The multiplier may drift up and down over time and may not necessarily converge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We will distinguish between rounds in which 휇푘,푡 lies above 훾푘 − 1 and rounds in which 휇푘,푡 lies below 훾푘 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Autobidders with Budget and ROI Constraints 11 Whenever 휇푘,푡 < 훾푘 − 1, our intuition is that agent 푘 is not bidding “too low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='" Indeed, the bid of agent 푘 must be at least 푣푘,푡/훾푘 on any such round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' So even if agent 푘 loses in this round, the winning bidder(s) must be paying at least 푣푘,푡/훾푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We can therefore charge any loss in liquid welfare against the total revenue collected, which (by Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1) is itself at most the liquid welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' On the other hand, in any round 푡 where 휇푘,푡 > 훾푘 − 1, we know that 휇푘,푡 = 휇퐵 푘,푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' This is because the ROI multiplier 휇푅 푘,푡 never lies above 훾푘 − 1 according to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Thus, over any contiguous interval of rounds in which 휇푘,푡 > 훾푘 − 1, it must be the budget multiplier that is determining the bid of agent 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Since any such interval must begin and end close to the threshold 훾푘 −1, the update rule for 휇퐵 푘,푡 implies that the total spend over the (say) 푡 rounds of that interval is very close to 푡 ×휌푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' As the optimal ex-ante liquid welfare cannot be more than 휌푘 per round, the obtained liquid welfare must be comparable to the optimal liquid welfare over this interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Thus, in every case, we can relate the obtained liquid welfare to the benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' There are some technical challenges to formalizing this intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Our very first step was to condition on whether agent 푘’s liquid welfare is determined by her budget or by her ROI-scaled gained value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' However, this conditioning introduces correlations between rounds, and in particular it impacts our assertion that the ex-ante liquid welfare is at most 휌푘 per round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We address this by explicitly bounding the impact of such correlations and arguing that they are small with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' This introduces the additive error term in the theorem statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We note that similar issues arise when analyzing the liquid welfare of the budget-pacing algorithm of Balseiro and Gur [10];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' our solution is a variation of an idea due to [23] for the budgeted case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Another technical issue that arises is specific to handling budget and ROI constraints simulta- neously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The intuition above does not carefully account for rounds in which 휇푘,푡 switches from lying strictly below 훾푘 − 1 to strictly above 훾푘 − 1 or vice-versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' It turns out that these transition rounds introduce error terms that can accumulate substantially;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' indeed, when we said above that the total spend over an interval is very close to 푡 ×휌푘, this approximation can be off by up to 푣 +휌푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We handle this by considering separately those rounds in which 휇푘,푡 is very close to the boundary 훾푘 − 1, and directly relate the outcomes to what would occur precisely on the boundary itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The resulting error terms are yet another source of the additive error in the theorem statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2 Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We now turn to the formal proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We prove the the- orem in three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We first show that with high probability, there is no significant correlation between the progression of our algorithm up to time 푡 and the ex-ante benchmark evaluated in future rounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We then condition on this event and bound the liquid welfare obtained on a per- instance basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Finally, we take an expectation over realizations to obtain the desired bound on expected liquid welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Step 1: Bounds on ex-ante Allocate Rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' First, it is without loss of generality to consider only the allocation rules y which satisfy Ev∼퐹 � 1 훾푘 푦푘 (v)푣푘 � ≤ 휌푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (3) This is because for any 푦 that violates this constraint, we can always decrease the allocation for agent 푘 without affecting 푊 (y, 퐹).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We would actually like to make a stronger claim that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (3) holds for every round 푡 in which 휇푘,푡 > 훾푘 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' To this end, we will show that, with high probability, the ex-ante optimal allocation rule 푦 does not generate significantly different outcomes in rounds where 휇푘,푡 > 훾푘 −1 and rounds where 휇푘,푡 ≤ 훾푘 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 12 Brendan Lucier, Sarath Patathil, Aleksandrs Slivkins, and Mengxiao Zhang For each agent 푘, define the following quantity: 푅푘 (v) ≜ � 푡 � 1{휇푘,푡 ≤ 훾푘 − 1} 1 훾푘 푦푘 (v)푣푘,푡 + 1{휇푘,푡 > 훾푘 − 1}휌푘 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (4) We can then use the theory of concentration of martingales to establish that the following bound on 푅푘 (v) holds with probability at least 1 − 1/(푣푛푇)2: 푅푘 (v) ≤ 휌푘 ·푇 + 푣 � 푇 log(푣푛푇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (5) We prove Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (5) in Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Now, taking a union bound over all agents푘 ∈ [푛], with probability at least 1 − 1/(푣푇)2, we have: 푅푘 (v) ≤ 휌푘 ·푇 + 푣 � 푇 log(푣푛푇), ∀푘 ∈ [푛] (6) We say that a value realization is “good” if it satisfies Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Step 2: Liquid Welfare of “Good” Value Realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Fix any “good” value profile realization 푣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' For any advertiser 푘 whose liquid welfare is 퐵푘, from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (6), we know that: 푊푘 (v) = 퐵푘 ≥ 푅푘 (v) − 푣 � 푇 log(푣푛푇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (7) Now we look at those agents 퐴 ⊆ [푛] for which the liquid welfare is strictly less than 퐵푘: 푊푘(v) = 1 훾푘 푇� 푡=1 푥푘,푡푣푘,푡 < 퐵푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' As we did in (7), we again wish to bound 푊푘 (v) with respect to 푅푘 (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' To that end, we will derive a bound on 푊푘(v) that accounts for variation in 휇푘,푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' For notational convenience, let 휂푘 = max{휂푘,푅,휂푘,퐵}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' For each round 푡, Let 푆푡 ⊆ 퐴 denote the agents for whom 휇푘,푡 ≤ 훾푘 −1, and푇푡 ⊆ 푆푡 for the agents for whom훾푘 −1−휂푘(푣 + 휌푘) < 휇푘,푡 ≤ 훾푘 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' That is, 푆푡 are the agents bidding “high enough,” and 푇푡 are the agents in 푆푡 that are “close to” the threshold value 훾푘 − 1 in round 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The following inequality is guaranteed if all agents 푘 ∈ [푛] apply Algorithm 2: � 푘 ∈퐴 1 훾푘 푇 � 푡=1 푥푘,푡푣푘,푡 ≥ � 푘 ∈퐴 푇 � 푡=1 � 1(푘 ∈ 푆푡) 1 훾푘 푥푘,푡푣푘,푡 − 1(푘 ∈ 푇푡)푝푘,푡 + 1(휇푘,푡 > 훾푘 − 1)휌푘 � (8) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Fix some agent 푘 ∈ 퐴.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Divide the time interval [1,푇] into intervals (퐼1, 퐼2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' ) in the following manner: each interval 퐼 = [푡1,푡2) is a minimal interval such that 휇푘,푡1 ≤ 훾푘 − 1 and 휇푘,푡2 ≤ 훾푘 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' That is, 휇푘,푡 > 훾푘 − 1 for all 푡1 < 푡 < 푡2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Note that according to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2, we know that 휇푘,푡 = 휇퐵 푘,푡 when 푡 ∈ (푡1,푡2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We wish to bound 1 훾푘 � 푡 ∈퐼 푥푘,푡푣푘,푡 for each such interval 퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Note that if 휇푘,푡1 ≤ 훾푘 −1−휂푘(푣 +휌푘), then we must have 푡2 = 푡1 + 1 (since 휇푘,푡1+1 ≤ 휇푡1 + 휂푘(푣 + 휌푘) ≤ 훾푘 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Thus, when 휇푘,푡1 ≤ 훾푘 − 1 − 휂푘 (푣 + 휌푘), we have 1 훾푘 �푡2−1 푡=푡1 푥푘,푡푣푘,푡 = 1 훾푘 푥푘,푡1푣푘,푡1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' On the other hand, if 훾푘 − 1 − 휂푘 (푣 + 휌푘) < 휇푘,푡1 ≤ 훾푘 − 1, we have 훾푘 − 1 < 휇푘,푡2−1 = 휇퐵 푘,푡2−1 = 휇퐵 푘,푡1 + 휂푘 푡2−2 � 휏=푡1 (푝푘,휏 − 휌푘) ≤ 훾푘 − 1 + 휂푘 푡2−2 � 휏=푡1 (푝푘,휏 − 휌푘), which means that �푡2−2 휏=푡1 (푝푘,휏 − 휌푘) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Since 푝푘,푡 ≤ 푏푘,푡푥푘,푡 < 1 훾푘 푥푘,푡푣푘,푡 for all 푡1 < 푡 < 푡2, we can conclude that 1 훾푘 푡2−1 � 푡=푡1 푥푘,푡푣푘,푡 ≥ 1 훾푘 푥푘,푡1푣푘,푡1 + 푡2−1 � 푡=푡1+1 푝푘,푡 ≥ 1 훾푘 푥푘,푡1푣푘,푡1 − 푝푘,푡1 + (푡2 − 푡1 − 1)휌푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Autobidders with Budget and ROI Constraints 13 Summing over all time steps, we conclude that 1 훾푘 푇� 푡=1 푥푘,푡푣푘,푡 ≥ 푇� 푡=1 � 1(휇푘,푡 ≤ 훾푘 − 1) 1 훾푘 푥푘,푡푣푘,푡 − 1(훾푘 − 1 − 휂푘(푣 + 휌푘) < 휇푘,푡 ≤ 훾푘 − 1)푝푘,푡 + 1(휇푘,푡 > 훾푘 − 1)휌푘 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Summing this inequality over all agents yields Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' □ Our next goal is to relate the terms in the right-hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (8) with the corresponding terms in 푅푘 (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Fix some round 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We will focus on the first two terms in the expression inside the summation on the right hand side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Consider the agents in 푇푡, which are the agents for whom 훾푘 − 1 − 휂푘(푣 + 휌푘) < 휇푘,푡 ≤ 훾푘 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We have � 푘 ∈푇푡 � 1 훾푘 푥푘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡푣푘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡 − 푝푘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡 � = � 푘 ∈푇푡 � 1 훾푘 푦푘 (v푡)푣푘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡 − 1 훾푘 푣푘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡 (푦푘 (v푡) − 푥푘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡) − 푝푘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡 � ≥ � 푘 ∈푇푡 � 1 훾푘 푦푘 (v푡)푣푘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡 − 푣푘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡 1 + 휇푘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡 (푦푘 (v푡) − 푥푘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡) − 휂푘(푣 + 휌푘)2 − 푝푘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡 � = � 푘 ∈푇푡 1 훾푘 푦푘 (v푡)푣푘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡 − � 푘 ∈푇푡 푣푘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡 1 + 휇푘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡 (푦푘 (v푡) − 푥푘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡) − � 푘 ∈푇푡 푝푘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡 − |푇푡 |휂푘 (푣 + 휌푘)2 where the inequality follows from the definition of 푇푡: if 푦푘 (v푡) ≥ 푥푘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡 we use that 휇푘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡 ≤ 훾푘 − 1 and hence 1 훾푘 ≤ 1 휇푘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' whereas if 푦푘 (v푡) < 푥푘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡 we use that 휇푘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡 ≥ 훾푘 − 1 − 휂푘 (푣 + 휌푘) and hence 1 훾푘 ≥ 1 휇푘,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡+1 − 휂푘(푣 + 휌푘).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' On the other hand, for agents in 푆푡 \\푇푡, we have � 푘 ∈푆푡 \\푇푡 1 훾푘 푥푘,푡푣푘,푡 ≥ � 푘 ∈푆푡 \\푇푡 � 1 훾푘 푦푘 (v푡)푣푘,푡 − 1 훾푘 푣푘,푡 (푦푘 (v푡) − 푥푘,푡)+ � ≥ � 푘 ∈푆푡 \\푇푡 � 1 훾푘 푦푘 (v푡)푣푘,푡 − 푣푘,푡 1 + 휇푘,푡 (푦푘 (v푡) − 푥푘,푡)+ � = � 푘 ∈푆푡 \\푇푡 1 훾푘 푦푘 (v푡)푣푘,푡 − � 푘 ∈푆푡 \\푇푡 푣푘,푡 1 + 휇푘,푡 (푦푘 (v푡) − 푥푘,푡)+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Let푈푡 be the set of all agents in푇푡, plus the agents in 푆푡 \\푇푡 such that 푥푘,푡 ≤ 푦푘 (v푡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Then, adding our two inequalities together gives � 푘 ∈푆푡 � 1 훾푘 푥푘,푡푣푘,푡 � − � 푘 ∈푇푡 푝푘,푡 ≥ � 푘 ∈푆푡 1 훾푘 푦푘 (v푡)푣푘,푡 − � 푘 ∈푈푡 푣푘,푡 1 + 휇푘,푡 (푦푘 (v푡) − 푥푘,푡) − � 푘 ∈푇푡 푝푘,푡 − |푇푡 |휂푘 (푣 + 휌푘)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We wish to bound the term � 푘 ∈푈푡 푣푘,푡 1+휇푘,푡 (푦푘 (v푡) −푥푘,푡) from the inequality above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Note that this is the exactly the difference in declared value (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=', bid) for 푦푘 (v푡) and 푥푘,푡 for agents in 푈푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' It is here where we use the fact that the underlying auction is a core auction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' From the definition of a core auction, this difference in bids is at most the sum of payments of agents not in 푈푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Therefore, 14 Brendan Lucier, Sarath Patathil, Aleksandrs Slivkins, and Mengxiao Zhang � 푘 ∈푆푡 � 1 훾푘 푥푘,푡푣푘,푡 � − � 푘 ∈푇푡 푝푘,푡 ≥ � 푘 ∈푆푡 1 훾푘 푦푘 (v푡)푣푘,푡 − � 푘∉푈푡 푝푘,푡 − � 푘 ∈푇푡 푝푘,푡 − |푇푡 |휂푘(푣 + 휌푘)2 ≥ � 푘 ∈푆푡 1 훾푘 푦푘 (v푡)푣푘,푡 − � 푘 푝푘,푡 − |푇푡 |휂푘(푣 + 휌푘)2, (9) where in the second inequality we used the fact that 푇푡 ⊆ 푈푡, so the two sums of over payments are over disjoint sets of agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Summing up Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (9) over all rounds and substituting into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (8) and using the definitions of 푊푘(v) and 푅푘 (v), we conclude that � 푘 ∈퐴 푊푘(v) ≥ � 푘 ∈퐴 푅푘 (v) − � 푡 푛 � 푘=1 푝푘,푡 − 휂푘(푣 + 휌푘)2푛푇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Summing over all agents 푘 ∈ [푛], we have that for every “good” value realization 푣, � 푘 푊푘 (v) ≥ � 푘 푅푘 (v) − � 푡 푛 � 푘=1 푝푘,푡 − 푣푛 � 푇 log(푣푛푇) − 휂푘(푣 + 휌푘)2푛푇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (10) Step 3: Bounding Expected Liquid Welfare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Recall from Observation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1 that the total rev- enue collected over all rounds will never be greater than the liquid welfare of the allocation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' In other words, � 푡 �푛 푘=1 푝푘,푡 ≤ � 푘 푊푘 (v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We can therefore rearrange Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (10) to conclude that 2 � 푘 푊푘(v) ≥ � 푘 푅푘 (v) − 푣푛 � 푇 log(푣푛푇) − 휂푘(푣 + 휌푘)2푛푇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Taking expectations over 푣 and conditioning on the good event, we conclude that our expected liquid welfare is at least half of the expected optimal liquid welfare with an error term that grows at a rate of O(푣푛 � 푇 log(푣푛푇)), as we take 휂푘 ≤ 푣 푣+휌푘 � log(푣푛푇) 푇 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' This completes the proof of Theo- rem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2 Individual Regret Guarantees of Algorithm 2 In this subsection, we consider the performance of an individual autobidder 푘 on its optimization problem when applying Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We abstract away the bids of other agents as supplied by an adversary (which may adapt to the outcomes of the previous rounds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' In particular, we do not assume that the other agents are controlled by any particular algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Our regret bound holds w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' a non-standard benchmark: the per-round pacing multipliers discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Since we focus on just agent 푘 throughout this subsection, we will drop the subscript 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' For each round 푡, we will write 푥푡 = 푥푡 (휇) and 푝푡 = 푝푡 (휇) for the allocation and payment if bidder 푘 picks multiplier 휇 in round 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Note that these depend on the realized value of bidder 푘 as well as the bids of the other auction participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Recall also that 푥푡 and 푝푡 are both weakly non-increasing in 휇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The expectations in this section are taken with respect to the randomness in value profiles of agent 푘 as well as the bids supplied by the adversary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Further, we define the following quantities: Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' For any round푡 ∈ [푇], define the expected budget expenditure 푍퐵 푡 (휇), the expected ROI expenditure 푍푅 푡 (휇), the expected ROI gain 휌푡 (휇), and the expected gained value 푉푡 (휇) when the bidder chooses multiplier 휇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Letting (푥)+ = max{푥, 0}, 푍퐵 푡 (휇) ≜ E [푝푡 (휇)] and 푍푅 푡 (휇) ≜ E � (훾푝푡 (휇) − 푣푡푥푡 (휇))+� , 휌푡 (휇) ≜ E � (푣푡푥푡 (휇) − 훾푝푡 (휇))+� and 푉푡 (휇) ≜ E [푣푡푥푡 (휇)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Autobidders with Budget and ROI Constraints 15 Note that 푍퐵 푡 (휇) and 푍푅 푡 (휇) − 휌푡 (휇) are both non-increasing functions when 휇 ≥ 0 and 휇 ∈ [0,훾 − 1], as is 푉푡 (휇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Formally, we have the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' For any 푡 ∈ [푇], 푍퐵 푡 (휇) is monotonically non-increasing for 휇 ≥ 0 and 푍푅 푡 (휇) − 휌푡 (휇) is monotonically non-increasing for 휇 ∈ [0,훾 − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Note also that 푍푅 푡 (휇)−휌푡 (휇) is precisely the expected value of (훾푝푡 (휇)−푣푡푥푡 (휇));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 푍푅 푡 (휇) captures the positive part of this random variable, and 휌푡 (휇) captures the negative part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We also assume Lipschitzness of allocations and payments with respect to 휇, which implies that 푍퐵 푡 , 푍푅 푡 , 휌푡, and 푉푡 are all Lipschitz as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' This can be interpreted as a requirement that the allocation and payment functions are sufficiently smooth as a function of an autobidder’s bid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='10 Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 푍퐵 푡 (휇), 푍푅 푡 (휇), 푉푡 (휇) and 휌푡 (휇) are all 휆-Lipschitz for all 푡 ∈ [푇], for some 휆 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Our regret bound holds with respect to a non-standard benchmark based on per-round pac- ing multipliers 휇∗ 푡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Informally, it is a value-optimizing multiplier for a given round 푡 subject to the time-averaged constraints applied to this round.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' While not necessarily optimal globally, these multipliers represent a reasonable goal for an online bidding algorithm in a complex, adversarial environment (recall that vanishing regret bounds with respect to the standard benchmark of the best-fixed-multiplier are impossible against an adversary [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The formal definition follows: Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='4 (Pacing multipliers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' For a given round 푡 ∈ [푇], the budget-pacing multiplier 휇퐵∗ 푡 is any 휇 ∈ [0, 푣 휌 − 1] with 푍퐵 푡 (휇) = 휌, or 0 if no such 휇 exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The ROI-pacing multiplier 휇푅∗ 푡 is any 휇 ∈ [0,훾 − 1] with 푍푅 푡 (휇) = 휌푡 (휇), or 0 if no such 휇 exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The pacing multiplier is 휇∗ 푡 = max{휇퐵∗ 푡 , 휇푅∗ 푡 } ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Thus, our notion of regret is defined as follows: Reg(푇) ≜ � 푡 ∈[푇 ] 푉푡 (휇∗ 푡 ) − 푉푡 (휇푡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Our regret bound depends on the amount of drift in the environment, captured (in a fairly weak way) by the path-lengths of the per-round pacing multipliers: 푃푅 푇 := � 푡 ∈[푇 ] ���휇푅∗ 푡 − 휇푅∗ 푡+1 ��� and 푃퐵 푇 := � 푡 ∈[푇 ] ���휇퐵∗ 푡 − 휇퐵∗ 푡+1 ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We are now ready to state our main individual guarantee: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Fix any distribution over the values of agent푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Assume Lipschitzness (Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Algorithm 2 with parameters 휂퐵 = √휌/ � 푇 (푣 + 휌) and 휂푅 = 1 √ 푇 (훾+1)푣 guarantees that E [Reg(푇)] ≤ O � (푃푅 푇 + 1) 1 4 휆 3 4 ((훾 + 1)푇) 7 8 + (푣 + 휌) 7 4 휌− 5 4 � 휆(1 + 푃퐵 푇 )푇 3 4 � , When 휆,훾,푣, 휌 are all constants, we have E[Reg] ≤ O((푃푅 푇 + 1) 1 4푇 7 8 + (푃퐵 푇 + 1) 1 2푇 3 4 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' In a stationary stochastic setting, where the competing bids and the agent’s value are drawn i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='d from a fixed distribution, our benchmark boils down to the best feasible multiplier, and the pathlengths are 푃퐵 푇 = 푃푅 푇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 10We note that this assumption could be satisfied by adding 푂 (휆) noise to the multiplier selected by any given autobidder, at a loss of welfare proportional to 휆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 16 Brendan Lucier, Sarath Patathil, Aleksandrs Slivkins, and Mengxiao Zhang Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' In the stationary stochastic setting, under the same conditions as stated in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2 with (휆,훾,푣, 휌) all constants, Algorithm 2 guarantees that max 휇 ∈Π � 푡 ∈[푇 ] 푉푡 (휇) − 푉푡 (휇푡) ≤ O(푇 7 8 ), where Π is the set of all constraint-feasible multipliers, Π = {휇 ≥ 0 : � 푡 ∈[푇 ] E[푍퐵 푡 (휇)] ≤ 퐵 and � 푡 ∈[푇 ] E[푍푅 푡 (휇) − 휌푡 (휇)] ≤ 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1 Proof intuition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Before proving Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2, we first provide the high-level idea of the proof and highlight some technical difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The high-level idea is to define auxiliary stochastic con- vex functions 퐻푅 푡 and 퐻퐵 푡 that achieve their minima at 휇푅∗ 푡 and 휇퐵∗ 푡 respectively, and interpret the update rules for 휇푅 푡 and 휇퐵 푡 as applying stochastic gradient descent (SGD) with respect to these auxiliary functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We will then relate the difference in obtained value 푉푡 (휇∗ 푡 ) − 푉푡 (휇푡) by the total loss in these auxiliary functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We would then ideally use facts about stochastic gradient descent to show that the total loss in value, relative to the optimal benchmark, is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Unfortunately, there are many technical problems with this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The first problem is rel- atively straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Recall that we have both a budget and an ROI constraint, but only one multiplier is used in each round;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' this can be either the budget-multiplier 휇퐵 푡 or the ROI-multiplier 휇푅 푡 , whichever is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' To avoid having to reason about which multiplier is being followed each round, we will actually bound the sum of regret experienced for both multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We can think of this as decomposing our experienced regret into the sum of two counterfactual regrets: one corresponding to the case where we have only the budget constraint and bid according to 휇퐵 푡 , and one corresponding to the case where we have only the ROI constraint and bid according to 휇푅 푡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' For each of these two cases, we can bound the total loss in value 푉푡 with respect to the differences |푍퐵 푡 (휇퐵 푡 ) −푍퐵 푡 (휇퐵∗ 푡 )| and |푍푅 푡 (휇푅 푡 ) −푍푅 푡 (휇푅∗ 푡 )|, respectively, which we can then relate to correspond- ing differences in our auxiliary functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Unfortunately, this raises a more fundamental problem with our proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We would like to argue that 휇퐵 푡 evolves according to stochastic gradient descent on our auxiliary function 퐻퐵 푡 , and similarly for 휇푅 푡 and 퐻푅 푡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' However, since we only receive feedback with respect to the larger multiplier, the smaller multiplier (휇퐵 푡 or 휇푅 푡 ) may not be updated according to the gradient of its corresponding loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' One thing we do know is that, since the auxiliary functions are convex, the gradient we use to update the smaller multiplier can only be more negative, in expectation, than its “correct” gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' At first this seems like an unacceptable source of error;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' if gradients are too negative, then (for example) 휇퐵 푡 could drift arbitrarily far from 휇퐵∗ 푡 in the negative direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' However, we are saved by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1: since we know that the budget and ROI constraints will necessarily be satisfied at the end of the 푇 rounds, our algorithm will not actually suffer any loss of value due to bids being too large (and hence, multipliers being too small).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We can therefore think of the evolution of one of the multipliers, say 휇퐵 푡 , as following a variant of stochastic gradient descent in which an adversary can, at will, perturb any given update step to be more negative;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' but in exchange, we only suffer losses when 휇퐵 푡 > 휇퐵∗ 푡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' As it turns out, the usual analysis of stochastic gradient descent extends to this variant, so we can conclude that our total loss is not too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' There are some additional technical challenges to handle as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Most notably, the auxiliary loss function for the ROI-multiplier is not convex in general but only convex when 휇 ∈ [0,훾 − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' This requires us to handle separately the case where the budget-multiplier is greater than훾 −1, and omit such rounds from our accounting of losses with respect to the ROI constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' This complicates our definition of counterfactual regret for ROI, but it turns out that the aggregate loss can still be bounded with some additional effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Autobidders with Budget and ROI Constraints 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2 Proof sketch of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We begin by formalizing our interpretation of Algorithm 2 as applying a form of stochastic gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We construct auxiliary loss functions 퐻퐵 푡 (휇) = 휌휇 − ∫ 휇 0 푍퐵 푡 (휏)푑휏 and 퐻푅 푡 (휇) = ∫ 휇 0 휌푡 (휏) − 푍푅 푡 (휏)푑휏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Based on Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2, we have the follow- ing lemma, which shows that if the ROI multiplier is larger than the budget multiplier, then the ROI-multiplier is updated by applying a stochastic gradient descent on function 퐻푅 푡 (휇), and if the budget-multiplier is larger than the ROI-multiplier, then the budget-multiplier is updated by applying a stochastic gradient descent on function 퐻퐵 푡 (휇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Algorithm 2 guarantees that: If 휇푅 푡 ≥ 휇퐵 푡 , E [훾푝푡 (휇푡) − 푣푡푥푡 (휇푡)] = 푍푅 푡 (휇푡) − 휌푡 (휇푡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' If 휇푅 푡 < 휇퐵 푡 , E [푝푡 (휇푡) − 휌] = 푍퐵 푡 (휇푡) − 휌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We next establish that the convexity and Lipschitzness of 퐻퐵 푡 (휇) and 퐻푅 푡 (휇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 퐻푅 푡 (휇) is 훾 ¯푣-Lipschitz and convex when 휇 ∈ [0,훾 − 1] and 퐻퐵 푡 (휇) is (¯푣 + 휌)-Lipschitz and convex in 휇 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The proofs of these two lemmas are deferred to Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Proof sketch of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' For conciseness, we omit all the problem dependent constants in the proof sketch and defer them to the formal proof in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' First, as mentioned in the previous section, we decompose the overall regret into the sum of the counterfactual regret with respect to budget-multiplier and ROI-multiplier respectively: Reg ≤ � 푡 ∈[푇 ] �� 푉푡 (휇퐵∗ 푡 ) − 푉푡 (휇퐵 푡 ) � 1{퐸퐵 푡 } � + � 푡 ∈[푇 ] �� 푉푡 (휇푅∗ 푡 ) − 푉푡 (휇푅 푡 ) � 1{퐸푅 푡 } � , (11) where 퐸퐵 푡 represents the event that 휇푡퐵 ≥ 휇퐵∗ 푡 and 퐸푅 푡 represents the event that 휇푅 푡 ≥ 휇푅∗ 푡 and 휇퐵 푡 ≤ 훾 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' In the following, we split the proof into four steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Step 1: Upper bounding the difference of 푉푡 by the difference of 푍퐵 푡 and 푍푅 푡 − 휌푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Using the monotonicity of 푝푡 (휇), we show in Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1 that for ROI-multiplier and budget- multiplier, we have the following inequalities, � 푉푡 (휇푅∗ 푡 ) − 푉푡 (휇푅 푡 ) � 1{퐸푅 푡 } ≤ O � 1 훽 � 푍푅 푡 (휇푅∗ 푡 ) − 휌푡 (휇푅∗ 푡 ) − 푍푅 푡 (휇푅 푡 ) + 휌푡 (휇푅 푡 ) � + 훽 � 1{퐸푅 푡 }, (12) � 푉푡 (휇퐵∗ 푡 ) − 푉푡 (휇퐵 푡 ) � 1{퐸퐵 푡 } ≤ O � 푍퐵 푡 (휇퐵∗ 푡 ) − 푍퐵 푡 (휇퐵 푡 ) � 1{퐸퐵 푡 }, (13) where 훽 > 0 is any positive number whose choice will be specified later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Step 2: Upper bounding the difference of 푍푅 푡 (푍퐵 푡 ) by the difference of 퐻푅 푡 (퐻퐵 푡 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Next, we need to relate 푍푅 푡 (휇푅∗ 푡 ) − 푍푅 푡 (휇푅 푡 ) with 퐻푅 푡 (휇푅∗ 푡 ) − 퐻푅 푡 (휇푅 푡 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Direct calculation shows that � 퐻푅 푡 (휇푅 푡 ) − 퐻푅 푡 (휇푅∗ 푡 ) � 1{퐸푅 푡 } = ∫ 휇푅 푡 −휇푅∗ 푡 0 � 휌푡 (휏 + 휇푅∗ 푡 ) − 푍푅 푡 (휏 + 휇푅∗ 푡 ) � 푑휏1{퐸푅 푡 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Note that 푓 푅(푥) = 휌푡 (휏 +휇푅∗ 푡 )−푍푅 푡 (휏 +휇푅∗ 푡 ) is a non-decreasing function of 푥 when 푥 ∈ [0, 휇푅 푡 −휇푅∗ 푡 ] according to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Also, we have 푓 (0) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Using a technical lemma Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='12 in [23] (we include this lemma in Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2 for completeness), we can show that � 푍푅 푡 (휇푅∗ 푡 ) − 휌푡 (휇푅∗ 푡 ) − 푍푅 푡 (휇푅 푡 ) + 휌푡 (휇푅 푡 ) � 1{퐸푅 푡 } ≤ O �� 퐻푅 푡 (휇푅 푡 ) − 퐻푅 푡 (휇푅∗ 푡 ) � 1{퐸푅 푡 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (14) 18 Brendan Lucier, Sarath Patathil, Aleksandrs Slivkins, and Mengxiao Zhang Similarly, for budget-constraint multiplier, we also have � 푍퐵 푡 (휇퐵∗ 푡 ) − 푍퐵 푡 (휇퐵 푡 ) � 1{퐸퐵 푡 } ≤ O �� 퐻퐵 푡 (휇퐵 푡 ) − 퐻퐵 푡 (휇퐵∗ 푡 ) � 1{퐸퐵 푡 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (15) Step 3: Upper bounding the regret with respect to 퐻푅 푡 and 퐻퐵 푡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Now we analyze the term 퐻푅 푡 (휇푅 푡 ) − 퐻푅 푡 (휇푅∗ 푡 ) (under the event 휇푅 푡 ≥ 휇푅∗ 푡 and 휇퐵 푡 ≤ 훾 − 1) and 퐻퐵 푡 (휇푅 푡 ) − 퐻퐵 푡 (휇퐵∗ 푡 ) (under the event 퐸퐵 푡 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' For the first term, note that � 퐻푅 푡 (휇푅 푡 ) − 퐻푅 푡 (휇푅∗ 푡 ) � 1{퐸푅 푡 } ≤ � 푔푅,푅 푡 , 휇푅 푡 − 휇푅∗ 푡 � 1{퐸푅 푡 }, where 푔푅,푅 푡 = ∇퐻푅 푡 (휇푅 푡 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' This is because 휇푅∗ 푡 ≤ 휇푅 푡 ≤ 훾 − 1 and 퐻푅 푡 (휇) is convex when 휇 ∈ [0,훾 − 1] according to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' According to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='5, if 휇푅 푡 ≥ 휇퐵 푡 , then 휇푅 푡 is updated by a stochastic gradient with mean 푔푅,푅 푡 = ∇퐻푅 푡 (휇푅 푡 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' However, note that 휇푅 푡 may not be updated using its own stochastic gradient on 퐻푅 푡 (휇), but may be updated by the gradient 푔푅,퐵 푡 = ∇퐻푅 푡 (휇퐵 푡 ) if 휇퐵 푡 ≥ 휇푅 푡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' However, using the convexity of 퐻푅 푡 (휇) when 휇 ∈ [0,훾 −1], we have 푔푅,퐵 푡 ≥ 푔푅,푅 푡 as 휇푅 푡 ≤ 휇퐵 푡 ≤ 훾 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Therefore, let 푔푅 푡 be the gradient that updates 휇푅 푡 and we have � 퐻푅 푡 (휇푅 푡 ) − 퐻푅 푡 (휇푅∗ 푡 ) � 1{퐸푅 푡 } ≤ � 푔푅 푡 , 휇푅 푡 − 휇푅∗ 푡 � 1{퐸푅 푡 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (16) Therefore, based on Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='5 and the classic analysis of online gradient descent, we have: E � 푇 � 푡=1 � 퐻푅 푡 (휇푅 푡 ) − 퐻푅 푡 (휇푅∗ 푡 ) � 1{퐸푅 푡 } � ≤ 푇� 푡=1 O � |휇푅 푡 − 휇푅∗ 푡 |2 − |휇푅 푡+1 − 휇푅∗ 푡+1|2 휂푅 + 휂푅 + |휇푅∗ 푡 − 휇푅∗ 푡+1| 휂푅 � 1{퐸푅 푡 }, While generally online gradient descent gives O( √ 푇) regret, the challenge in bounding the term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (28) is that with the condition 1{퐸푅 푡 }, the term |휇푅 푡 − 휇푅∗ 푡 |2 − |휇푅 푡+1 − 휇푅∗ 푡+1|2 can not be tele- scoped after summation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Therefore, we decompose the total horizon [푇], into 푆 intervals 퐼1 = [1,푒1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' , 퐼푆 = [푤푆,푒푆], where each interval is a maximal sequence of consecutive rounds such that 휇푅 푡 ≥ 휇푅∗ 푡 and 휇퐵 푡 ≤ 훾 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Then we have E � 푇� 푡=1 � 퐻푅 푡 (휇푅 푡 ) − 퐻푅 푡 (휇푅∗ 푡 ) � 1{퐸푅 푡 } � ≤ 푆 � 푠=1 O � |휇푅푤푠 − 휇푅∗ 푤푠 |2 − |휇푅 푒푠+1 − 휇푅∗ 푒푠+1|2 휂푅 � + O � 푃푅 푇 휂푅 + 휂푅푇 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' With a more careful analysis on the dynamic of 휇푅 푡 , we can show that the terms |휇푅 푤푠 − 휇푅∗ 푤푠 |2 − |휇푅 푒푠+1 − 휇푅∗ 푒푠+1|2 indeed telescope after summation over 푠 = 1 to 푆 and we obtain that E � 푇� 푡=1 � 퐻푅 푡 (휇푅 푡 ) − 퐻푅 푡 (휇푅∗ 푡 ) � 1{퐸푅 푡 } � ≤ O � 1 + 푃푅 푇 휂푅 + 휂푅푇 � = O � (푃푅 푇 + 1) √ 푇 � , (17) where the final equality is by choosing 휂푅 = Θ( 1 √ 푇 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Similarly for budget-multiplier, with 휂퐵 = Θ( 1 √ 푇 ), we can also obtain that E �� 푡 ∈[푇 ] � 퐻퐵 푡 (휇퐵 푡 ) − 퐻푅 푡 (휇퐵∗ 푡 ) � 1{퐸퐵 푡 } � ≤ O � (1 + 푃퐵 푇 ) √ 푇 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (18) Autobidders with Budget and ROI Constraints 19 Step 4: Combining all the above analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Finally, we combine Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (11), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (12), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (13), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (14), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (15), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (17), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='and obtain the following ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡 ∈[푇 ] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='�푉푡 (휇∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡 ) − 푉푡 (휇푡)�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='≤ E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='� 푇 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='훽 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푍푅 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡 (휇푅∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡 ) − 휌푡 (휇푅∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡 ) − 푍푅 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡 (휇푅 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡 ) + 휌푡 (휇푅 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡 ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='+ 훽휆 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1{퐸푅 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡 } ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='+ E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='� 푇 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푍퐵 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡 (휇퐵∗ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡 ) − 푍퐵 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡 (휇퐵 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푡 ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1{퐸퐵 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='≤ O ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='훽 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푇 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='5(푃푅 푇 + 1) + 훽푇 + � 푇 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='5(1 + 푃퐵 푇 ) � ≤ O � (푃푅 푇 + 1) 1 4푇 7 8 + � (1 + 푃퐵 푇 )푇 3 4 � , where the third inequality is because of Jensen’s inequality and the last inequality is by picking the optimal 훽 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' This completes the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' □ 5 CONCLUSIONS AND OPEN PROBLEMS We consider the problem of online bidding with both budget and ROI constraints, under a broad class of auction formats including first-price and second-price auction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We set out to achieve both aggregate and individual guarantees, as expressed, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=', by liquid welfare and vanishing regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We accomplish this with a novel variant of constraint-pacing, achieving (i) the best possible guarantee in expected liquid welfare, (ii) vanishing individual regret against an adversary, and (iii) satisfying the budget and ROI constraints with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The regret bound holds against a non-standard (albeit reasonable) benchmark, side-stepping impossibility results from prior work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Our work opens up several directions for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' First, can we obtain similar results for other algorithms or classes thereof?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Second, while any vanishing regret satisfies our desiderata, can we improve regret rates while maintaining a similar aggregate guarantee?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' This is a non- standard question for the literature on online bidding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Particularly interesting are regret bounds that go beyond a stationary stochastic environment (since the auction environment is often/typically not stationary in practice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Another open direction is to analyze other aggregate market metrics such as platform revenue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' For the latter, it would be helpful to have a more complete understanding of the interaction between autobidders and tunable parameters like reserve prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 20 Brendan Lucier, Sarath Patathil, Aleksandrs Slivkins, and Mengxiao Zhang REFERENCES [1] Gagan Aggarwal, Ashwinkumar Badanidiyuru, and Aranyak Mehta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Autobidding with constraints.' metadata={'source': 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2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' [32] Juncheng Li and Pingzhong Tang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Auto-bidding equilibrium in roi-constrained online advertising markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' arXiv preprint arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='06107, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' [33] Aranyak Mehta and Andres Perlroth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Auctions without commitment in the auto-bidding world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' arXiv preprint arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='07312, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' [34] Panayotis Mertikopoulos, Christos H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Papadimitriou, and Georgios Piliouras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Cycles in adversarial regularized learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' In 29th ACM-SIAM Symp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' on Discrete Algorithms (SODA), pages 2703–2717, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' [35] Herve Moulin and Jean-Paul Vial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Strategically zero-sum games: the class of games whose completely mixed equilibria cannot be improved upon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Intl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' of Game Theory, 7(3):201–221, 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' [36] Leon Musolff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Algorithmic pricing facilitates tacit collusion: Evidence from e-commerce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' In Proceedings of the 23rd ACM Conference on Economics and Computation, pages 32–33, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' [37] Tim Roughgarden, Vasilis Syrgkanis, and Éva Tardos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The price of anarchy in auctions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Artif.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Intell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=', 59:59–101, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1613/jair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='5272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' URL https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1613/jair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='5272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' [38] Aleksandrs Slivkins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Introduction to multi-armed bandits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Foundations and Trends® in Machine Learn- ing, 12(1-2):1–286, November 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Published with Now Publishers (Boston, MA, USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Also available at https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='org/abs/1904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='07272.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Latest online revision: Jan 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' [39] Chen-Yu Wei, Chung-Wei Lee, Mengxiao Zhang, and Haipeng Luo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Linear last-iterate convergence in constrained saddle-point optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' In 9th International Conference on Learning Representations (ICLR), 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 22 Brendan Lucier, Sarath Patathil, Aleksandrs Slivkins, and Mengxiao Zhang A NOTATION SUMMARY Here we list the notations used throughout the paper for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Summary of Notations 푛 the total number of agents 푇 the total number of rounds x푡 = (푥1,푡, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' ,푥푛,푡) allocation profile at round 푡 ∈ [푇] v푡 = (푣1,푡, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' , 푣푛,푡) value profile at round 푡 ∈ [푇] b푡 = (푏1,푡, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' ,푏푛,푡) bid profile at round 푡 ∈ [푇] p푡 = (푝1,푡, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' ,푝푛,푡) payment profile at round 푡 ∈ [푇] 훾푘 ROI constraint parameter for agent 푘 퐵푘 = 휌푘 ·푇 total budget for agent 푘 푣 upper bound of the value 푊푘(x) liquid value of agent 푘 of allocation sequence x (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1) 푊 (x) liquid welfare of allocation sequence x (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1) 푊 푘 (x, 퐹) agent 푘’s ex-ante liquid value of allocation x and value dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 퐹 (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1) 푊 (x, 퐹) ex-ante liquid welfare of allocation x and value dist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 퐹 (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1) 휇퐵 푘,푡 agent 푘’s budget-multiplier at round 푡 휇푅 푘,푡 agent 푘’s ROI-multiplier at round 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 휂푘,퐵 learning rate of agent 푘’s budget-multiplier 휂푘,푅 learning rate of agent 푘’s ROI-multiplier 푍퐵 푡 expected budget expenditure at round 푡 (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='3) 푍푅 푡 expected ROI expenditure at round 푡 (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='3) 휌푡 expected ROI gain at round 푡 (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='3) 푉푡 expected value gain at round 푡 (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='3) 휇퐵∗ 푡 budget-pacing multiplier at round 푡 (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='4) 휇푅∗ 푡 ROI-pacing multiplier at round 푡 (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='4) 휇∗ 푡 pacing multiplier at round 푡 (Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='4) 푃퐵 푇 = �푇−1 푡=1 |휇퐵∗ 푡 −휇퐵∗ 푡+1| path-length of budget-pacing multiplier 푃푅 푇 = �푇 −1 푡=1 |휇푅∗ 푡 −휇푅∗ 푡+1| path-length of ROI-pacing multiplier Autobidders with Budget and ROI Constraints 23 B OMITTED DETAILS IN SECTION 3 In this section, we provide the omitted details and proofs in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We first prove Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1, which shows that monotonicity of 푣푡푥푡 (휇) − 훾푝푡 (휇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Let 푑 = max푗≠푘 푏 푗, and let 휇′ be such that 푣푡/(1 + 휇′) = 푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Then for all 휇 > 휇′ we have 푥푡 (휇) = 푝푡 (휇) = 0 (which is weakly increasing in 휇), and for all 휇 < 휇′ we have that 푥푡 (휇) = 1 and 푝푡 (휇) is weakly decreasing in 휇, so 푣푡푥푡 (휇) − 훾푝푡 (휇) is weakly increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' It only remains to establish what happens at the threshold 휇 = 휇′, and then only when 휇′ ≤ 훾 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Note however that when 휇 = 휇′, the first and second highest bids are equal, so the payment of agent 푘 is determined to be 푥푡 (휇)푣푡/(1 + 휇) ≥ 푥푡 (휇)푣푡/훾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' This implies 푣푡푥푡 (휇′) − 훾푝푡 (휇′) ≤ 0, and hence 푣푡푥푡 (휇) − 훾푝푡 (휇) ≤ 0 for all 휇 < 휇′ as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Since 푣푡푥푡 (휇) − 훾푝푡 (휇) = 0 for all 휇 > 휇′, we conclude that the difference is monotone in 휇 as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' □ Next, we show that Algorithm 1 guarantees that we never violate the ROI constriant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We prove this using induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The base case follows trivially (since the multiplier is initialized to 훾 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Now, suppose this is true for all time up to 푡 − 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=', 푡′ � 휏=1 푣휏푥휏 ≥ 훾 푡′ � 휏=1 푝휏, ∀푡 ′ ≤ 푡 − 1 Now, consider time 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' From the update rule, we have 휇푡 ≥ 휇0 + 휂(�푡−1 휏=1 푣휏푥휏 − 훾푝휏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We split the proof into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Suppose 휂(�푡−1 휏=1 푣휏푥휏 − 훾푝휏) < 훾 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' This gives us: 푝푡 ≤ 푏푡 ≤ 푣푡 훾−휂(�푡−1 휏=1 푣휏푥휏−훾푝휏) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Now, using the fact that 휂푝푡 ≤ 1 since 휂 < 1/¯푣, we have: 훾푝푡 + �푡−1 휏=1 푝휏 ≤ 푣푡푥푡 + �푡−1 휏=1 푣휏푥휏, which gives us the required claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Next, if 휂(�푡−1 휏=1 푣휏푥휏 − 훾푝휏) > 훾 − 1, we have 푝푡 ≤ 푏푡 ≤ 푣푡 which gives us �푡 휏=1 푣휏푥휏 − 훾푝휏 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' □ C OMITTED DETAILS IN SECTION 4 In this section, we present the omitted proofs in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' First, we prove Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1, which shows that Algorithm 2 never violates either the ROI-constraint or the budget-constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We first show that the ROI constraint �푇 푡=1 푣푘,푡푥푘,푡 ≥ 훾푘 �푇 푡=1 푝푘,푡 is satis- fied if the agent is applying Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The proof is basically the same as the one of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We prove this using induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We consider a fixed individual bidder and omit the subscript 푘 in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The base case still follows trivially, since the multiplier 휇1 is initialized to max{훾 −1, 푣 휌 −1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Now, suppose this is true for all time up to 푡 − 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=', 푡′ � 휏=1 푣휏푥휏 ≥ 훾 푡′ � 휏=1 푝휏, ∀푡 ′ ≤ 푡 − 1 Now, consider time 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' From the update rule, we have 휇푅 푡 ≥ 휇푅 0 + 휂푅 �푡−1 휏=1 (훾푝휏 − 푣휏푥휏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We split the proof into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Suppose 휂푅 �푡−1 휏=1(푣휏푥휏 − 훾푝휏) < 훾 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' This gives us: 푝푡 ≤ 푏푡푥푡 = 푣푡푥푡 1 + 휇푡 ≤ 푣푡푥푡 1 + 휇푅 푡 ≤ 푣푡푥푡 훾 − 휂푅 �푡−1 휏=1(푣휏푥휏 − 훾푝휏) 24 Brendan Lucier, Sarath Patathil, Aleksandrs Slivkins, and Mengxiao Zhang Now, using the fact that 휂푅푝푡 ≤ 휂푅푣푡 ≤ 휂푣 ≤ 1, we have: 훾 푡� 휏=1 푝휏 = 훾 � 푝푡 + 푡−1 � 휏=1 푝휏 � ≤ 휂푅푝푡 푡−1 � 휏=1 (푥휏푣휏 − 훾푝휏) + 푣푡푥푡 + 훾 푡−1 � 휏=1 푝휏 ≤ 푡� 휏=1 푥휏푣휏, where the last inequality uses the induction hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Now, if 휂푅 �푡−1 휏=1(푣휏푥휏 − 훾푝휏) > 훾 − 1, we have �푡−1 휏=1(푣휏푥휏 − 훾푝휏) ≥ (훾 − 1)푣 ≥ (훾 − 1)푣푡푥푡 and 푝푡 ≤ 푏푡 ≤ 푣푡, which means that 훾 푡� 휏=1 푝휏 ≤ 훾 푡−1 � 휏=1 푝휏 + 훾푣푡푥푡 ≤ 푡� 휏=1 푣휏푥휏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Combining the above two claims finishes the proof for the ROI constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' For budget constraint, similarly, we prove the following inequality using induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 푡� 휏=1 푝휏 ≤ 휌푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (19) The base case holds as 푝1 ≤ 푏1 ≤ 푣1 1+휇퐵 1 ≤ 휌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Suppose that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (19) holds up to time 푡 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' According to the update rule, we know that 휇퐵 푡 ≥ 휇퐵 0 + 휂퐵 �푡−1 휏=1(푥휏푝휏 − 휌).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We also split the proof into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Suppose 휂퐵 �푡−1 휏=1(푥휏푝휏 − 휌) > 1 − 푣 휌 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Then we have 푝푡 ≤ 푏푡푥푡 = 푣푡푥푡 1 + 휇푡 ≤ 푣푡푥푡 1 + 휇퐵 푡 ≤ 푣푡푥푡 휂퐵 �푡−1 휏=1(푥휏푝휏 − 휌) + 푣 휌 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Therefore, we have: 푡� 휏=1 푝휏 = 푡−1 � 휏=1 푝휏 + 휌 푣 � 푣푡푥푡 + 휂퐵푝푡 푡−1 � 휏=1 (휌 − 푝휏) � ≤ 푡−1 � 휏=1 푝휏 + 휌 + 휂퐵휌 푡−1 � 휏=1 (휌 − 푝휏) = 휌푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' If 휂퐵 �푡−1 휏=1(푥휏푝휏 − 휌) < 1 − 푣 휌 , then we have 푝푡 ≤ 푏푡 ≤ 푣푡, which gives us 푡� 휏=1 (푥휏푝휏) ≤ 휌(푡 − 1) + 푣 ≤ 휌푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Combining the above two inequalities finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' □ Next, we provide the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2 which guarantees that 휇푅 푡 ≤ 훾 − 1 under auctions between first-price and second price, when we update the multipliers using Algorithm 2 Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We prove this by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' For conciseness, we omit the subscript of the agent index 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Base case trivially holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Suppose that up to round 푡, 휇푅 푡 ≤ 훾 − 1 and 휇퐵 푡 ≤ 푣 휌 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' First, consider 휇푅 푡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' At round 푡 + 1, if the bidder does not win an auction, then 휇푅 푡+1 = 휇푅 푡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Otherwise, we have 휇푅 푡+1 = 휇푅 푡 + 휂푅 (훾푝푡 (휇푡) − 푣푡) ≤ 휇푅 푡 + 휂푅 � 훾푝푡 (휇푅 푡 ) − 푣푡 � (휇푡 = max{휇퐵 푡 , 휇푅 푡 , 0} and 푝푡 (휇) is decreasing in 휇) ≤ 휇푅 푡 + 휂푅 � 훾 푣푡 1 + 휇푅 푡 − 푣푡 � (payment does not exceed bid) ≤ 휇푅 푡 + 휂푅 (훾 − 1 − 휇푅 푡 )푣푡 1 + 휇푅 푡 ≤ 휇푅 푡 + 훾 − 1 − 휇푅 푡 = 훾 − 1, Autobidders with Budget and ROI Constraints 25 where the last inequality uses the fact that 휂푅 ≤ 1 ¯푣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' This proves the result for 휇푅 푡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Consider the budget-multiplier 휇퐵 푡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' At round 푡 +1, similarly, if the bidder does not win an auction, then 휇퐵 푡+1 = 휇푅 푡 − 휂퐵휌 ≤ 푣 휌 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Otherwise, we have 휇퐵 푡+1 = 휇퐵 푡 + 휂퐵 (푝푡 (휇푡) − 휌) ≤ 휇퐵 푡 + 휂퐵 � 푣푡 1 + 휇퐵 푡 − 휌 � (푝푡 (휇푡) ≤ 푣푡 1+휇푡 ≤ 푣푡 1+휇퐵 푡 ) ≤ 1 + 휇퐵 푡 + 휂퐵푣 1 + 휇퐵 푡 − 휌휂퐵 − 1 ≤ 푣 휌 + 휂퐵푣 푣/휌 − 휌휂퐵 − 1 ≤ 푣 휌 − 1, where the third inequality is because 휂퐵푣 ≤ 1 and ℎ(푥) = 푥 + 1 푥 is increasing for 푥 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' □ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1 Omited Details in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2 In this section, we provide the proofs for the individual regret guarantee for Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' First, we prove Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='4, which shows the monotonicity of 푍퐵 푡 (휇) and 푍푅 푡 (휇) − 휌푡 (휇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' For any 푡 ∈ [푇], as 푝푡 is non-decreasing in the bid, which means that 푝푡 (휇) is non-increasing in 휇, 푍퐵 푡 (휇) is non-increasing for 휇 ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Next, consider 휌푡 (휇) − 푍푅 푡 (휇) = E[푣푡푥푡 (휇) −훾푝푡 (휇)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Let 푑푡 denote the competing bid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Also note that the bid 푏푡 = 푣푡/(1+ 휇) > 푣푡/훾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We split the proof into three cases: (1) If 푑푡 < 푣푡/훾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We have 푥푡 (휇) = 1 (since 푏푡 > 푑푡) and 푝푡 (휇) is decreasing in 휇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (2) If 푣푡/훾 ≤ 푑푡 ≤ 푣푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Suppose the competing bid푑푡 = 푣푡/(1+휇′) for some constant 휇′ ∈ [1,훾 −1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' In this case, 푥푡 (휇) = 1 for 휇 ∈ [0, 휇′ − 1), and we have 푣푡 − 훾푝푡 (휇) < 0 and increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' At 휇 = 휇′, we have 푏푡 = 푑푡, and therefore 푥푡 (휇) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Furthermore, we have 푝푡 (휇) = 푥푡 (휇)푣푡/(1+휇′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Therefore 푣푡푥푡 (휇) −훾푝푡 (휇) = 푥푡 (휇) (푣푡 − 훾푣푡/(1 + 휇′)) ≤ 0, and also greater than 푣푡푥푡 (휇) − 훾푝푡 (휇), 휇 < 휇′ (since 푝푡 (휇) decreases with 휇 and 푥(휇′) ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' This shows that there will be an increase in 푣푡푥푡 (휇) − 훾푝푡 (휇) when we move from 휇 < 휇′ to 휇 = 휇′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Finally, for 휇 ∈ (휇′,훾 − 1], 푣푡푥푡 (휇) −훾푝푡 (휇) will identically equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Therefore, we have 푣푡푥푡 (휇) − 훾푝푡 (휇) is increasing in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (3) If 푑푡 > 푣푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' In this case, since 푏푡 < 푑푡, 푥푡 (휇) and 푝푡 (휇) will both be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Combining these three cases, and taking expectations gives the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' □ Next, we prove the following lemma, which bounds the expected difference in value by the expected difference in budget and ROI payoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' For any 0 ≤ 휇1 ≤ 휇2 ≤ 훾 − 1, and any 훽 > 0, we have 푉푡 (휇1) − 푉푡 (휇2) ≤ 훾 훽 � 푍푅 푡 (휇1) − 휌푡 (휇1) − 푍푅 푡 (휇2) + 휌푡 (휇2) � + 훽휆, where 휆 > 0 is the Lipschitz constant defined in Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' In addition, for any 0 ≤ 휇1 ≤ 휇2 ≤ 푣 휌 − 1, we have 푉푡 (휇1) − 푉푡 (휇2) ≤ 푣 휌 � 푍퐵 푡 (휇1) − 푍퐵 푡 (휇2) � , Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' For any 훽 > 0, as 푉푡 (휇) is 휆-Lipschitz based on Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1, we have 푉푡 (휇1) − 푉푡 (휇2) = 푉푡 (휇1) − 푉푡 (min{휇2,훾 − 1 − 훽}) + 푉푡 (min{휇2,훾 − 1 − 훽}) − 푉푡 (휇2) ≤ 푉푡 (휇1) − 푉푡 (min{휇2,훾 − 1 − 훽}) + 훽휆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (20) 26 Brendan Lucier, Sarath Patathil, Aleksandrs Slivkins, and Mengxiao Zhang Now we show that 푉푡 (휇1) − 푉푡 (min{휇2,훾 − 1 − 훽}) ≤ 훾 훽 � 푍푅 푡 (휇1) − 휌푅 푡 (휇1) − 푍푅 푡 (min{휇2,훾 − 1 − 훽}) + 휌푡 (min{휇2,훾 − 1 − 훽}) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' For any 휇 ∈ [휇1,훾 − 1 − 훽], ∇[푍푅 푡 (휇) − 휌푡 (휇)] = ∇E [((훾푝푡 (휇) − 푣푡)푥푡 (휇))] = E [훾∇푝푡 (휇)푥푡 (휇)}] + ∇E [훾푝푡 (휇)푥푡 (휇)}] − ∇E [푣푡푥푡 (휇)] ≤ ∇E � 훾 푣푡 1 + 휇 푥푡 (휇) � − ∇푉푡(휇) (∇푝푡 (휇) ≤ 0, ∇푥푡 (휇) ≤ 0 and 푝푡 (휇) ≤ 푣푡 1+휇 ) = 훾 − 휇 − 1 1 + 휇 ∇푉푡 (휇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' In addition, note that 푍푅 푡 (휇1) − 휌푡 (휇1) − 푍푅 푡 (휇) + 휌푡 (휇) = ∫ 휇1 휇 ∇(푍푅 푡 (휏) − 휌푡 (휏))푑휏 and 푉푡 (휇1) − 푉푡 (휇) = ∫ 휇1 휇 ∇푉푡 (휏)푑휏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Therefore, we have 푉푡 (휇1) − 푉푡 (min{휇2,훾 − 1 − 훽}) = ∫ min{휇2,훾−1−훽 } 휇1 −∇푉푡 (휏)푑휏 ≤ 1 + min{휇2,훾 − 1 − 훽} 훾 − min{휇2,훾 − 1 − 훽} − 1 ∫ min{휇2,훾−1−훽 } 휇1 −∇(푍푅 푡 (휏) − 휌푡 (휇))푑휏 = 1 + min{휇2,훾 − 1 − 훽} 훾 − min{휇2,훾 − 1 − 훽} − 1 � 푍푅 푡 (휇1) − 푍푅 푡 (min{휇2,훾 − 1 − 훽} − 휌푡 (휇1) + 휌푡 (min{휇2,훾 − 1 − 훽}) � ≤ 훾 훽 � 푍푅 푡 (휇1) − 휌푡 (휇1) − 푍푅 푡 (휇2) + 휌푡 (휇2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (푍푅 푡 (휇) − 휌푡 (휇) is non-increasing in 휇 ∈ [0,훾 − 1]) Plugging the above into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (20) gives 푉푡 (휇1) − 푉푡 (휇2) ≤ 훾 훽 � 푍푅 푡 (휇1) − 휌푡 (휇1) − 푍푅 푡 (휇2) + 휌푡 (휇2) � + 훽휆, (21) which finishes the proof of the first inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' For the second inequality, note that for any 휇 ∈ [0, 푣 휌 − 1] ∇푍퐵 푡 (휇) = ∇E [푝푡 (휇)푥푡 (휇)] = E [∇푝푡 (휇)푥푡 (휇)] + E [푝푡 (휇)∇푥푡 (휇)] ≤ ∇E � 푣푡 1 + 휇 푥푡 (휇) � (∇푝푡 (휇) ≤ 0, ∇푥푡 (휇) ≤ 0 and 푝푡 (휇) ≤ 푣푡 1+휇 ) = 1 1 + 휇 ∇푉푡 (휇) ≤ 휌 푣 ∇푉푡 (휇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Therefore, we have 푉푡 (휇1) − 푉푡 (휇2) = ∫ 휇2 휇1 −∇푉푡 (휏)푑휏 ≤ 푣 휌 ∫ 휇2 휇1 −∇푍퐵 푡 (휏)푑휏 = 푣 휌 � 푍퐵 푡 (휇1) − 푍푅 푡 (휇2) � ≤ 푣 휌 � 푍퐵 푡 (휇1) − 푍퐵 푡 (휇2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' □ Autobidders with Budget and ROI Constraints 27 Next, we prove Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='5, which shows that Algorithm 2 can be viewed as applying SGD on the auxiliary loss function 퐻푅 푡 and 퐻퐵 푡 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' For ROI-multiplier, direct calculation shows that E [(훾푝푡 (휇푡) − 푣푡)푥푡 (휇푡)] = E � (훾푝푡 (휇푡) − 푣푡푥푡 (휇푡))+� − E � (푣푡푥푡 (휇푡) − 훾푝푡 (휇푡))+� = 푍푅 푡 (휇푡) − 휌푡 (휇푡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' For budget-multiplier, direct calculation shows that E [푥푡푝푡 (휇푡) − 휌] = 푍퐵 푡 (휇푡) − 휌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Combining the above two equations completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' □ Next we prove Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='6, which shows the Lipshitzness and convexity of auxiliary functions 퐻푅 푡 and 퐻퐵 푡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' The results for 퐻퐵 푡 (휇) is proven in Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2 in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' For the function 퐻푅 푡 (휇) we have ∇퐻푅 푡 (휇) = 휌푡 (휇) − 푍푅 푡 (휇) = E [푣푡푥푡 (휇) − 훾푝푡 (휇)] , which we show is increasing over 휇 ∈ [0,훾 − 1] according to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' In addition, we have for all 휇 ∈ [0,훾 − 1], |∇퐻푅 푡 (휇)| ≤ max{훾푝푡 (휇), 푣푡 } ≤ 훾 ¯푣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' □ Now we provide the proof of our main result Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2, which shows the individual regret of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' First,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' we decompose the overall regret into the regret with respect to budget-multiplier and ROI-multiplier as follows: Reg = 푇 � 푡=1 �푉푡 (휇∗ 푡 ) − 푉푡 (휇푡)� ≤ 푇 � 푡=1 �푉푡 (휇∗ 푡 ) − 푉푡 (휇푡)� 1{휇푡 ≥ 휇∗ 푡 } (푉푡 (휇) is decreasing in 휇) ≤ 푇 � 푡=1 �� 푉푡 (휇∗ 푡 ) − 푉푡 (휇퐵 푡 ) � 1{휇퐵 푡 ≥ max{휇푅 푡 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 휇∗ 푡 } � + 푇 � 푡=1 �� 푉푡 (휇∗ 푡 ) − 푉푡 (휇푅 푡 ) � 1{휇푅 푡 ≥ max{휇퐵 푡 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 휇∗ 푡 }} � ≤ 푇 � 푡=1 �� 푉푡 (휇∗ 푡 ) − 푉푡 (휇퐵 푡 ) � 1{휇퐵 푡 ≥ 휇∗ 푡 } � + 푇� 푡=1 �� 푉푡 (휇∗ 푡 ) − 푉푡 (휇푅 푡 ) � 1{휇푅 푡 ≥ 휇∗ 푡 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 휇퐵 푡 ≤ 훾 − 1} � ≤ 푇 � 푡=1 �� 푉푡 (휇퐵∗ 푡 ) − 푉푡 (휇퐵 푡 ) � 1{휇퐵 푡 ≥ 휇퐵∗ 푡 } � + 푇 � 푡=1 �� 푉푡 (휇푅∗ 푡 ) − 푉푡 (휇푅 푡 ) � 1{휇푅 푡 ≥ 휇푅∗ 푡 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 휇퐵 푡 ≤ 훾 − 1} � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (22) where the third inequality is because 휇푅 푡 ≤ 훾 −1 according to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2, meaning that 휇퐵 푡 ≤ 휇푅 푡 ≤ 훾 − 1 in the second term;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' the fourth inequality is because 푉푡 (휇) is non-increasing in 휇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' We split the rest of the proof into four steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Step 1: Upper bounding the difference of 푉푡 by the difference of 푍퐵 푡 and 푍푅 푡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 28 Brendan Lucier, Sarath Patathil, Aleksandrs Slivkins, and Mengxiao Zhang According to Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1, for ROI-multiplier and budget-multiplier, we have � 푉푡 (휇푅∗ 푡 ) − 푉푡 (휇푅 푡 ) � 1{휇푅 푡 ≥ 휇푅∗ 푡 , 휇퐵 푡 ≤ 훾 − 1} ≤ �훾 훽 � 푍푅 푡 (휇푅∗ 푡 ) − 휌푡 (휇푅∗ 푡 ) − 푍푅 푡 (휇푅 푡 ) + 휌푡 (휇푅 푡 ) � + 훽휆 � 1{휇푅 푡 ≥ 휇푅∗ 푡 , 휇퐵 푡 ≤ 훾 − 1}, (23) � 푉푡 (휇퐵∗ 푡 ) − 푉푡 (휇퐵 푡 ) � 1{휇퐵 푡 ≥ 휇퐵∗ 푡 } ≤ 푣 휌 � 푍퐵 푡 (휇퐵∗ 푡 ) − 푍퐵 푡 (휇퐵 푡 ) � 1{휇퐵 푡 ≥ 휇퐵∗ 푡 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (24) Step 2: Upper bounding the difference of 푍푅 푡 (푍퐵 푡 ) by the difference of 퐻푅 푡 (퐻퐵 푡 ) Next, we need to relate 푍푅 푡 (휇푅∗ 푡 ) −푍푅 푡 (휇푅 푡 ) with 퐻푅 푡 (휇푅∗ 푡 ) −퐻푅 푡 (휇푅 푡 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Direct calculation shows that � 퐻푅 푡 (휇푅 푡 ) − 퐻푅 푡 (휇푅∗ 푡 ) � 1{휇푅 푡 ≥ 휇푅∗ 푡 , 휇퐵 푡 ≤ 훾 − 1} = ∫ 휇푅 푡 휇푅∗ 푡 � 휌푡 (휏) − 푍푅 푡 (휏) � 푑휏1{휇푅 푡 ≥ 휇푅∗ 푡 , 휇퐵 푡 ≤ 훾 − 1} = ∫ 휇푅 푡 −휇푅∗ 푡 0 � 휌푡 (휏 + 휇푅∗ 푡 ) − 푍푅 푡 (휏 + 휇푅∗ 푡 ) � 푑휏1{휇푅 푡 ≥ 휇푅∗ 푡 , 휇퐵 푡 ≤ 훾 − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Note that 푔푅(푥) = 휌푡 (푥 +휇푅∗ 푡 )−푍푅 푡 (푥 +휇푅∗ 푡 ) is a non-decreasing function of 푥 when 푥 ∈ [0, 휇푅 푡 −휇푅∗ 푡 ] according to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Also, we have 푔푅(0) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Therefore, let 푓 푅(푥) = 푔푅(푥) − 푔푅(0) and according to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='13 in [23] (we include this lemma in Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2 for completeness), we know that �� 휌푡 (휇푅 푡 ) − 푍푅 푡 (휇푅 푡 ) � − � 휌푡 (휇푅∗ 푡 ) − 푍푅 푡 (휇푅∗ 푡 ) �� 1{휇푅 푡 ≥ 휇푅∗ 푡 , 휇퐵 푡 ≤ 훾 − 1} ≤ � 4휆 �퐻푅 푡 (휇푅 푡 ) − 퐻푅 푡 (휇푅∗ 푡 ) − (휇푅 푡 − 휇푅∗ 푡 )(휌푡 (휇푅∗ 푡 ) − 푍푅 푡 (휇푅∗ 푡 ))�1{휇푅 푡 ≥ 휇푅∗ 푡 , 휇퐵 푡 ≤ 훾 − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' When 휇푅 푡 ≥ 휇푅∗ 푡 , we know that (휇푅 푡 − 휇푅∗ 푡 )(휌푡 (휇푅∗ 푡 ) − 푍푅 푡 (휇푅∗ 푡 )) is non-negative and we can obtain that � 푍푅 푡 (휇푅∗ 푡 ) − 휌푡 (휇푅∗ 푡 ) − 푍푅 푡 (휇푅 푡 ) + 휌푡 (휇푅 푡 ) � 1{휇푅 푡 ≥ 휇푅∗ 푡 , 휇퐵 푡 ≤ 훾 − 1} ≤ � 4휆(퐻푅 푡 (휇푅 푡 ) − 퐻푅 푡 (휇푅∗ 푡 ))1{휇푅 푡 ≥ 휇푅∗ 푡 , 휇퐵 푡 ≤ 훾 − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (25) For budget-constraint multiplier 휇퐵 푡 , similarly we define 푓 퐵(푥) = 휌 − 푍퐵 푡 (휇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Note that 푓 퐵(푥) is also a non-decreasing function and 휆-Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Applying Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2 on 푓 퐵(푥) − 푓 퐵(0), we also have � 푍퐵 푡 (휇퐵∗ 푡 ) − 푍퐵 푡 (휇퐵 푡 ) � 1{휇퐵 푡 ≥ 휇퐵∗ 푡 } ≤ � 2휆(퐻퐵 푡 (휇퐵 푡 ) − 퐻퐵 푡 (휇퐵∗ 푡 ))1{휇퐵 푡 ≥ 휇퐵∗ 푡 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (26) Step 3-1: Upper bounding the regret with respect to 퐻푅 푡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Now we analyze the term 퐻푅 푡 (휇푅 푡 ) − 퐻푅 푡 (휇푅∗ 푡 ) (under the event 휇푅 푡 ≥ 휇푅∗ 푡 and 휇퐵 푡 ≤ 훾 − 1) and 퐻퐵 푡 (휇푅 푡 ) − 퐻퐵 푡 (휇퐵∗ 푡 ) (under the event 휇퐵 푡 ≥ 휇퐵∗ 푡 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' For the first term, note that � 퐻푅 푡 (휇푅 푡 ) − 퐻푅 푡 (휇푅∗ 푡 ) � 1{휇푅 푡 ≥ 휇푅∗ 푡 , 휇퐵 푡 ≤ 훾 − 1} ≤ � 푔푅,푅 푡 , 휇푅 푡 − 휇푅∗ 푡 � 1{휇푅 푡 ≥ 휇푅∗ 푡 , 휇퐵 푡 ≤ 훾 − 1}, where 푔푅,푅 푡 = ∇퐻푅 푡 (휇푅 푡 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' This is because 휇푅∗ 푡 ≤ 휇푅 푡 ≤ 훾 − 1 and 퐻푅 푡 (휇) is convex when 휇 ∈ [0,훾 − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' According to Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='5, if 휇푅 푡 ≥ 휇퐵 푡 , then 휇푅 푡 is updated by a stochastic gradient with mean 푔푅,푅 푡 = ∇퐻푅 푡 (휇푅 푡 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' However, note that 휇푅 푡 may not be updated using its own stochastic gradient on 퐻푅 푡 (휇), but may be updated by the gradient 푔푅,퐵 푡 = ∇퐻푅 푡 (휇퐵 푡 ) if 휇퐵 푡 ≥ 휇푅 푡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' However, using the convexity of 퐻푅 푡 (휇) when 휇 ∈ [0,훾 − 1], we have 푔푅,퐵 푡 ≥ 푔푅,푅 푡 as 휇푅 푡 ≤ 휇퐵 푡 ≤ 훾 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Autobidders with Budget and ROI Constraints 29 Let 푔푅 푡 be the gradient that updates 휇푅 푡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Now we have � 퐻푅 푡 (휇푅 푡 ) − 퐻푅 푡 (휇푅∗ 푡 ) � 1{휇푅 푡 ≥ 휇푅∗ 푡 , 휇퐵 푡 ≤ 훾 − 1} ≤ � 푔푅,푅 푡 , 휇푅 푡 − 휇푅∗ 푡 � 1{휇푅 푡 ≥ 휇푅∗ 푡 , 휇퐵 푡 ≤ 훾 − 1} ≤ � 푔푅 푡 , 휇푅 푡 − 휇푅∗ 푡 � 1{휇푅 푡 ≥ 휇푅∗ 푡 , 휇퐵 푡 ≤ 훾 − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (27) When 휇퐵 푡 ≤ 휇푅 푡 , this inequality directly holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' If 휇퐵 푡 ≥ 휇푅 푡 , then 푔푅 푡 > 푔푅,푅 푡 and 휇푅 푡 ≥ 휇푅∗ 푡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Let �푔푟 푡 denote the empirical gradient of 휇푅 푡 at round 푡 with E[�푔푅 푡 ] = 푔푅 푡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' from the analysis of online gradient descent we have: E � 푇 � 푡=1 � 퐻푅 푡 (휇푅 푡 ) − 퐻푅 푡 (휇푅∗ 푡 ) � 1{휇푅 푡 ≥ 휇푅∗ 푡 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 휇퐵 푡 ≤ 훾 − 1} � ≤ E �� �푔푅 푡 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 휇푅 푡 − 휇푅∗ 푡 � 1{휇푅 푡 ≥ 휇푅∗ 푡 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 휇퐵 푡 ≤ 훾 − 1} � ≤ 푇 � 푡=1 � |휇푅 푡 − 휇푅∗ 푡 |2 − |휇푅 푡+1 − 휇푅∗ 푡+1|2 2휂푅 + 휂푅 2 (훾 + 1)2푣2 + (훾 − 1)|휇푅∗ 푡 − 휇푅∗ 푡+1| 휂푅 � 1{휇푅 푡 ≥ 휇푅∗ 푡 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 휇퐵 푡 ≤ 훾 − 1},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (28) where the last inequality is because: |휇푅 푡+1 − 휇푅∗ 푡 |2 ≤ |휇푅 푡 − 휂푅�푔푅 푡 − 휇푅∗ 푡 |2 = |휇푅 푡 − 휇푅∗ 푡 |2 − 2휂푅 � �푔푅 푡 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 휇푅 푡 − 휇푅∗ 푡 � + 휂2 푅|�푔푅 푡 |2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' |휇푅 푡+1 − 휇푅∗ 푡 |2 = |휇푅 푡+1 − 휇푅∗ 푡+1|2 + 2 � 휇푅 푡+1 − 휇푅∗ 푡+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 휇푅∗ 푡 − 휇푅∗ 푡+1 � + |휇푅∗ 푡 − 휇푅∗ 푡+1|2 ≥ |휇푅 푡+1 − 휇푅∗ 푡+1|2 − 2|휇푅 푡+1 − 휇푅∗ 푡+1| · |휇푅∗ 푡 − 휇푅∗ 푡+1| ≥ |휇푅 푡+1 − 휇푅∗ 푡+1|2 − 2(훾 − 1)|휇푅∗ 푡 − 휇푅∗ 푡+1| and |�푔푅 푡 | ≤ (훾 +1)¯푣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Next, we decompose the total horizon [푇], into 푆 intervals 퐼1 = [1,푒1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' , 퐼푆 = [푤푆,푒푆], where each interval is a maximal sequence of consecutive rounds such that 휇푅 푡 ≥ 휇푅∗ 푡 and 휇퐵 푡 ≤ 훾 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Then we have E � 푇� 푡=1 � 퐻푅 푡 (휇푅 푡 ) − 퐻푅 푡 (휇푅∗ 푡 ) � 1{휇푅 푡 ≥ 휇푅∗ 푡 , 휇퐵 푡 ≤ 훾 − 1} � ≤ 푆 � 푠=1 � |휇푅 푤푠 − 휇푅∗ 푤푠 |2 − |휇푅 푒푠+1 − 휇푅∗ 푒푠+1|2 2휂푅 + 휂푅 2 (훾 + 1)2푣2푇 + �푇−1 푡=1 (훾 − 1)|휇푅∗ 푡 − 휇푅∗ 푡+1|2 휂푅 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (29) For 푠 ≥ 2, consider the most recent round 휎푠 before round 푤푠 such that 휇퐵 휎푠 ≤ 훾 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' As 휇푅 푡 ≤ 훾 − 1, we know that 휇퐵 푡 ≥ 휇푅 푡 when 푡 ∈ [휎푠 + 1,푤푠 − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' In addition, according to the update rule of 휇푅 푡 , we know that when 푡 ∈ [휎푠 + 1,푤푠 − 1], 휇푅 푡+1 = 휇푅 푡 + 휂푅 (훾푝푡 (휇푡) − 푣푡) 푥푡 ≤ 휇푅 푡 + 휂푅 � 훾푣푡 1 + (훾 − 1) � 푥푡 ≤ 휇푅 푡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 30 Brendan Lucier, Sarath Patathil, Aleksandrs Slivkins, and Mengxiao Zhang Next, consider the round 휎푠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' If 휎푠 belongs to some interval 퐼푖, according to the definition of 휎푠, 휎푠 must be the end of 퐼푠−1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 휎푠 = 푒푠−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' In this case, we have |휇푅 푤푠 − 휇푅∗ 푤푠 |2 ≤ |휇푅 휎푠+1 − 휇푅∗ 푤푠 |2 (휇푅∗ 푤푠 ≤ 휇푅 푤푠 ≤ 휇푅 휎푠 +1) = |휇푅 휎푠 +1 − 휇푅∗ 휎푠+1|2 + 2(휇푅 휎푠+1 − 휇푅∗ 휎푠 +1)(휇푅∗ 휎푠 +1 − 휇푅∗ 푤푠 ) + |휇푅∗ 휎푠 +1 − 휇푅∗ 푤푠 |2 ≤ |휇푅 푒푠−1+1 − 휇푅∗ 푒푠−1+1|2 + 3(훾 − 1) �� � � 푡 ∈[푒푠−1+1,푤푠−1] |휇푅∗ 푡 − 휇푅∗ 푡+1|�� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Otherwise, 휎푠 is outside the interval and 휇푅 휎푠 < 휇푅∗ 휎푠 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' From the update of 휇푅 푡 , we know that 휇푅 휎푠+1 ≤ 휇푅 휎푠 + 휂푅(훾 + 1)푣 < 휇푅∗ 휎푠 + 휂푅(훾 + 1)푣.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Therefore we know that |휇푅 푤푠 − 휇푅∗ 푤푠 |2 ≤ |휇푅 휎푠 +1 − 휇푅∗ 푤푠 |2 ≤ |휇푅∗ 휎푠 + 휂푅(훾 + 1)푣 − 휇푅∗ 푤푠 |2 ≤ |휇푅∗ 휎푠 − 휇푅∗ 푤푠 |2 + 2휂푅(훾 + 1)푣 · � 휏 ∈[휎푠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푤푠−1] |휇푅∗ 휏 − 휇푅∗ 휏+1| + 휂2 푅(훾 + 1)2푣2 ≤ (훾 − 1)|휇푅∗ 휎푠 − 휇푅∗ 푤푠 | + 2휂푅(훾 + 1)푣 · � 휏 ∈[휎푠,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='푤푠−1] |휇푅∗ 휏 − 휇푅∗ 휏+1| + 휂2 푅(훾 + 1)2푣2 Combining the above two cases and noticing that in the second case,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 휎푠 does not belong to an interval 퐼푖,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' we have for any 휂푅 ≤ min{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 1 푣 },' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 푆 � 푠=1 � |휇푅 푤푠 − 휇푅∗ 푤푠 |2 − |휇푅 푒푠 +1 − 휇푅∗ 푒푠+1|2 휂푅 � ≤ O � 1 + (훾 + 1) �푇−1 푡=1 |휇푅∗ 푡 − 휇푅∗ 푡+1| 휂푅 + 휂푅(훾 + 1)2푣2푇 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (30) Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (30) with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (29), along with the definition of 푃푅 푇 , we have E � 푇� 푡=1 � 퐻푅 푡 (휇푅 푡 ) − 퐻푅 푡 (휇푅∗ 푡 ) � 1{휇푅 푡 ≥ 휇푅∗ 푡 , 휇퐵 푡 ≤ 훾 − 1} � ≤ O � 1 + (훾 + 1)푃푅 푇 휂푅 + 휂푅(훾 + 1)2푣2푇 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Choosing 휂푅 = 1 √ 푇 (훾+1)푣, we know that E � 푇� 푡=1 � 퐻푅 푡 (휇푅 푡 ) − 퐻푅 푡 (휇푅∗ 푡 ) � 1{휇푅 푡 ≥ 휇푅∗ 푡 , 휇퐵 푡 ≤ 훾 − 1} � ≤ O � (훾 + 1)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='5푣(푃푅 푇 + 1) √ 푇 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (31) Step 3-2: Upper bounding the regret with respect to 퐻퐵 푡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Next, we consider 퐻퐵 푡 (휇퐵 푡 ) − 퐻퐵 푡 (휇퐵∗ 푡 ) under the condition that 휇퐵 푡 ≥ 휇퐵∗ 푡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Let 푔퐵,퐵 푡 = ∇퐻퐵 푡 (휇퐵 푡 ) and 푔퐵,푅 푡 = ∇퐻퐵 푡 (휇푅 푡 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Similar to Step 3-1, because of the convexity of 퐻퐵 푡 (휇) when 휇 ∈ [0, +∞), we know that if 휇퐵 푡 ≤ 휇푅 푡 , we have 푔퐵,푅 푡 ≥ 푔퐵,퐵 푡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Therefore, we have the following inequality similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (27): � 퐻퐵 푡 (휇퐵 푡 ) − 퐻퐵 푡 (휇퐵∗ 푡 ) � 1{휇퐵 푡 ≥ 휇퐵∗ 푡 } ≤ � 푔퐵 푡 , 휇퐵 푡 − 휇퐵∗ 푡 � 1{휇퐵 푡 ≥ 휇퐵∗ 푡 }, (32) Autobidders with Budget and ROI Constraints 31 where 푔퐵 푡 = ∇퐻퐵 푡 (휇푡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Then, similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (28), using the fact that 휇퐵∗ 푡 ≤ 푣 휌 for all 푡 ∈ [푇], we have E � 푇� 푡=1 � 퐻퐵 푡 (휇퐵 푡 ) − 퐻퐵 푡 (휇퐵∗ 푡 ) � 1{휇퐵 푡 ≥ 휇퐵∗ 푡 } � ≤ 푇� 푡=1 � |휇퐵 푡 − 휇퐵∗ 푡 |2 − |휇퐵 푡+1 − 휇퐵∗ 푡+1|2 2휂퐵 + 휂퐵 2 (휌 + 푣)2 + 푣|휇퐵∗ 푡 − 휇퐵∗ 푡+1|2 휌휂퐵 � 1{휇퐵 푡 ≥ 휇퐵∗ 푡 }, (33) where (휌 + ¯푣) is a universal upper bound of the empirical gradient for 휇퐵 푡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Next, we similarly decompose the total horizon [푇], into 푆푏 intervals 퐼1 = [1,푒′ 1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' , 퐼 ′ 푆푏 = [푤 ′ 푆푏,푒′ 푆푏], where each interval 퐼 contains a maximal sequence of consecutive rounds such that 휇퐵 푡 ≥ 휇퐵∗ 푡 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Then we have E � 푇� 푡=1 � 퐻퐵 푡 (휇퐵 푡 ) − 퐻푅 푡 (휇퐵∗ 푡 ) � 1{휇퐵 푡 ≥ 휇퐵∗ 푡 } � ≤ 푆푏 � 푠=1 � |휇퐵 푤′푠 − 휇퐵∗ 푤′푠 |2 − |휇퐵 푒′푠+1 − 휇퐵∗ 푒′푠+1|2 2휂퐵 � + 휂퐵 2 (휌 + 푣)2푇 + 푣 �푇−1 푡=1 |휇퐵∗ 푡 − 휇퐵∗ 푡+1|2 휌휂퐵 , (34) For 푠 ≥ 2, Note that we have 휇퐵 푤′푠−1 ≤ 휇퐵∗ 푤′푠−1 and according to the update of 휇퐵 푡 , we also have 휇퐵 푤′푠 ≤ 휇퐵 푤′푠−1 + 휂퐵(푣 + 휌).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Combining the fact that 휇퐵 푤′푠 ≥ 휇퐵∗ 푤′푠 , we have |휇퐵 푤′푠 − 휇퐵∗ 푤′푠 |2 ≤ |휇퐵 푤′푠−1 + 휂퐵(푣 + 휌) − 휇퐵∗ 푤′푠 |2 ≤ |휇퐵∗ 푤′푠−1 + 휂퐵(푣 + 휌) − 휇퐵∗ 푤′푠 |2 ≤ � 푣 휌 + 2휂퐵(푣 + 휌) � |휇퐵∗ 푤′푠−1 − 휇퐵∗ 푤′푠 | + 휂2 퐵(푣 + 휌)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Combining the above with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (34) and choosing 휂퐵 = √휌 √ 푇 (푣+휌) , we have E � 푇� 푡=1 � 퐻퐵 푡 (휇퐵 푡 ) − 퐻푅 푡 (휇퐵∗ 푡 ) � 1{휇퐵 푡 ≥ 휇퐵∗ 푡 } � ≤ O � 푣 휌휂퐵 (1 + 푃퐵 푇 ) + 휂퐵(푣 + 휌)2푇 � ≤ O � 휌−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='5(푣 + 휌)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='5(1 + 푃퐵 푇 ) √ 푇 � (35) Step 4: Combining all the above analysis 32 Brendan Lucier, Sarath Patathil, Aleksandrs Slivkins, and Mengxiao Zhang Finally, we are ready to prove the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (22), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (23), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (24), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (25), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (26), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (31), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (35), we have E � 푇� 푡=1 �푉푡 (휇∗ 푡 ) − 푉푡 (휇푡)� � ≤ E � 푇� 푡=1 �훾 훽 � 푍푅 푡 (휇푅∗ 푡 ) − 휌푡 (휇푅∗ 푡 ) − 푍푅 푡 (휇푅 푡 ) + 휌푡 (휇푅 푡 ) � + 훽휆 � 1{휇푅 푡 ≥ 휇푅∗ 푡 , 휇퐵 푡 ≤ 훾 − 1} � + E � 푇� 푡=1 푣 + 휌 휌 � 푍퐵 푡 (휇퐵∗ 푡 ) − 푍퐵 푡 (휇퐵 푡 ) � 1{휇퐵 푡 ≥ 휇퐵∗ 푡 } � (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (23), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (24)) ≤ E � 푇� 푡=1 �훾 훽 � 4휆(퐻푅 푡 (휇푅 푡 ) − 퐻푅 푡 (휇푅∗ 푡 )) + 훽휆 � 1{휇푅 푡 ≥ 휇푅∗ 푡 , 휇퐵 푡 ≤ 훾 − 1} � + E � 푇� 푡=1 푣 + 휌 휌 � 2휆(퐻퐵 푡 (휇퐵 푡 ) − 퐻퐵 푡 (휇퐵∗ 푡 ))1{휇퐵 푡 ≥ 휇퐵∗ 푡 } � (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (25), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (26)) ≤ E \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 ��� � 훾 훽 � � � 4푇 푇� 푡=1 휆(퐻푅 푡 (휇푅 푡 ) − 퐻푅 푡 (휇푅∗ 푡 )) + 훽휆 ��� � 1{휇푅 푡 ≥ 휇푅∗ 푡 , 휇퐵 푡 ≤ 훾 − 1} \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb + E \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 푣 + 휌 휌 � � � 2푇휆 푇� 푡=1 (퐻퐵 푡 (휇퐵 푡 ) − 퐻퐵 푡 (휇퐵∗ 푡 ))1{휇퐵 푡 ≥ 휇퐵∗ 푡 } \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb (Jensen’s inequality) ≤ O �훾 훽 � 휆푇 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='5(훾 + 1)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='5푣(푃푅 푇 + 1) + 훽휆푇 + (푣 + 휌)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='75 휌1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='25 � 푇 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='5휆(1 + 푃퐵 푇 ) � (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (31), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (35)) ≤ O � (푃푅 푇 + 1) 1 4 휆 3 4 ((훾 + 1)푇) 7 8 + (푣 + 휌) 7 4 휌− 5 4 � 휆(1 + 푃퐵 푇 )푇 3 4 � where we obtain the last inequality is by picking the optimal 훽 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' This completes the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' □ Finally, we provide the proof of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1, which shows that Algorithm 2 ensures a �O(푇 7/8) individual regret bound in the stationary stochastic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Proof of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' In the stationary stochastic setting, define 푍퐵(휇) ≜ 푍퐵 푡 (휇), 푍푅(휇) ≜ 푍푅 푡 (휇), 휌(휇) ≜ 휌푡 (휇), and 푉 (휇) ≜ 푉푡 (휇) for all 푡 ∈ [푇].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Also, we have for all 푡 ∈ [푇], 휇푅∗ 푡 = 휇푅∗ and 휇퐵∗ 푡 = 휇퐵∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Here 휇푅∗ is any 휇 ∈ [0,훾 − 1] such that E[푍푅(휇) − 휌(휇)] = 0, or 0 if no such 휇 exists;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' 휇퐵∗ is any 휇 ∈ [0, 푣 휌 − 1] such that E[푍퐵(휇) − 휌] = 0, or 0 if no such 휇 exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Now consider any 휇 ∈ Π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' As E[푍퐵(휇)] ≤ 퐵 푇 = 휌 and E[푍푅(휇) − 휌(휇)] ≤ 0, according to the monotonicity of 푍퐵(휇) and 푍푅(휇) − 휌(휇) proven in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='4, there exists 휇퐵∗ ∈ [0, 푣 휌 − 1] and 휇푅∗ ∈ [0,훾 − 1] such that 휇 ≥ 휇퐵∗ and 휇 ≥ 휇푅∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Therefore, according to Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2 and the monotonicity of 푉 (휇), we know that 푇� 푡=1 (푉 (휇) − 푉 (휇푡)) ≤ 푇� 푡=1 (푉 (max{휇퐵∗ 푡 , 휇푅∗ 푡 }) − 푉 (휇푡)) ≤ O(푇 7 8 ), where the last inequality is because Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2 and 푃푅 푇 = 푃퐵 푇 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' □ Autobidders with Budget and ROI Constraints 33 D AUXILIARY LEMMAS In this section, we show the auxiliary lemmas that are useful in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1 (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='7 of [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Let 푌1, · · · ,푌푇 be random variables and F0 ⊆ · · · ⊆ F푇 be a filtration such that: (1) 0 ≤ 푌푡 ≤ 푣 with probability 1 for some parameter 푣 ≥ 0 for all 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (2) E[푌푡] ≤ 휌 for some parameter 휌 for all 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' (3) For all 푡, 푌푡 is F푡-measurable but is independent of F푡−1-measurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Then: P( 푇� 푡=1 푋푡푌푡 + (1 − 푋푡)휌 ≥ 휌 ·푇 + 휃) ≤ exp �−2휃2 푇푣2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='2 (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='12 of [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Let 푓 : R → R be an increasing 휆−Lipschitz function such that 푓 (0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Let 푅 = ∫ 푥 0 푓 (푦)푑푦 for some 푥 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Then |푓 (푥)| ≤ √ 2휆푅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='3 (Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content='1 of [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Let �y : [0,푣]푛푇 → 푋푇 be an allocation sequence rule that takes in the entire sequence v1, v2, · · · , v푇 and allocates �푦푘,푡 (v1, v2, · · · , v푇 ) units to agent 푘 at time 푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} +page_content=' Then there exists a (single-round) allocation rule 푦 : [0,푣]푛 → 푋 such that � 푊 (�y, 퐹) := 푛 � 푘=1 min � 퐵푘, 1 훾푘 Ev1,v2,···,v푇 ∼퐹 � 푇� 푡=1 �푦푘,푡 (v1, v2, · · · , v푇 )푣푘,푡 �� = 푛 � 푘=1 푇 · min � 휌푘, 1 훾푘 Ev∼퐹 [푦푘 (v)푣푘] � = 푊 (y, 퐹).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/N9FQT4oBgHgl3EQfWzb0/content/2301.13306v1.pdf'} diff --git a/NtFRT4oBgHgl3EQf4DgA/content/2301.13667v1.pdf b/NtFRT4oBgHgl3EQf4DgA/content/2301.13667v1.pdf new file mode 100644 index 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b/ONAzT4oBgHgl3EQfIPvm/content/tmp_files/2301.01060v1.pdf.txt @@ -0,0 +1,2137 @@ +Noname manuscript No. +(will be inserted by the editor) +Further Improving Weakly-supervised Object Localization via Causal +Knowledge Distillation +Feifei Shao +Yawei Luo* +Shengjian Wu +Qiyi Li +Fei Gao +Yi Yang +Jun Xiao +Received: date / Accepted: date +Abstract Weakly-supervised object localization aims to in- +dicate the category as well as the scope of an object in an im- +age given only the image-level labels. Most of the existing +works are based on Class Activation Mapping (CAM) and +endeavor to enlarge the discriminative area inside the acti- +vation map to perceive the whole object, yet ignore the co- +occurrence confounder of the object and context (e.g., fish +and water), which makes the model inspection hard to dis- +tinguish object boundaries. Besides, the use of CAM also +brings a dilemma problem that the classification and local- +ization always suffer from a performance gap and can not +reach their highest accuracy simultaneously. In this paper, +we propose a casual knowledge distillation method, dubbed +KD-CI-CAM, to address these two under-explored issues in +one go. More specifically, we tackle the co-occurrence con- +text confounder problem via causal intervention (CI), which +explores the causalities among image features, contexts, and +categories to eliminate the biased object-context entangle- +ment in the class activation maps. Based on the de-biased +object feature, we additionally propose a multi-teacher causal +distillation framework to balance the absorption of classifi- +cation knowledge and localization knowledge during model +training. Extensive experiments on several benchmarks demon- +strate the effectiveness of KD-CI-CAM in learning clear ob- +ject boundaries from confounding contexts and addressing +the dilemma problem between classification and localization +performance.1 +∗ Yawei Luo is the corresponding author. +Feifei Shao, Yawei Luo, Yi Yang, and Jun Xiao are with the +Zhejiang University, Hangzhou, China. E-mail: sff@zju.edu.cn, +yaweiluo329@gmail.com, yangyics@zju.edu.cn, junx@cs.zju.edu.cn. +Shengjian Wu and Qiyi Li are with the Finvolution Group, Shanghai, +China. E-mail: wushengjian@xinye.com, liqiyi@xinye.com. +Fei Gao is with the Zhejiang University of Technology, Hangzhou, +China. E-mail: feig@zjut.edu.cn. +1 Our code is publicly available at https://github.com/ +shaofeifei11/KD-CI-CAM. +Keywords Object Localization · Weakly-supervised +Learning · Knowledge Distillation · Causal Intervention +1 Introduction +Object localization [5, 9, 39, 50] aims to indicate the cate- +gory and the scope of an object in a given image, in forms +of bounding box [12,25,38]. This task has been studied ex- +tensively in the computer vision community [46, 50] due to +its broad applications, such as medical diagnosis [1,37] and +autonomous driving [4, 44]. Recently, the techniques based +on deep convolutional neural networks (DCNNs) [17,28,30, +45,47] promote the localization performance to a new level. +However, this performance promotion is at the price of huge +amounts of fine-grained human annotations [6, 27, 29, 42]. +To alleviate such a heavy burden, weakly-supervised object +localization (WSOL) has been proposed by only resorting to +image-level labels. +To capitalize on the image-level labels, the existing stud- +ies [10, 21, 40, 41, 43, 53] follow the Class Activation Map- +ping (CAM) approach [72] to generate class activation maps +first and then segment the highest activation area for a coarse +localization. Albeit, CAM is initially designed for the clas- +sification task and tends to focus only on the most discrimi- +native feature to increase its classification accuracy. To deal +with this issue, recent prevailing works [10, 14, 21, 31, 53, +62,67] endeavor to perceive the whole objects instead of the +shrunken and sparse “discriminative regions”. On one hand, +they refine the network structure to make the detector more +tailored for object localization in a weak supervision set- +ting. For example, some methods [10, 53] use a three-stage +structure to continuously optimize the prediction results by +training the current stage using the output of the previous +stages as supervision. Other works [14, 21, 31, 67] leverage +two parallel branches where the first branch is designed for +arXiv:2301.01060v1 [cs.CV] 3 Jan 2023 + +2 +Feifei Shao +Yawei Luo* +Shengjian Wu +Qiyi Li +Fei Gao +Yi Yang +Jun Xiao +Image +CAM +NL-CCAM +CI-CAM +(a) +(b) +KD-CI-CAM +使用了center crop， using fish +Fig. 1: (a) Visualization comparison between vanilla CAM, NL-CCAM, CI-CAM, and KD-CI-CAM. The yellow arrows +indicate the regions suffer from entangled contexts. (b) The classification-localization dilemma faced by prior CAM-based +methods (e.g., CI-CAM), where the classification and localization suffer from a performance gap and can not achieve their +highest accuracy simultaneously. +digging out the most discriminative small regions while the +second one is responsible for detecting the less discrimina- +tive large regions. On the other hand, they also make full +use of the image information to improve the prediction re- +sults. For instance, TS2C [53] and NL-CCAM [62] utilize +the contextual information of surrounding pixels and the ac- +tivation maps of low probability class, respectively. +Although the works along the vein of CAM have led +to some impressive early results, we have discovered multi- +faceted problems that remain unexplored by those prior arts, +which emerged from our experimental analysis. Firstly, vanilla +CAM mechanism is incapable of reasoning about the co- +occurrence confounder (e.g., fish and water) in the images, +which makes the model inspection hard to distinguish be- +tween the object and context and causes biased activation +maps, as shown in Figure 1 (a). We dub this problem as “en- +tangled context” for convenience. Secondly, we noticed that +the classification and localization always suffer from a per- +formance gap and can not reach their highest accuracy si- +multaneously in previous CAM-based methods. Taking CI- +CAM [43] as an example, we set up multiple independent +experiments using different training settings, however, none +of the results can achieve the best classification and localiza- +tion performance at the same time, as reported in Figure 1 +(b). We summarise this issue as a classification-localization +dilemma (“C-L dilemma” for short). We argue that these +two problems severely hinder the WSOL performance and +heretofore yet to be well studied, despite the existence of a +vast body of WSOL literature [10,31,52,56,67,69,72]. +Taking one step further into these two issues, we found +the internal reasons behind them are different. The “entan- +gled context” derives from the fact that objects usually co- +occur with a certain context background, e.g., the most “fish” +appears concurrently with “water” in the images. Conse- +quently, these two concepts would be inevitably entangled +and a classification model would wrongly generate an am- +biguous boundary between “fish” and “water” with image- +level supervision. In contrast to the vanilla CAM which yields +a relatively shrunken bounding box on the small discrimina- +tive region, we notice that the “entangled context” problem +would cause a biased expanding bounding box that includes +the wrongly entangled background, which impairs the lo- +calization accuracy in terms of the object range. While the +“C-L dilemma” is derived from a hallmark of a classification +network that tends to focus on discriminative region infor- +mation (e.g., the head of an animal) over the integral profile +of an object to better distinguish various categories. Forc- +ing a classification model to pay more attention to the un- +representative area (e.g., the fur of an animal) conceding to +the integral contour perception would inevitably cause a bi- +ased categorical prediction, and vise versa for a localization +model. Prior approaches sidestep such a dilemma by simply +trading off the classification and localization performances, +i.e., choosing a mutually acceptable model or only reporting +localization results [22,35,56,59]. However, the intrinsic is- +sue behind this dilemma remains under-explored. +In this paper, we propose a casual knowledge distilla- +tion framework to solve both “entangled context” and “C-L + +Perio20LwdPajons2o17@Antonio Hidalgo@Antonio HidalgoAntonio Hidalgo@Antonio Hidalgo@Antonio Hidalgo(VGG16) CI-CAM +(VGG16) KD-CI-CAM +(InceptionV3) CI-CAM +(InceptionV3) KD-CI-CAM +79 +78 +16 +75 +66 +68 +70 +72 +74 +76 +Top-1 Loc Accuracy (%)TOFurther Improving Weakly-supervised Object Localization via Causal Knowledge Distillation +3 +dilemma” problems in one go, dubbed KD-CI-CAM. Specif- +ically, we first explore the causalities among image features, +contexts, and labels by establishing a structural causal model +(SCM) [36] and pinpoint the context as a confounder as +shown in Figure 2. According to the causal analysis, we +propose a causal context pool to eliminate the biased co- +occurrence in the class activation maps to solve the “entan- +gled context” problem as shown in Figure 3. Based on dis- +entangled object features from the wrongly contextual in- +formation, we further tackle the “C-L dilemma” problem +faced by the CAM-based model via a multi-teacher causal +distillation framework as shown in Figure 4. The first kind +of teacher focuses on classification, which aims to provide +good logit knowledge. The other one is a localization teacher, +which is responsible for providing high-quality class activa- +tion map knowledge. These two teachers endow our student +model with the ability to balance the absorption of classifica- +tion knowledge and localization knowledge. To prevent the +degradation of knowledge transfer from the model capacity +gap between the student and teachers [13,34], these teachers +adopt the same network structure with the student except for +their objective loss functions. With these tailored designs, +KD-CI-CAM can not only effectively eliminate the spurious +correlations between pixels and labels from the “entangled +context” problem, but also alleviates the “C-L dilemma” prob- +lem resulting from the interesting bias between the classifi- +cation information and localization information. +It is noteworthy that some pertinent results of this work +have been published in an earlier version [43]. This paper +goes beyond [43] by contributing to introducing a novel ca- +sual knowledge distillation framework to further boost the +WSOL by solving the notorious but under-explored “entan- +gled context” and “C-L dilemma” problems in one go. We +also make significant improvements in experimental valida- +tion and analysis compared to the early version. +Generally, the contributions of this paper can be summa- +rized as follows: +– We are among the pioneers to concern and reveal the +“entangled context” problem and “C-L dilemma” prob- +lem of WSOL that remain unexplored by prevailing ef- +forts. +– We propose a novel casual knowledge distillation method, +dubbed KD-CI-CAM, to solve the under-explored “en- +tangled context” and “C-L dilemma” problems in one +go. We first design a student model by using causal in- +tervention to address the confounder context, and then +balance its absorption of classification knowledge and +localization knowledge by using a knowledge distilla- +tion framework. +– Extensive experiments show that both “entangled con- +text” and “C-L dilemma” problems are effectively solved +by our proposed methods. +2 Related Work +2.1 Weakly-supervised Object Localization +Since CAM [72] is prone to bias to the most discriminative +part of the object rather than the integral object, the research +attention of most of the current methods is how to improve +the accuracy of object localization. These methods can be +broadly categorized into two groups: enlarging the proposal +region and discriminative region removal. +1) Enlarging proposal region: enlarging the box size +appropriately of the initial prediction box [10,53]. WCCN [10] +introduces three cascaded networks trained in an end-to-end +pipeline. The latter stage continuously enlarges and refines +the output proposals of its previous stage. TS2C [53] selects +the box by comparing the mean pixel confidence values of +the initial prediction region and its surrounding region. If +the gap between the mean values of the two regions is large, +the initial prediction region is the final prediction box; oth- +erwise, the surrounding region. +2) Discriminative region removal: detecting the big- +ger region after removing the most discriminative region [8, +21, 31, 67]. TP-WSL [21] first detects the most discrimina- +tive region in the first network, Then, it erases this region of +the conv5-3 feature maps in the second network (e.g., zero). +ACoL [67] uses the masked feature maps by erasing the +most discriminative region discovered in the first classifier +as the input feature maps of the second classifier. ADL [8] +stochastically produces an erased mask or an importance +map at each iteration as a final attention map projected in the +feature maps of images. MEIL [31] is an adversarial erasing +method that simultaneously computes the erased branch and +the unerased branch by sharing one classifier. +The above methods basically focus on the poor localiza- +tion caused by the most discriminative part of the object. +However, they ignore the problem of the fuzzy boundary +between the objects and the co-occur certain context back- +ground. For example, if most “fish” appears concurrently +with “water” in the images, these two concepts would be in- +evitably entangled and wrongly generate ambiguous bound- +aries using only image-level supervision. +2.2 Causal Inference +Structural Causal Model (SCM) [36] is an important analy- +sis tool in causal inference scenes [49,63,66], which is a di- +rected graph in which each node represents each participant +of the model, and each link denotes the causalities between +the two nodes [66]. Specifically, Zhang et al. [66] utilize a +SCM [11, 36] to deeply analyze the causalities among im- +age features, contexts, and class labels and propose Context +Adjustment (CONTA) that achieves the new state-of-the- +art in weakly-supervised semantic segmentation task. Yue + +4 +Feifei Shao +Yawei Luo* +Shengjian Wu +Qiyi Li +Fei Gao +Yi Yang +Jun Xiao +et al. [63] use the causal intervention in few-shot learning +and uncover the pre-trained knowledge is indeed a confounder +that limits the performance. Finally, they propose a novel +FSL paradigm: Interventional Few-Shot Learning (IFSL). +Tang et al. [49] show the SGD momentum is essentially a +confounder in long-tailed classification by using a SCM. +Inspired by CONTA [66], we also leverage a SCM [36] +to analyze the causalities among image features, contexts, +and class labels and find context is a confounder factor in +§3.2.1. But, our work does not follow the strategy of al- +ternating training of multiple models. Instead, we propose +an end-to-end model embedding the causal inference into +the WSOL pipeline, which is capable of making the feature +boundary clearer by using a causal context pool in §3.2.3. +2.3 Knowledge Distillation +Knowledge distillation [15, 19, 57] aims to transfer knowl- +edge from the cumbersome teacher model to a small student +model. It can be categorized into two parts: knowledge and +distillation. First, knowledge can be grouped into response- +based knowledge [19] and feature-based knowledge [71]. +The response-based knowledge refers to the prediction log- +its, which are simple yet effective in the distillation of the +classification task. However, using only the teacher’s logits +information in complex distillation is insufficient. Thus, re- +searchers pay attention to utilizing feature-based knowledge +to boost the effect of distillation in complex scenes. Sec- +ond, distillation is the mode of transferring the knowledge +from the teacher model to the student model, which also can +be divided into offline distillation [18, 20], online distilla- +tion [7, 55], and self-distillation [24, 64, 70]. To make the +student model learn knowledge smoothly, it usually adopts +a “soft target”—using a “softmax” with a temperature T on +the knowledge of teachers, as the extra supervision of the +student model. +In this work, to harness the absorption of classification +and localization knowledge in the CAM-based model, we +derive a multi-teacher causal distillation framework. The clas- +sification knowledge and localization knowledge are selected +stochastically at each distillation process to supervise the +learning of the student, which will be detailed in §3.3. +3 Methodology +In this section, we first introduce the preliminaries of prob- +lem settings and baseline method in §3.1. Then we will con- +centrate on solving the “entangled context” issue and pro- +pose our causal intervention method in §3.2. Based on the +casual model, we finally tackle the “C-L dilemma” problem +and construct the multi-teacher casual distillation framework +(KD-CI-CAM) in §3.3. +3.1 Preliminaries +3.1.1 Problem Settings +Before presenting our method, we first introduce the prob- +lem settings of WSOL formally. Given an image I, WSOL +targets classifying and locating one object in terms of the +class label and the bounding box. However, only image-level +labels Y can be accessed during the training phase. +3.1.2 Baseline Method +Class activation maps (CAMs) are widely employed for gen- +erating the object boxes in the WSOL task. Yang et al. [62] +argue that using only one activation map of the highest prob- +ability class for segmenting object boxes is problematic since +it often biases to over-small regions or sometimes even high- +lights background area. Based on such observation, they pro- +pose the NL-CCAM [62] method to combine all activation +maps from the highest to the lowest probability class to a lo- +calization map using a specific combinational function and +achieve good localization performance. +Based on the vanilla fully convolutional network (FCN)- +based backbone, e.g., VGG16 [45], NL-CCAM [62] inserts +four non-local blocks before every bottleneck layer exclud- +ing the first bottleneck layer simultaneously to produce a +non-local fully convolutional network (NL-FCN). Given an +image I, it is fed into the NL-FCN to produce its feature +maps X ∈ Rc×h×w, where c is the number of channels and +h × w is the spatial size. Then, the feature maps X are for- +warded to a global average pooling (GAP) layer followed +by a classifier with a fully connected layer. The prediction +scores S = {s1, s2, . . . , sn} are computed by using a soft- +max layer on the top of the classifier for classification. The +weight matrix of the classifier is denoted as W ∈ Rn×c, +where n is the number of image classes. Thus, the activation +maps Mi of class i among class activation maps (CAMs) +M ∈ Rn×h×w proposed in [72] are given as follows. +Mi = +c +� +k +Wi,k · Xk, +(1) +where i ∈ {1, 2, . . . , n}. +NL-CCAM [62] produces a localization map by using a +combinational function in CAMs instead of using the acti- +vation map of the highest probability class among CAMs. +Firstly, it ranks the activation maps from the highest proba- +bility class to the lowest and uses Mtk to denote the activa- +tion map of the k highest probability class. The class label +with the highest probability t1 is computed as follows. +t1 = argmaxk({Sk}), +(2) + +Further Improving Weakly-supervised Object Localization via Causal Knowledge Distillation +5 +������������ +������������ +������������ +������������ +(a) +(b) +������������ +������������ +������������ +������������ +Fig. 2: (a) Building the structural causal model (SCM) in +WSOL. (b) Cutting off the confounding effect of C → X in +WSOL. X: feature maps. C: confounder context. V : image +representation. Y : image label. +where k ∈ {1, 2, . . . , n}. Then it combines M to a localiza- +tion heatmap H ∈ Rh×w as follows. +H = +n +� +k +γ(k) · Mk, +(3) +where γ(·) is a combinational function. Finally, it segments +the localization heatmap H using a threshold proposed in [72] +to generate a bounding box for object localization. +Our method is based on NL-CCAM [62] but introduces +substantial improvements. We not only equip the baseline +network with the ability of causal inference to tackle the “en- +tangled context” problem but also address the “C-L dilemma” +problem suffered from the traditional CAM-based models. +3.2 Causal Intervention for Student Network +In this section, we target the “entangled context” problem. +Specifically, we first reveal the reason why the confounder +context hurts the object localization quality using a struc- +tural causal model [36] in §3.2.1. Then, we use the causal +intervention to solve the “entangled context” problem theo- +retically via the backdoor adjustment [36]. Finally, we im- +plement the causal module in our model in §3.2.3. +3.2.1 Structural Causal Model +Inspired by CONTA [66], we utilize a structural causal model +(SCM) [36] to analyze the causalities among image features +X, confounder context C, and image-level labels Y . The +direct links shown in Figure 2 (a) denote the causalities be- +tween the two nodes: cause → effect [66]. +C → X: This link indicates that the backbone gen- +erates feature maps X under the effect of context C. Al- +though the confounder context C is helpful for a better as- +sociation between the image features X and labels Y via a +model P(Y |X), e.g., it is likely a “fish” when perceiving a +“water” region, P(Y |X) mistakenly associates non-causal +but positively correlated pixels to labels, e.g., the “water” +region wrongly belongs to “fish”. This is a vital reason for +the inaccurate localization in WSOL. Fortunately, as we will +introduce later in §3.2.3, we can avoid it by using a causal +context pool in causal intervention. +C → V ← X: V is an image specific-representation +using the contextual templates from C [66]. For example, +V tells us the shape and location of a “fish” (foreground) +in a scene (background) [66]. In this paper, V denotes the +activation map of the highest probability class in the CAM +1 module as shown in Figure 3. +X → Y ← V : These links indicate that image feature +X and image representation V together affect the image la- +bel Y of an image. We consider that the shape and location +information of object instance contained in image represen- +tation V also directly affects the image label Y . As a conse- +quence, though V is not an input factor in the WSOL model, +it exists [66]. +3.2.2 Theoretical Analysis +To remove the confounding effect of C → X as shown in +Figure 2 (b), we take inspiration from CONTA [66], follow- +ing the same rule to use P(Y |do(X)) based on the backdoor +adjustment [36] as the new image-level classifier. The key +idea is that X and C = {c1, c2, . . . , cn} are independent +events. As C does not affect X, it guarantees X to have a +fair opportunity to incorporate every context c into Y ’s pre- +diction, subject to a prior P(c) [66]. Formally, we have the +following probability formula: +P(Y |do(X)) = +n +� +i +P(Y |do(X), ci) · P(ci|do(X)) += +n +� +i +P(Y |X, ci) · P(ci) += +n +� +i +P(Y |X, ci, V ) · P(V |X, ci) · P(ci) += +n +� +i +P(Y |X, ci, V = f(X, ci)) · P(ci) += +n +� +i +P(Y |X, V = f(X, ci)) · P(ci) += +n +� +i +P(Y |X = x, V = f(x, ci)) · P(ci), +(4) +where P(Y |do(X)) = �n +i P(Y |do(X), ci) · P(ci|do(X)) +follows the law of total probability. n is the number of im- +age classes. Since X and C = {c1, c2, . . . , cn} are indepen- +dent events, P(ci|do(X)) = P(ci). f(X, ci) abstractly rep- +resents that V is effected by X and ci, so P(V |X, ci) = 1. + +6 +Feifei Shao +Yawei Luo* +Shengjian Wu +Qiyi Li +Fei Gao +Yi Yang +Jun Xiao +Because Y is only directly affected by X and V in Figure 2 +(b), P(Y |X, ci, V = f(X, ci)) = P(Y |X, V = f(X, ci)). +Inspired by CONTA [66], we adopt the NWGM [60] to op- +timize Eq. (4) by moving the outer sum �n +i P(ci) into the +feature level +P(Y |do(X)) ≈ P(Y |X = x, V = +n +� +i +f(x, ci) · P(ci)). +(5) +Since the number of samples for each class in the dataset is +roughly the same, we set P(c) to uniform 1/n. After further +optimizing Eq. (5), we have +P(Y |do(X)) ≈ P(Y |x ⊕ 1 +n · +n +� +i +f(x, ci)), +(6) +where ⊕ denotes projection. So far, the “entangled context” +issue has been transferred into calculating �n +i f(x, ci). We +will introduce a causal context pool Q to represent �n +i f(x, ci) +in §3.2.3. +3.2.3 Student Network Architecture +We implement a causal model for the “entangled context” +problem, dubbed CI-CAM, at the core of which is a causal +context pool. The main idea of the causal context pool is to +accumulate all contexts of each class, and then re-project the +contexts to the feature maps of convolutional layers shown +in Eq. (6) to pursue the pure causality between the cause +X and the effect Y . Figure 3 illustrates the overview of +CI-CAM that includes four parts: backbone, CAM module, +causal context pool, and combinational part. +Backbone. Inherited from the baseline method, we de- +sign our backbone by inserting multiple non-local blocks at +both low- and high-level layers of a FCN-based network +simultaneously. It acts as a feature extractor that takes the +RGB images as input and produces high-level position-aware +feature maps. +CAM module. It includes a global average pooling (GAP) +layer and a classifier with a fully connected layer [72]. Im- +age feature maps X generated by the backbone are forwarded +into GAP and classifier to produce prediction scores S = +{s1, s2, . . . , sn}. The CAM network multiplies the weight +W of the classifier to X to produce class activation maps +M ∈ Rn×h×w shown in Eq. (1). In our model, we use two +CAM modules with shared weights. The first CAM mod- +ule is designed to produce initial prediction scores S and +class activation maps M, and the second CAM network is +responsible for producing more accurate prediction scores +Se = {se +1, se +2, . . . , se +n} and class activation maps M e ∈ +Rn×h×w using the feature maps Xe ∈ Rc×h×w enhanced +by the causal context pool. +Causal context pool. We maintain a causal context pool +Q ∈ Rn×h×w during the network training phase, where Qi +denotes the context of all ith class images. Q ceaselessly +stores all contextual information maps (e.g., V ) of each class +by accumulating the activation map of the highest probabil- +ity class. Then, it projects all contexts of each class as at- +tention onto the feature maps of the last convolutional layer +to produce enhanced feature maps. The idea behind using +a causal context pool is not only to cut off the negative ef- +fect of entangled context on image feature maps but also to +spotlight the positive region of the image feature maps for +boosting localization performance. +Combinational part. The input of the combinational +part is class activation maps M e generated from the CAM +2 module, and the corresponding output is a localization +heatmap H ∈ Rh×w calculated by Eq. (3): First, the com- +binational part ranks the activation maps from the highest +probability class to the lowest. Second, it combines these +sorted activation maps by a combinational function as Eq. (3). +With all the key modules presented above, we would +give a brief illustration of the data flow in our network. Given +an image I, we first forward I to the backbone to produce +feature maps X. X is then fed into the following two par- +allel CAM branches. The first CAM branch produces initial +prediction scores S and class activation maps M. Then, the +causal context pool Q would be updated by fusing the acti- +vation map of the highest probability class in M as follows: +Qπ = bn(Qπ + λ · bn(V )), V = Mπ, +(7) +where π = argmax({s1, s2, . . . , sn}), λ denotes the up- +date rate, and bn denotes the batch normalization. The sec- +ond branch is responsible for producing more accurate pre- +diction scores Se and class activation maps M e. The input +of the second branch is enhanced feature maps Xe projected +by the context among causal context pool Q of the highest +probability class generated from the first branch. More con- +cretely, the feature enhancement can be calculated as +Xe = X + X ⊙ Conv1×1(Qπ), +(8) +where ⊙ denotes the matrix dot product. In the combina- +tional part, we first build a localization heatmap H ∈ Rh×w +by aggregating all activation maps from the highest to the +lowest probability class using a specific combinational func- +tion [62] in Eq. (3). Then, we use the simple thresholding +technique proposed by [72] to generate a bounding box B +from the localization map. Finally, the bounding box B and +prediction scores Se as the final prediction of CI-CAM. +3.2.4 Student Network Training Objective +During the phase of training, our proposed student network +learns to minimize image classification losses for both clas- +sification branches. Given an image I, we can obtain initial + +Further Improving Weakly-supervised Object Localization via Causal Knowledge Distillation +7 +VGG16 +Non-local +modules +Feature maps ������������ +Enhanced +feature maps ������������������������ +element +-wise +CAM 1 +GAP +Classifier +Causal context pool ������������ +cat +airliner +barn +beer +taxi +necklace +… +… +camera +bridge +Causal context pool +CAM 2 +Shared +Normalization +CAMs ������������������������ +Combination +all classes +Prediction ������������ +CAMs ������������ +Highest +probability +class ������������ +⊙ +Image representation ������������ +GAP +Classifier +Prediction ������������������������ +Label ������������ +Label ������������ +Fig. 3: Overview of the proposed CI-CAM (Student network). CI-CAM consists of four parts: a backbone to extract the +feature maps, the share-weighted CAM modules to generate the class activation maps, a causal context pool (which is the core +of the CI-CAM method) to enhance the feature maps by eliminating the negative effect of confounder, and a combinational +module to generate the final bounding box. +prediction scores S = {s1, s2, . . . , sn} and more accurate +prediction scores Se = {se +1, se +2, . . . , se +n} of the two classi- +fiers as shown in Figure 3. We follow a naive scheme to train +the two classifiers together in an end-to-end pipeline using +the following loss function Lstu (Stu Loss). +Lstu = ρ · +� +− +n +� +i=1 +yi · log(si) +� ++ +� +− +n +� +i=1 +yi · log(se +i) +� +, +(9) +where y is the ground-truth label of an image. If Stu Loss +Lstu is used in Eq. 19 and Eq. 21, ρ = 1; otherwise, ρ = 0. +3.3 Multi-Teacher Causal Knowledge Distillation +3.3.1 Distillation Framework +To make the model simultaneously retain good classifica- +tion performance and localization performance, we design +a multi-teacher causal distillation framework that randomly +selects the knowledge of the pre-trained classification teacher +and localization teacher to supervise the learning of our stu- +dent as shown in Figure 4 (a). +First, the pre-trained classification teacher provides logit +knowledge to help the student network to achieve good clas- +sification performance. Inspired by [19], we design our log- +its distillation loss function Lld between classification teacher +and student as follows. +Lld = KL( ˜Zstu, ˜Zcls tch), +(10) +˜Zstu +i += +exp(Zstu +i +/T cls) +�n +j=1 exp(Zstu +j +/T cls), +(11) +˜Zcls tch +i += +exp(Zcls tch +i +/T cls) +�n +j=1 exp(Zcls tch +j +/T cls), +(12) +where KL is the Kullback-Leibler divergence function. Zstu +and Zcls tch respectively denote the output logits Z ∈ Rn of +the student network and the classification teacher network. +n is the number of classes and i ∈ {1, 2, . . . , n}. T cls is the +logits distillation temperature. +Second, the pre-trained localization teacher brings the +localization knowledge in the class activation maps to help +the student network to accurately localize the whole object +rather than the most discriminative part of the object. We +design the activation distillation loss function Lad between +localization teacher and student as follows. +Lad = MSE( ˜Astu, ˜Aloc tch), +(13) +˜Astu +i += +exp(Astu +i +/T loc) +�nhw +j=1 exp(Astu +j +/T loc) +, +(14) +˜Aloc tch +i += +exp(Aloc tch +i +/T loc) +�nhw +j=1 exp(Aloc tch +j +/T loc) +, +(15) +where MSE is the Mean Squared Error function. Astu and +Aloc tch respectively denote the class activation maps A ∈ + +[]8 +Feifei Shao +Yawei Luo* +Shengjian Wu +Qiyi Li +Fei Gao +Yi Yang +Jun Xiao +Class activation maps +Pre-trained +classification +teacher +Pre-trained +localization +teacher +Student +Logits Loss +ℒ������������������������ +Student +network +Fore Loss +ℒ ������������������������������������������������ +Diff Loss +ℒ������������������������������������������������ +Student +network +Student +network +Student +network +Student +network +(b) Classification teacher +(c) Localization teacher +Original image logits ������������ +Original image logits ������������ +(a) Distillation +Heatmap H +Stu Loss +ℒ������������������������������������ +������������ +∫ +∫ +Random +selection +Threshold +Background +threshold +Area Loss ℒ������������������������������������������������ +Area Loss ℒ������������������������������������������������ +Activation Loss +ℒ������������������������ +Original image ������������ +Foreground +image ������������fore +Stu Loss ℒ������������������������������������ +Stu Loss ℒ������������������������������������ +Foreground image +logits ������������������������������������������������������������ +Background image +logits ������������������������������������������������������������ +Original image ������������ +Background +image ������������������������������������������������������������ +Fig. 4: Overview of the multi-teacher causal distillation framework. (a) The classification teacher and localization teacher are +applied stochastically during each distillation. (b) The classification teacher adopts the student network as its model structure +but adds Area Loss and Fore Loss. The two orange student networks in the classification teacher are the same model. (c) +The localization teacher uses the student network as its model structure but adds Area Loss and Diff Loss. The three orange +student networks in the localization teacher are the same model. +Rnhw of the student network and the localization teacher +network. h and w are the height and width of the activation +map. T loc is the activation distillation temperature. Finally, +the total loss function Ltotal of the multi-teacher causal dis- +tillation framework in Figure 4 (a) is given as follows. +Ltotal =α · [(ϵ == 1) · Lld + (ϵ == 2) · Lad] ++ (1 − α) · Lstu, +(16) +where α is the distillation strength hyper-parameter. ϵ ∈ +{0, 1, 2} denotes the controller of the random selection in +Figure 4 (a). +3.3.2 Classification Teacher Network +As shown in Figure 4 (b), to prevent the degradation of knowl- +edge transfer from the model capacity gap between student +and teacher [13, 34], we adopt the same network structure +with the student as our classification teacher but add two +extra loss functions. Specifically, given an image I, it is +first forwarded into the model to produce the original im- +age logits S = {s1, s2, . . . , sn} and localization heatmap +H ∈ Rh×w, where h and w respectively denote the height +and width of the original image I. Then, we obtain a binary +foreground mask M fore by segmenting the heatmap H us- +ing a threshold. Next, we generate a foreground image Ifore +by projecting the M fore into the original image I. Finally, +Ifore is fed into the model again to produce the foreground +image logits Sfore = {sfore +1 +, sfore +2 +, . . . , sfore +n +}. +To clearly distinguish the foreground area from the back- +ground area in heatmap H, we first introduce the Area Loss +of [32, 58] to reduce the activation value of the foreground +area and the background area in heatmap H. Then, we ap- +ply a Fore Loss to activate the object areas in the image by +classifying the foreground image. As a result, Area Loss and +Fore Loss terms together can suppress the background areas +and highlight the foreground areas [32]. Area Loss Larea +and Fore Loss Lfore are given as follows. +Larea = +1 +h · w +h +� +i=1 +w +� +j=1 +Hi,j, +(17) +Lfore = − +n +� +i=1 +yi · log(sfore +i +), +(18) +where y is the ground-truth label of an image. Finally, our +classification teacher learns to minimize the Area Loss Larea, +Fore Loss Lfore, and Stu Loss Lstu. +Lcls tch = Lstu + µ · Lfore + η · Larea, +(19) +where Lcls tch is the total loss of our classification teacher. +µ and η are the hyper-parameters. +3.3.3 Localization Teacher Network +As shown in Figure 4 (c), similar to the classification teacher +network, we also use the same network structure with the +student as our localization teacher but add two extra loss +functions. Given an image I, it is first forwarded into the +model to produce the original image logits S = {s1, . . . , sn} + +Further Improving Weakly-supervised Object Localization via Causal Knowledge Distillation +9 +Vanilla student +Localization teacher +Input image +Bohemian_Waxwing_0057_177784 +Brandt_Cormorant_0091_22825 +Fig. 5: Visualization comparison between vanilla student +(without knowledge distillation) and localization teacher. +and localization heatmap H ∈ Rh×w, where h and w re- +spectively denote the height and width of the original im- +age I. Then, we respectively obtain the binary foreground +mask M fore and background mask M back by segmenting +the heatmap H using a threshold. Next, we generate the +foreground image Ifore and background image Iback by re- +spectively projecting the M fore and M back into the origi- +nal image I. Finally, the foreground image Ifore and back- +ground image Iback are fed into the model again to yield the +foreground image logits Sfore = {sfore +1 +, sfore +2 +, . . . , sfore +n +} +and background image logits Sback = {sback +1 +, . . . , sback +n +}, +respectively. +To make the foreground image contain the integral ob- +ject as much as possible while reducing the object informa- +tion in the background area, we design a difference loss (Diff +Loss) Ldiff, which is given as follows. +Ldiff = − +n +� +i=1 +yi · log(sfore +i +− sback +i +) +− +n +� +i=1 +yi · log(si + (sfore +i +− sback +i +) +2 +), +(20) +where y and s are the ground-truth label of an image and +the original image prediction, respectively. Finally, our lo- +calization teacher learns to minimize the Area Loss Larea, +Diff Loss Ldiff, and Stu Loss Lstu. +Lloc tch = Lstu + β · Ldiff + δ · Larea, +(21) +where Lloc tch denotes the total loss of the localization teacher. +β and δ are the hyper-parameters. Figure 5 shows the com- +parison result between our vanilla student and localization +teacher, which verifies our localization teacher can produce +a high-quality class activation map and alleviate the discrim- +inative region problem. +4 Experiments +4.1 Datasets and Evaluation Metrics +Datasets. The proposed KD-CI-CAM was evaluated on two +public datasets: CUB-200-2011 [51] and ILSVRC 2016 [38]. +1) CUB-200-2011 is an extended version of Caltech-UCSD +Birds 200 (CUB-200) [54] containing 200 bird species which +focuses on the study of subordinate categorization. Based on +the CUB-200, CUB-200-2011 adds more images for each +category and labels new part localization annotations. CUB- +200-2011 contains 5, 994 images in the training set and 5, 794 +images in the test set. Each image of CUB-200-2011 is an- +notated by the bounding boxes, part locations, and attribute +labels. 2) ILSVRC 2016 is the dataset originally prepared +for the ImageNet Large Scale Visual Recognition Challenge +(ILSVRC). It contains 1.2 million images of 1, 000 cate- +gories in the training set, 50, 000 in the validation set, and +100, 000 images in the test set. For both datasets, we only +utilize the image-level classification labels for training, as +constrained by the problem setting in WSOL. +Evaluation Metrics. We leverage the classification ac- +curacy (Cls) and localization accuracy (Loc) as the eval- +uation metrics for WSOL. The former includes Top-1 and +Top-5 classification accuracy, while the latter includes Top- +1, Top-5, and GT-known localization accuracy. Top-1 classi- +fication accuracy denotes the accuracy of the highest predic- +tion score (likewise for localization accuracy). Top-5 classi- +fication accuracy denotes that if one of the five predictions +with the highest score is correct, it counts as correct (like- +wise for localization accuracy). GT-known localization ac- +curacy is the accuracy that only considers localization re- +gardless of classification result compared to Top-1 localiza- +tion accuracy [31]. +4.2 Implementation Details +We adopt the VGG16 [45] and InceptionV3 [48] pre-trained +on the ImageNet [38] as our backbone. We augment the +training images with RandAugment [48] and use Adam [23] +to optimize our network with β1 = 0.9 and β2 = 0.99 on +the CUB-200-2011 [51] and ILSVRC 2016 [38] datasets. If +we use VGG16 as our backbone and train it on the CUB- +200-2011, we insert four non-local blocks to the backbone +before every bottleneck layer excluding the first one. Oth- +erwise, we insert three non-local blocks into the backbone. +The newly added non-local blocks are randomly initialized +except for the batch normalization layers, which are initial- +ized as zero [62]. +On the CUB-200-2011 dataset [51], on one hand, we use +VGG16 [45] as the backbone and train our student model +with the learning rate lr = 0.0005, batch size bz = 6, up- +date rate λ = 0.01, epoch e = 100, distillation temperature + +10 +Feifei Shao +Yawei Luo* +Shengjian Wu +Qiyi Li +Fei Gao +Yi Yang +Jun Xiao +T cls = 15 and T loc = 15, distillation strength α = 0.8. The +hyper-parameters of classification and localization teachers +are consistent with the student model except for µ = 1.0, +η = 0.04, β = 0.5, and δ = 0.0. At the test stage, we +first resize images to 344 × 344 and then centrally crop it +to 224 × 224 inspired by [52, 56, 65]. Then, we generate +the bounding box by segmenting the localization map us- +ing a threshold θ = 0.14. On the other hand, we use In- +ceptionV3 [48] as the backbone and train our student model +with the learning rate lr = 0.0001, batch size bz = 6, up- +date rate λ = 0.01, epoch e = 100, distillation temperature +T cls = 15 and T loc = 15, distillation strength α = 0.8. The +hyper-parameters of classification and localization teachers +are consistent with the student model except for µ = 0.6, +η = 0.0, β = 0.2, and δ = 2e − 08. In the testing phase, +we first resize images to 500 × 500 and then centrally crop +it to 299 × 299 inspired by [52, 56, 65]. Then, we generate +the bounding box by segmenting the localization map using +a threshold θ = 0.21. +On the ILSVRC 2016 dataset [38], we use VGG16 as the +backbone and train our student model with the learning rate +lr = 7.8e − 05, batch size bz = 48, update rate λ = 0.001, +and epoch e = 20, distillation temperature T cls = 2 and +T loc = 2, distillation strength α = 0.8. We train classifi- +cation and localization teachers using µ = 1.0, η = 0.04, +β = 1.0, and δ = 1e − 06. The other hyper-parameters of +teachers are consistent with the student model. At test time, +we first resize images to 288 × 288 and then centrally crop +it to 224 × 224 inspired by [52, 56, 65]. Then, we generate +the bounding box by segmenting the localization map using +a threshold θ = 0.11. +4.3 Comparison with State-of-the-Art Methods +We compare KD-CI-CAM with other state-of-the-art (SOTA) +methods on the CUB-200-2011 [51] and ILSVRC 2016 [38] +datasets shown in Table 1 and Table 2. +To validate the robustness of our solutions, we imple- +ment our approach with different backbones on the CUB- +200-2011 dataset. We observe that KD-CI-CAM significantly +outperforms the current SOTA method under multiple evalu- +ation metrics shown in Table 1. More concretely, if the back- +bone is VGG16 [45], KD-CI-CAM achieves 79.2% Top-1 +classification accuracy that is 1.9% higher than the current +SOTA FAM [32] and outperforms it by 3.7% and 2.3% in the +Top-1 localization accuracy and GT-known localization ac- +curacy, respectively. Besides, KD-CI-CAM reaches 73.0% +Top-1 localization accuracy that is 1.7% higher than the cur- +rent SOTA BAS [56] and outperforms it by 0.5% in the +GT-known localization accuracy. Compared with the GT- +known localization SOTA BridgeGap [22], KD-CI-CAM is +in a narrow margin that 1.6% lower for the GT-known lo- +calization accuracy, but it brings a significant performance +Table 1: Performance (%) on the CUB-200-2011 test set. +IncepV3: InceptionV3 +Methods +Backbone Top-1 Cls Top-1 Loc GT-known Loc +CAM [72]16 +VGG16 +76.6 +44.2 +- +ACoL [67]18 +VGG16 +71.9 +45.9 +59.3 +ADL [8]19 +VGG16 +65.3 +52.4 +75.4 +DANet [61]19 +VGG16 +75.4 +52.5 +- +NL-CCAM [62]20 +VGG16 +73.4 +52.4 +- +MEIL [31]20 +VGG16 +74.8 +57.5 +73.8 +PSOL [65]20 +VGG16 +- +66.3 +- +GCNet [26]20 +VGG16 +76.8 +63.2 +- +RCAM [3]20 +VGG16 +74.9 +61.3 +80.7 +MCIR [2]21 +VGG16 +72.6 +58.1 +- +SLT-Net [16]21 +VGG16 +76.6 +67.8 +87.6 +SPA [35]21 +VGG16 +- +60.3 +77.3 +ORNet [58]21 +VGG16 +77.0 +67.7 +- +FAM [32]21 +VGG16 +77.3 +69.3 +89.3 +PDM [33]22 +VGG16 +76.9 +67.3 +82.2 +BAS [56]22 +VGG16 +- +71.3 +91.1 +BridgeGap [22]22 +VGG16 +- +70.8 +93.2 +CREAM [59]22 +VGG16 +- +70.4 +91.0 +KD-CI-CAM +VGG16 +79.2 +73.0 +91.6 +SPG [68]18 +IncepV3 +- +46.6 +- +ADL [8]19 +IncepV3 +74.6 +53.0 +- +DANet [61]19 +IncepV3 +71.2 +49.5 +- +PSOL [65]20 +IncepV3 +- +65.5 +- +I2C [69]20 +IncepV3 +- +55.0 +72.6 +SLT-Net [16]21 +IncepV3 +76.4 +66.1 +86.5 +SPA [35]21 +IncepV3 +- +53.5 +72.1 +FAM [32]21 +IncepV3 +81.3 +70.7 +87.3 +PDM [33]22 +IncepV3 +- +64.3 +79.6 +BAS [56]22 +IncepV3 +- +73.3 +92.2 +CREAM [59]22 +IncepV3 +- +71.8 +90.4 +KD-CI-CAM +IncepV3 +79.7 +76.3 +95.3 +gain of 2.2% over BridgeGap [22] in the Top-1 localiza- +tion accuracy. When we use InceptionV3 [48] as the back- +bone, compared with the current SOTA method in terms +of Top-1 localization accuracy and GT-known localization, +i.e., BAS [56], KD-CI-CAM respectively outperforms it by +3.0% and 3.1% in the Top-1 localization and GT-known +localization accuracy. Although compared with the current +Top-1 classification SOTA FAM [32], KD-CI-CAM yields a +slightly lower Top-1 classification accuracy, but it brings a +significant performance gain of 5.6% and 8.0% over FAM in +the Top-1 localization accuracy and GT-known localization +accuracy, respectively. +For more general scenarios as on the ILSVRC 2016 [38] +which suffers less from the “entangled context” due to the +huge amount of images and various backgrounds, KD-CI- +CAM can also perform on par with the state of the arts, espe- +cially in the Top-1 classification accuracy shown in Table 2. +Specifically, we observe that KD-CI-CAM reaches 72.5% +Top-1 classification accuracy that is 0.1% higher than the +current SOTA SLT-Net [16] and has the same performance +with SLT-Net [16] in the Top-1 localization accuracy. Com- +pared with the current GT-known localization SOTA FAM, + +Further Improving Weakly-supervised Object Localization via Causal Knowledge Distillation +11 +Table 2: Performance(%) on the ILSVRC 2016 validation +set. +Methods +Backbone Top-1 Cls Top-1 Loc GT-known Loc +CAM [72]16 +VGG16 +66.6 +42.8 +59.0 +ACoL [67]18 +VGG16 +67.5 +45.8 +63.0 +ADL [8]19 +VGG16 +69.5 +44.9 +- +NL-CCAM [62]20 +VGG16 +72.3 +50.2 +65.2 +MEIL [31]20 +VGG16 +70.3 +46.8 +- +PSOL [65]20 +VGG16 +- +50.9 +64.0 +RCAM [3]20 +VGG16 +67.2 +45.4 +62.7 +MCIR [2]21 +VGG16 +71.2 +51.6 +66.3 +SLT-Net [16]21 +VGG16 +72.4 +51.2 +67.2 +SPA [35]21 +VGG16 +- +49.6 +65.1 +ORNet [58]21 +VGG16 +71.6 +52.1 +- +FAM [32]21 +VGG16 +70.9 +52.0 +71.7 +PDM [33]22 +VGG16 +68.7 +51.1 +69.3 +BAS [56]22 +VGG16 +- +53.0 +69.6 +BridgeGap [22]22 +VGG16 +- +49.9 +68.9 +CREAM [59]22 +VGG16 +- +52.4 +68.3 +KD-CI-CAM +VGG16 +72.5 +51.2 +66.3 +KD-CI-CAM yields a lower GT-known localization accu- +racy but it brings a significant performance gain of 1.6% +over FAM [32] in the Top-1 classification accuracy. +4.4 Ablation Study +We conduct two groups of ablation experiments on the CUB- +200-2011 [51] and ILSVRC 2016 [38] datasets. The first +one is the importance of each module in the architecture, +which is designed to demonstrate the effectiveness of the +causal context pool and knowledge distillation in the “en- +tangled context” problem and “C-L dilemma” as shown in +Table 3 and Figure 1. The other one is the importance of +diverse knowledge in the multi-teacher causal distillation +framework, which is reported in Table 4. +Importance of each module in the architecture. Our +proposed architecture has two components: causal context +pool and knowledge distillation. 1) Causal Context Pool. +To validate the importance of the causal context pool, we im- +plement our approach with different backbones on the CUB- +200-2011 dataset [51] and ILSVRC 2016 dataset [38]. In Ta- +ble 3, causal context pool significantly brings 11.1%, 12.9% +and 13.6% improvement in the Top-1 localization, Top-5 lo- +calization, and GT-known localization when the backbone +is InceptionV3 [48] on the CUB-200-2011 dataset. Mean- +while, if the backbone is VGG16 [45], the causal context +pool also brings comprehensive improvement in the Top- +1 localization, Top-5 localization, and GT-known localiza- +tion on both CUB-200-2011 [51] and ILSVRC 2016 [38] +datasets. 2) Knowledge Distillation. Employing knowledge +distillation can comprehensively improve classification and +localization accuracy at all backbones and datasets. For ex- +ample, knowledge distillation respectively brings extra 4.1%, +6.5%, and 3.5% improvement in the Top-1 classification, +Top-1 localization, and GT-known localization accuracy when +the backbone is VGG16 [45] on the CUB-200-2011 dataset +as shown in Table 3. We are surprised to find that knowl- +edge distillation also performs well on the ILSVRC 2016 +dataset that it brings extra 0.4%, 2.2%, and 2.7% improve- +ment in the Top-1 classification, Top-1 localization, and GT- +known localization accuracy. 3) Quantitative Analysis. To +gain more insights into the effectiveness of the causal con- +text pool and knowledge distillation in the “entangled con- +text” and “C-L dilemma” problems, we plot the visualiza- +tion results in Figure 1. For example, Figure 1 (a) depicts the +effect of causal context pool by comparing different meth- +ods. We observe that CI-CAM and KD-CI-CAM employing +causal context pool can distinguish the boundary between +object and co-occurrence background clearer than vanilla +CAM [72] and NL-CCAM [62]. Besides, Figure 1 (b) pro- +vides multiple independent experiments using different train- +ing hyperparameters of CI-CAM and KD-CI-CAM. We find +that KD-CI-CAM with knowledge distillation is capable of +simultaneously reaching the highest classification and local- +ization accuracy on both VGG16 [45] and InceptionV3 [48]. +Importance of diverse knowledge in distillation. To +reveal the importance of diverse knowledge in distillation, +we set up multiple distillation experiments by using differ- +ent teachers, which are reported in Table 4. We observe that +employing a classification teacher can bring comprehensive +improvement in classification performance at all backbones +and datasets. And using a localization teacher also com- +prehensively improves localization performance. Compared +with only using a single classification teacher or localiza- +tion teacher, employing both classification and localization +teachers together in distillation can bring larger improve- +ment in both classification and localization performance. Tak- +ing VGG16 [45] as the backbone for example, on the CUB- +200-2011 dataset, we obtain 2.3% and 0.9% improvement +in the Top-1 classification and Top-5 classification accuracy +by only using the classification teacher. When only using +the localization teacher, it brings 5.3%, 4.2%, and 3.4% im- +provement in the Top-1 localization, Top-5 localization, and +GT-known localization. If we use both classification and lo- +calization teachers together, our method respectively further +improves the performance and brings 4.1%, 6.5%, and 3.4% +improvement in the Top-1 classification, Top-1 localization, +and GT-known localization. On the ILSVRC 2016 dataset, +we obtain 0.1% and 0.1% improvement in the Top-1 clas- +sification and Top-5 classification by only using the classi- +fication teacher, respectively. When only using the localiza- +tion teacher, we respectively obtain 0.9%, 1.7%, and 2.1% +improvement in the Top-1 localization, Top-5 localization, +and GT-known localization. Employing both classification +and localization teachers together can comprehensively im- +prove the classification and localization performance more + +12 +Feifei Shao +Yawei Luo* +Shengjian Wu +Qiyi Li +Fei Gao +Yi Yang +Jun Xiao +Table 3: Ablation studies using different components on the CUB-200-2011 and ILSVRC 2016 datasets. Baseline perfor- +mance is our re-implemented results. 1) Base: Baseline method (NL-CCAM), 2) ConPool: Causal context pool. 3) KD: +Knowledge distillation. 4) IncepV3: InceptionV3 +Dataset +Backbone +Base +ConPool +KD +Top-1 Cls(%) +Top-5 Cls(%) +Top-1 Loc(%) +Top-5 Loc(%) +GT-known Loc(%) +CUB-200-2011 +test set +VGG16 +√ +74.7 +92.6 +65.6 +81.1 +87.5 +√ +√ +75.1 (+0.4) +93.0 (+0.4) +66.5 (+0.9) +82.4 (+1.3) +88.1 (+0.6) +√ +√ +√ +79.2 (+4.5) +94.1 (+1.5) +73.0 (+7.4) +86.5 (+5.4) +91.6 (+4.1) +IncepV3 +√ +76.8 +93.8 +62.8 +75.9 +80.2 +√ +√ +78.7 (+1.9) +94.6 (+0.8) +73.9 (+11.1) +88.8 (+12.9) +93.8 (+13.6) +√ +√ +√ +79.7 (+2.9) +95.3 (+1.5) +76.3 (+13.5) +91.0 (+5.1) +95.3 (+15.1) +ILSVRC 2016 +val set +VGG16 +√ +72.3 +90.4 +48.6 +58.8 +62.9 +√ +√ +72.1 (-0.2) +90.4 (+0.0) +49.0 (+0.4) +59.4 (+0.6) +63.6 (0.7) +√ +√ +√ +72.5 (+0.2) +90.6 (+0.2) +51.2 (+2.6) +62.1 (+3.3) +66.3 (+3.4) +Table 4: Ablation experiments using diverse knowledge on the CUB-200-2011 and ILSVRC 2016 datasets. 1) Stu: Student +(CI-CAM). 2) ClsTea: Classification teacher. 3) LocTea: Localization teacher. 4) IncepV3: InceptionV3. +Dataset +Backbone +Stu +ClsTea +LocTea +Top-1 Cls(%) +Top-5 Cls(%) +Top-1 Loc(%) +Top-5 Loc(%) +GT-known Loc(%) +CUB-200-2011 +test set +VGG16 +√ +75.1 +93.0 +66.5 +82.4 +88.1 +√ +√ +77.4 (+2.3) +93.9 (+0.9) +66.5 (+0.0) +80.2 (-2.2) +85.0 (-3.1) +√ +√ +78.1 (+3.0) +94.4 (+1.4) +71.8 (+5.3) +86.6 (+4.2) +91.5 (+3.4) +√ +√ +√ +79.2 (+4.1) +94.1 (+1.1) +73.0 (+6.5) +86.5 (+4.1) +91.6 (+3.5) +IncepV3 +√ +78.7 +94.6 +73.9 +88.8 +93.8 +√ +√ +79.8 (+1.1) +95.1 (+0.5) +74.3 (+0.4) +88.4 (-0.4) +92.7 (-1.1) +√ +√ +78.6 (-0.1) +94.7 (+0.1) +74.9 (+1.0) +90.1 (+1.3) +94.9 (+1.1) +√ +√ +√ +79.7 (+1.0) +95.3 (+0.7) +76.3 (+2.4) +91.0 (+2.2) +95.3 (+1.5) +ILSVRC 2016 +val set +VGG16 +√ +72.1 +90.4 +49.0 +59.4 +63.6 +√ +√ +72.2 (+0.1) +90.5 (+0.1) +49.8 (+0.8) +60.4 (+1.0) +64.5 (+0.9) +√ +√ +71.5 (-0.6) +90.1 (-0.3) +49.9 (+0.9) +61.1 (+1.7) +65.7 (+2.1) +√ +√ +√ +72.5 (+0.4) +90.6 (+0.2) +51.2 (+2.2) +62.1 (+2.7) +66.3 (+2.7) +Table 5: Analysis experiments on the CUB-200-2011 and ILSVRC 2016 datasets by using different localization teachers. 1) +Stu: Student (CI-CAM). 2) ClsTea: Classification teacher. 3) LocTea: Localization teacher. 4) IncepV3: InceptionV3. +Dataset +Backbone Stu ClsTea +Two kinds of LocTea +KD-CI-CAM +Top-1 Cls(%) Top-1 Loc(%) GT-known Loc(%) Top-1 Cls(%) Top-1 Loc(%) GT-known Loc(%) +CUB-200-2011 +test set +VGG16 +√ +√ +77.0 +68.3 +87.9 +78.3 +71.5 +90.5 +73.3 +65.8 +88.8 +79.2 +73.0 +91.6 +IncepV3 +√ +√ +73.5 +68.6 +92.9 +79.1 +75.0 +94.5 +72.1 +68.2 +94.1 +79.7 +76.3 +95.3 +ILSVRC 2016 +val set +VGG16 +√ +√ +71.4 +49.2 +64.6 +72.6 +50.4 +65.3 +60.5 +42.2 +64.7 +72.5 +51.2 +66.3 +than only using a single classification teacher or localization +teacher. +4.5 Discussion +In this section, we will discuss how to select our localiza- +tion teachers according to their Top-1 localization and GT- +known localization performance. We conduct three groups +of experiments with different backbones on the CUB-200- +2011 dataset and ILSVRC 2016 dataset. In Table 5, there +are one classification teacher and two kinds of localization +teachers in every group experiment. The first localization +teacher has the highest Top-1 localization accuracy, and the +second one has the highest GT-known localization accuracy. +Taking the CUB-200-2011 dataset for example, we train +KD-CI-CAM twice using the same classification teacher and +two different localization teachers. If the backbone is VGG16, +we first train the first-row KD-CI-CAM using a classifica- +tion teacher and the first-row localization teacher with higher +Top-1 localization accuracy. Then, we train the second-row +KD-CI-CAM using the same classification teacher and the +second-row localization teacher with higher GT-known lo- +calization accuracy. From Table 5, we find that despite the +Top-1 classification and Top-1 localization of the second- +row localization teacher being significantly lower than that +of the first-row localization teacher, the second-row KD-CI- +CAM significantly outperforms the first-row KD-CI-CAM +under all the evaluation metrics. To validate the generality +of this phenomenon, we set up two repeated experiments on + +Further Improving Weakly-supervised Object Localization via Causal Knowledge Distillation +13 +CI-CAM +heat map +NL-CCAM +boxes +NL-CCAM +heat map +CAM +boxes +CAM +heat map +CI-CAM +boxes +Image +(a) CUB-200-2011 +(b) ILSVRC 2016 +KD-CI-CAM +heat map +KD-CI-CAM +boxes +CI-CAM +heat map +NL-CCAM +boxes +NL-CCAM +heat map +CAM +boxes +CAM +heat map +CI-CAM +boxes +Image +KD-CI-CAM +heat map +KD-CI-CAM +boxes +使用了center crop， using fish +Fig. 6: Qualitative object localization results compared with the CAM and NL-CCAM methods. The predicted bounding +boxes are in green, and the ground-truth boxes are in red. The yellow arrows indicate the regions suffer from entangled +contexts. +the ILSVRC 2016 dataset and InceptionV3 [48] backbone. +Similar to the above observation, KD-CI-CAM can achieve +higher accuracy by using the localization teacher with higher +GT-known localization performance. As a consequence, we +advocate that the selection of localization teacher should be +based on its GT-known localization performance rather than +the Top-1 localization performance. +5 Conclusions +In this paper, we target the “entangled context” and “C-L +dilemma” problems in the WSOL task, which remain un- +noticed and unexplored by existing efforts. To this end, we +propose a principled framework for solving both “entan- +gled context” and “C-L dilemma” problems in one go. Con- +cretely, we first address the “entangled context” via causal +intervention by analyzing the causal relationship between +image features, context, and image labels, and cutting off the +effect from confounder context to image features. In terms +of the model network, we propose a causal context pool to +accumulate all contexts of each class, and then re-project +the fused contexts to the feature maps of convolutional lay- +ers to make the feature boundary clearer. Second, we de- +sign a multi-teacher casual distillation framework as shown + +14 +Feifei Shao +Yawei Luo* +Shengjian Wu +Qiyi Li +Fei Gao +Yi Yang +Jun Xiao +in Figure 4 (a) for solving the “C-L dilemma” problem by +balancing the absorption of classification knowledge and lo- +calization knowledge during model training. To our knowl- +edge, we have made a very early attempt to apprehend and +approach the “entangled context” and “C-L dilemma” prob- +lems for WSOL. Extensive experiments have demonstrated +that the “entangled context” and “C-L dilemma” are practi- +cal issues within the WSOL task and our proposed methods +are effective towards them as shown in Figure 1 and Fig- +ure 6. +6 Acknowledge +This work was supported by the National Key Research & +Development Project of China (2021ZD0110700), Zhejiang +Innovation Foundation (2019R52002), the National Natu- +ral Science Foundation of China (U19B2043, 61976185), +China Postdoctoral Science Foundation (2022T150567), and +the Fundamental Research Funds for the Central Universi- +ties. +References +1. H Alaskar, A Hussain, B Almaslukh, T Vaiyapuri, Z Sbai, and +Arun Kumar Dubey. +Deep learning approaches for automatic +localization in medical images. Computational Intelligence and +Neuroscience, 2022, 2022. +2. Sadbhavana Babar and Sukhendu Das. Where to look?: Mining +complementary image regions for weakly supervised object local- +ization. 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In CVPR, 2016. + diff --git a/ONAzT4oBgHgl3EQfIPvm/content/tmp_files/load_file.txt b/ONAzT4oBgHgl3EQfIPvm/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c008ad57d57093f0c3d7ee5bd7f10c1891dfa674 --- /dev/null +++ b/ONAzT4oBgHgl3EQfIPvm/content/tmp_files/load_file.txt @@ -0,0 +1,1319 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf,len=1318 +page_content='Noname manuscript No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' (will be inserted by the editor) Further Improving Weakly-supervised Object Localization via Causal Knowledge Distillation Feifei Shao Yawei Luo* Shengjian Wu Qiyi Li Fei Gao Yi Yang Jun Xiao Received: date / Accepted: date Abstract Weakly-supervised object localization aims to in- dicate the category as well as the scope of an object in an im- age given only the image-level labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Most of the existing works are based on Class Activation Mapping (CAM) and endeavor to enlarge the discriminative area inside the acti- vation map to perceive the whole object, yet ignore the co- occurrence confounder of the object and context (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=', fish and water), which makes the model inspection hard to dis- tinguish object boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Besides, the use of CAM also brings a dilemma problem that the classification and local- ization always suffer from a performance gap and can not reach their highest accuracy simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' In this paper, we propose a casual knowledge distillation method, dubbed KD-CI-CAM, to address these two under-explored issues in one go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' More specifically, we tackle the co-occurrence con- text confounder problem via causal intervention (CI), which explores the causalities among image features, contexts, and categories to eliminate the biased object-context entangle- ment in the class activation maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Based on the de-biased object feature, we additionally propose a multi-teacher causal distillation framework to balance the absorption of classifi- cation knowledge and localization knowledge during model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Extensive experiments on several benchmarks demon- strate the effectiveness of KD-CI-CAM in learning clear ob- ject boundaries from confounding contexts and addressing the dilemma problem between classification and localization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1 ∗ Yawei Luo is the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Feifei Shao, Yawei Luo, Yi Yang, and Jun Xiao are with the Zhejiang University, Hangzhou, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' E-mail: sff@zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='cn, yaweiluo329@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='com, yangyics@zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='cn, junx@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='zju.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Shengjian Wu and Qiyi Li are with the Finvolution Group, Shanghai, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' E-mail: wushengjian@xinye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='com, liqiyi@xinye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Fei Gao is with the Zhejiang University of Technology, Hangzhou, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' E-mail: feig@zjut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 1 Our code is publicly available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='com/ shaofeifei11/KD-CI-CAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Keywords Object Localization · Weakly-supervised Learning · Knowledge Distillation · Causal Intervention 1 Introduction Object localization [5, 9, 39, 50] aims to indicate the cate- gory and the scope of an object in a given image, in forms of bounding box [12,25,38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' This task has been studied ex- tensively in the computer vision community [46, 50] due to its broad applications, such as medical diagnosis [1,37] and autonomous driving [4, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Recently, the techniques based on deep convolutional neural networks (DCNNs) [17,28,30, 45,47] promote the localization performance to a new level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' However, this performance promotion is at the price of huge amounts of fine-grained human annotations [6, 27, 29, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' To alleviate such a heavy burden, weakly-supervised object localization (WSOL) has been proposed by only resorting to image-level labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' To capitalize on the image-level labels, the existing stud- ies [10, 21, 40, 41, 43, 53] follow the Class Activation Map- ping (CAM) approach [72] to generate class activation maps first and then segment the highest activation area for a coarse localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Albeit, CAM is initially designed for the clas- sification task and tends to focus only on the most discrimi- native feature to increase its classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' To deal with this issue, recent prevailing works [10, 14, 21, 31, 53, 62,67] endeavor to perceive the whole objects instead of the shrunken and sparse “discriminative regions”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' On one hand, they refine the network structure to make the detector more tailored for object localization in a weak supervision set- ting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' For example, some methods [10, 53] use a three-stage structure to continuously optimize the prediction results by training the current stage using the output of the previous stages as supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Other works [14, 21, 31, 67] leverage two parallel branches where the first branch is designed for arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='01060v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='CV] 3 Jan 2023 2 Feifei Shao Yawei Luo* Shengjian Wu Qiyi Li Fei Gao Yi Yang Jun Xiao Image CAM NL-CCAM CI-CAM (a) (b) KD-CI-CAM 使用了center crop， using fish Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 1: (a) Visualization comparison between vanilla CAM, NL-CCAM, CI-CAM, and KD-CI-CAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' The yellow arrows indicate the regions suffer from entangled contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' (b) The classification-localization dilemma faced by prior CAM-based methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=', CI-CAM), where the classification and localization suffer from a performance gap and can not achieve their highest accuracy simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' digging out the most discriminative small regions while the second one is responsible for detecting the less discrimina- tive large regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' On the other hand, they also make full use of the image information to improve the prediction re- sults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' For instance, TS2C [53] and NL-CCAM [62] utilize the contextual information of surrounding pixels and the ac- tivation maps of low probability class, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Although the works along the vein of CAM have led to some impressive early results, we have discovered multi- faceted problems that remain unexplored by those prior arts, which emerged from our experimental analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Firstly, vanilla CAM mechanism is incapable of reasoning about the co- occurrence confounder (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=', fish and water) in the images, which makes the model inspection hard to distinguish be- tween the object and context and causes biased activation maps, as shown in Figure 1 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' We dub this problem as “en- tangled context” for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Secondly, we noticed that the classification and localization always suffer from a per- formance gap and can not reach their highest accuracy si- multaneously in previous CAM-based methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Taking CI- CAM [43] as an example, we set up multiple independent experiments using different training settings, however, none of the results can achieve the best classification and localiza- tion performance at the same time, as reported in Figure 1 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' We summarise this issue as a classification-localization dilemma (“C-L dilemma” for short).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' We argue that these two problems severely hinder the WSOL performance and heretofore yet to be well studied, despite the existence of a vast body of WSOL literature [10,31,52,56,67,69,72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Taking one step further into these two issues, we found the internal reasons behind them are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' The “entan- gled context” derives from the fact that objects usually co- occur with a certain context background, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=', the most “fish” appears concurrently with “water” in the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Conse- quently, these two concepts would be inevitably entangled and a classification model would wrongly generate an am- biguous boundary between “fish” and “water” with image- level supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' In contrast to the vanilla CAM which yields a relatively shrunken bounding box on the small discrimina- tive region, we notice that the “entangled context” problem would cause a biased expanding bounding box that includes the wrongly entangled background, which impairs the lo- calization accuracy in terms of the object range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' While the “C-L dilemma” is derived from a hallmark of a classification network that tends to focus on discriminative region infor- mation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=', the head of an animal) over the integral profile of an object to better distinguish various categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Forc- ing a classification model to pay more attention to the un- representative area (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=', the fur of an animal) conceding to the integral contour perception would inevitably cause a bi- ased categorical prediction, and vise versa for a localization model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Prior approaches sidestep such a dilemma by simply trading off the classification and localization performances, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=', choosing a mutually acceptable model or only reporting localization results [22,35,56,59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' However, the intrinsic is- sue behind this dilemma remains under-explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' In this paper,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' we propose a casual knowledge distilla- tion framework to solve both “entangled context” and “C-L Perio20LwdPajons2o17@Antonio Hidalgo@Antonio HidalgoAntonio Hidalgo@Antonio Hidalgo@Antonio Hidalgo(VGG16) CI-CAM (VGG16) KD-CI-CAM (InceptionV3) CI-CAM (InceptionV3) KD-CI-CAM 79 78 16 75 66 68 70 72 74 76 Top-1 Loc Accuracy (%)TOFurther Improving Weakly-supervised Object Localization via Causal Knowledge Distillation 3 dilemma” problems in one go,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' dubbed KD-CI-CAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Specif- ically, we first explore the causalities among image features, contexts, and labels by establishing a structural causal model (SCM) [36] and pinpoint the context as a confounder as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' According to the causal analysis, we propose a causal context pool to eliminate the biased co- occurrence in the class activation maps to solve the “entan- gled context” problem as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Based on dis- entangled object features from the wrongly contextual in- formation, we further tackle the “C-L dilemma” problem faced by the CAM-based model via a multi-teacher causal distillation framework as shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' The first kind of teacher focuses on classification, which aims to provide good logit knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' The other one is a localization teacher, which is responsible for providing high-quality class activa- tion map knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' These two teachers endow our student model with the ability to balance the absorption of classifica- tion knowledge and localization knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' To prevent the degradation of knowledge transfer from the model capacity gap between the student and teachers [13,34], these teachers adopt the same network structure with the student except for their objective loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' With these tailored designs, KD-CI-CAM can not only effectively eliminate the spurious correlations between pixels and labels from the “entangled context” problem, but also alleviates the “C-L dilemma” prob- lem resulting from the interesting bias between the classifi- cation information and localization information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' It is noteworthy that some pertinent results of this work have been published in an earlier version [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' This paper goes beyond [43] by contributing to introducing a novel ca- sual knowledge distillation framework to further boost the WSOL by solving the notorious but under-explored “entan- gled context” and “C-L dilemma” problems in one go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' We also make significant improvements in experimental valida- tion and analysis compared to the early version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Generally, the contributions of this paper can be summa- rized as follows: – We are among the pioneers to concern and reveal the “entangled context” problem and “C-L dilemma” prob- lem of WSOL that remain unexplored by prevailing ef- forts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' – We propose a novel casual knowledge distillation method, dubbed KD-CI-CAM, to solve the under-explored “en- tangled context” and “C-L dilemma” problems in one go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' We first design a student model by using causal in- tervention to address the confounder context, and then balance its absorption of classification knowledge and localization knowledge by using a knowledge distilla- tion framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' – Extensive experiments show that both “entangled con- text” and “C-L dilemma” problems are effectively solved by our proposed methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 2 Related Work 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1 Weakly-supervised Object Localization Since CAM [72] is prone to bias to the most discriminative part of the object rather than the integral object, the research attention of most of the current methods is how to improve the accuracy of object localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' These methods can be broadly categorized into two groups: enlarging the proposal region and discriminative region removal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 1) Enlarging proposal region: enlarging the box size appropriately of the initial prediction box [10,53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' WCCN [10] introduces three cascaded networks trained in an end-to-end pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' The latter stage continuously enlarges and refines the output proposals of its previous stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' TS2C [53] selects the box by comparing the mean pixel confidence values of the initial prediction region and its surrounding region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' If the gap between the mean values of the two regions is large, the initial prediction region is the final prediction box;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' oth- erwise, the surrounding region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 2) Discriminative region removal: detecting the big- ger region after removing the most discriminative region [8, 21, 31, 67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' TP-WSL [21] first detects the most discrimina- tive region in the first network, Then, it erases this region of the conv5-3 feature maps in the second network (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=', zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' ACoL [67] uses the masked feature maps by erasing the most discriminative region discovered in the first classifier as the input feature maps of the second classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' ADL [8] stochastically produces an erased mask or an importance map at each iteration as a final attention map projected in the feature maps of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' MEIL [31] is an adversarial erasing method that simultaneously computes the erased branch and the unerased branch by sharing one classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' The above methods basically focus on the poor localiza- tion caused by the most discriminative part of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' However, they ignore the problem of the fuzzy boundary between the objects and the co-occur certain context back- ground.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' For example, if most “fish” appears concurrently with “water” in the images, these two concepts would be in- evitably entangled and wrongly generate ambiguous bound- aries using only image-level supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2 Causal Inference Structural Causal Model (SCM) [36] is an important analy- sis tool in causal inference scenes [49,63,66], which is a di- rected graph in which each node represents each participant of the model, and each link denotes the causalities between the two nodes [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Specifically, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' [66] utilize a SCM [11, 36] to deeply analyze the causalities among im- age features, contexts, and class labels and propose Context Adjustment (CONTA) that achieves the new state-of-the- art in weakly-supervised semantic segmentation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Yue 4 Feifei Shao Yawei Luo* Shengjian Wu Qiyi Li Fei Gao Yi Yang Jun Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' [63] use the causal intervention in few-shot learning and uncover the pre-trained knowledge is indeed a confounder that limits the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Finally, they propose a novel FSL paradigm: Interventional Few-Shot Learning (IFSL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' [49] show the SGD momentum is essentially a confounder in long-tailed classification by using a SCM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Inspired by CONTA [66], we also leverage a SCM [36] to analyze the causalities among image features, contexts, and class labels and find context is a confounder factor in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' But, our work does not follow the strategy of al- ternating training of multiple models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Instead, we propose an end-to-end model embedding the causal inference into the WSOL pipeline, which is capable of making the feature boundary clearer by using a causal context pool in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 Knowledge Distillation Knowledge distillation [15, 19, 57] aims to transfer knowl- edge from the cumbersome teacher model to a small student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' It can be categorized into two parts: knowledge and distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' First, knowledge can be grouped into response- based knowledge [19] and feature-based knowledge [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' The response-based knowledge refers to the prediction log- its, which are simple yet effective in the distillation of the classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' However, using only the teacher’s logits information in complex distillation is insufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Thus, re- searchers pay attention to utilizing feature-based knowledge to boost the effect of distillation in complex scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Sec- ond, distillation is the mode of transferring the knowledge from the teacher model to the student model, which also can be divided into offline distillation [18, 20], online distilla- tion [7, 55], and self-distillation [24, 64, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' To make the student model learn knowledge smoothly, it usually adopts a “soft target”—using a “softmax” with a temperature T on the knowledge of teachers, as the extra supervision of the student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' In this work, to harness the absorption of classification and localization knowledge in the CAM-based model, we derive a multi-teacher causal distillation framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' The clas- sification knowledge and localization knowledge are selected stochastically at each distillation process to supervise the learning of the student, which will be detailed in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 3 Methodology In this section, we first introduce the preliminaries of prob- lem settings and baseline method in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Then we will con- centrate on solving the “entangled context” issue and pro- pose our causal intervention method in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Based on the casual model, we finally tackle the “C-L dilemma” problem and construct the multi-teacher casual distillation framework (KD-CI-CAM) in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1 Preliminaries 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1 Problem Settings Before presenting our method, we first introduce the prob- lem settings of WSOL formally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Given an image I, WSOL targets classifying and locating one object in terms of the class label and the bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' However, only image-level labels Y can be accessed during the training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2 Baseline Method Class activation maps (CAMs) are widely employed for gen- erating the object boxes in the WSOL task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' [62] argue that using only one activation map of the highest prob- ability class for segmenting object boxes is problematic since it often biases to over-small regions or sometimes even high- lights background area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Based on such observation, they pro- pose the NL-CCAM [62] method to combine all activation maps from the highest to the lowest probability class to a lo- calization map using a specific combinational function and achieve good localization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Based on the vanilla fully convolutional network (FCN)- based backbone, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=', VGG16 [45], NL-CCAM [62] inserts four non-local blocks before every bottleneck layer exclud- ing the first bottleneck layer simultaneously to produce a non-local fully convolutional network (NL-FCN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Given an image I, it is fed into the NL-FCN to produce its feature maps X ∈ Rc×h×w, where c is the number of channels and h × w is the spatial size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Then, the feature maps X are for- warded to a global average pooling (GAP) layer followed by a classifier with a fully connected layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' The prediction scores S = {s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' , sn} are computed by using a soft- max layer on the top of the classifier for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' The weight matrix of the classifier is denoted as W ∈ Rn×c, where n is the number of image classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Thus, the activation maps Mi of class i among class activation maps (CAMs) M ∈ Rn×h×w proposed in [72] are given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Mi = c � k Wi,k · Xk, (1) where i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' NL-CCAM [62] produces a localization map by using a combinational function in CAMs instead of using the acti- vation map of the highest probability class among CAMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Firstly, it ranks the activation maps from the highest proba- bility class to the lowest and uses Mtk to denote the activa- tion map of the k highest probability class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' The class label with the highest probability t1 is computed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' t1 = argmaxk({Sk}), (2) Further Improving Weakly-supervised Object Localization via Causal Knowledge Distillation 5 ������������ ������������ ������������ ������������ (a) (b) ������������ ������������ ������������ ������������ Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 2: (a) Building the structural causal model (SCM) in WSOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' (b) Cutting off the confounding effect of C → X in WSOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' X: feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' C: confounder context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' V : image representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Y : image label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' where k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Then it combines M to a localiza- tion heatmap H ∈ Rh×w as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' H = n � k γ(k) · Mk, (3) where γ(·) is a combinational function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Finally, it segments the localization heatmap H using a threshold proposed in [72] to generate a bounding box for object localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Our method is based on NL-CCAM [62] but introduces substantial improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' We not only equip the baseline network with the ability of causal inference to tackle the “en- tangled context” problem but also address the “C-L dilemma” problem suffered from the traditional CAM-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2 Causal Intervention for Student Network In this section, we target the “entangled context” problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Specifically, we first reveal the reason why the confounder context hurts the object localization quality using a struc- tural causal model [36] in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Then, we use the causal intervention to solve the “entangled context” problem theo- retically via the backdoor adjustment [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Finally, we im- plement the causal module in our model in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1 Structural Causal Model Inspired by CONTA [66], we utilize a structural causal model (SCM) [36] to analyze the causalities among image features X, confounder context C, and image-level labels Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' The direct links shown in Figure 2 (a) denote the causalities be- tween the two nodes: cause → effect [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' C → X: This link indicates that the backbone gen- erates feature maps X under the effect of context C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Al- though the confounder context C is helpful for a better as- sociation between the image features X and labels Y via a model P(Y |X), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=', it is likely a “fish” when perceiving a “water” region, P(Y |X) mistakenly associates non-causal but positively correlated pixels to labels, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=', the “water” region wrongly belongs to “fish”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' This is a vital reason for the inaccurate localization in WSOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Fortunately, as we will introduce later in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3, we can avoid it by using a causal context pool in causal intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' C → V ← X: V is an image specific-representation using the contextual templates from C [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' For example, V tells us the shape and location of a “fish” (foreground) in a scene (background) [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' In this paper, V denotes the activation map of the highest probability class in the CAM 1 module as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' X → Y ← V : These links indicate that image feature X and image representation V together affect the image la- bel Y of an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' We consider that the shape and location information of object instance contained in image represen- tation V also directly affects the image label Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' As a conse- quence, though V is not an input factor in the WSOL model, it exists [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2 Theoretical Analysis To remove the confounding effect of C → X as shown in Figure 2 (b), we take inspiration from CONTA [66], follow- ing the same rule to use P(Y |do(X)) based on the backdoor adjustment [36] as the new image-level classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' The key idea is that X and C = {c1, c2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' , cn} are independent events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' As C does not affect X, it guarantees X to have a fair opportunity to incorporate every context c into Y ’s pre- diction, subject to a prior P(c) [66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Formally, we have the following probability formula: P(Y |do(X)) = n � i P(Y |do(X), ci) · P(ci|do(X)) = n � i P(Y |X, ci) · P(ci) = n � i P(Y |X, ci, V ) · P(V |X, ci) · P(ci) = n � i P(Y |X, ci, V = f(X, ci)) · P(ci) = n � i P(Y |X, V = f(X, ci)) · P(ci) = n � i P(Y |X = x, V = f(x, ci)) · P(ci), (4) where P(Y |do(X)) = �n i P(Y |do(X), ci) · P(ci|do(X)) follows the law of total probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' n is the number of im- age classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Since X and C = {c1, c2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' , cn} are indepen- dent events, P(ci|do(X)) = P(ci).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' f(X, ci) abstractly rep- resents that V is effected by X and ci, so P(V |X, ci) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 6 Feifei Shao Yawei Luo* Shengjian Wu Qiyi Li Fei Gao Yi Yang Jun Xiao Because Y is only directly affected by X and V in Figure 2 (b), P(Y |X, ci, V = f(X, ci)) = P(Y |X, V = f(X, ci)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Inspired by CONTA [66], we adopt the NWGM [60] to op- timize Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' (4) by moving the outer sum �n i P(ci) into the feature level P(Y |do(X)) ≈ P(Y |X = x, V = n � i f(x, ci) · P(ci)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' (5) Since the number of samples for each class in the dataset is roughly the same, we set P(c) to uniform 1/n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' After further optimizing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' (5), we have P(Y |do(X)) ≈ P(Y |x ⊕ 1 n · n � i f(x, ci)), (6) where ⊕ denotes projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' So far, the “entangled context” issue has been transferred into calculating �n i f(x, ci).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' We will introduce a causal context pool Q to represent �n i f(x, ci) in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 Student Network Architecture We implement a causal model for the “entangled context” problem, dubbed CI-CAM, at the core of which is a causal context pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' The main idea of the causal context pool is to accumulate all contexts of each class, and then re-project the contexts to the feature maps of convolutional layers shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' (6) to pursue the pure causality between the cause X and the effect Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Figure 3 illustrates the overview of CI-CAM that includes four parts: backbone, CAM module, causal context pool, and combinational part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Inherited from the baseline method, we de- sign our backbone by inserting multiple non-local blocks at both low- and high-level layers of a FCN-based network simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' It acts as a feature extractor that takes the RGB images as input and produces high-level position-aware feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' CAM module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' It includes a global average pooling (GAP) layer and a classifier with a fully connected layer [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Im- age feature maps X generated by the backbone are forwarded into GAP and classifier to produce prediction scores S = {s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' , sn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' The CAM network multiplies the weight W of the classifier to X to produce class activation maps M ∈ Rn×h×w shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' In our model, we use two CAM modules with shared weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' The first CAM mod- ule is designed to produce initial prediction scores S and class activation maps M, and the second CAM network is responsible for producing more accurate prediction scores Se = {se 1, se 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' , se n} and class activation maps M e ∈ Rn×h×w using the feature maps Xe ∈ Rc×h×w enhanced by the causal context pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Causal context pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' We maintain a causal context pool Q ∈ Rn×h×w during the network training phase, where Qi denotes the context of all ith class images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Q ceaselessly stores all contextual information maps (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=', V ) of each class by accumulating the activation map of the highest probabil- ity class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Then, it projects all contexts of each class as at- tention onto the feature maps of the last convolutional layer to produce enhanced feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' The idea behind using a causal context pool is not only to cut off the negative ef- fect of entangled context on image feature maps but also to spotlight the positive region of the image feature maps for boosting localization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Combinational part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' The input of the combinational part is class activation maps M e generated from the CAM 2 module, and the corresponding output is a localization heatmap H ∈ Rh×w calculated by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' (3): First, the com- binational part ranks the activation maps from the highest probability class to the lowest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Second, it combines these sorted activation maps by a combinational function as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' With all the key modules presented above, we would give a brief illustration of the data flow in our network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Given an image I, we first forward I to the backbone to produce feature maps X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' X is then fed into the following two par- allel CAM branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' The first CAM branch produces initial prediction scores S and class activation maps M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Then, the causal context pool Q would be updated by fusing the acti- vation map of the highest probability class in M as follows: Qπ = bn(Qπ + λ · bn(V )), V = Mπ, (7) where π = argmax({s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' , sn}), λ denotes the up- date rate, and bn denotes the batch normalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' The sec- ond branch is responsible for producing more accurate pre- diction scores Se and class activation maps M e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' The input of the second branch is enhanced feature maps Xe projected by the context among causal context pool Q of the highest probability class generated from the first branch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' More con- cretely, the feature enhancement can be calculated as Xe = X + X ⊙ Conv1×1(Qπ), (8) where ⊙ denotes the matrix dot product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' In the combina- tional part, we first build a localization heatmap H ∈ Rh×w by aggregating all activation maps from the highest to the lowest probability class using a specific combinational func- tion [62] in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Then, we use the simple thresholding technique proposed by [72] to generate a bounding box B from the localization map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Finally, the bounding box B and prediction scores Se as the final prediction of CI-CAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4 Student Network Training Objective During the phase of training, our proposed student network learns to minimize image classification losses for both clas- sification branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Given an image I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' we can obtain initial ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Further Improving Weakly-supervised Object Localization via Causal Knowledge Distillation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='VGG16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Non-local ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='modules ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Feature maps ������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Enhanced ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='feature maps ������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='element ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='wise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='CAM 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='GAP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Classifier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Causal context pool ������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='cat ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='airliner ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='barn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='beer ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='taxi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='necklace ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='… ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='… ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='camera ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='bridge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Causal context pool ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='CAM 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Shared ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Normalization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='CAMs ������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Combination ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='all classes ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Prediction ������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='CAMs ������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Highest ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='probability ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='class ������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='⊙ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Image representation ������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='GAP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Classifier ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Prediction ������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Label ������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Label ������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 3: Overview of the proposed CI-CAM (Student network).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' CI-CAM consists of four parts: a backbone to extract the feature maps, the share-weighted CAM modules to generate the class activation maps, a causal context pool (which is the core of the CI-CAM method) to enhance the feature maps by eliminating the negative effect of confounder, and a combinational module to generate the final bounding box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' prediction scores S = {s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' , sn} and more accurate prediction scores Se = {se 1, se 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' , se n} of the two classi- fiers as shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' We follow a naive scheme to train the two classifiers together in an end-to-end pipeline using the following loss function Lstu (Stu Loss).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Lstu = ρ · � − n � i=1 yi · log(si) � + � − n � i=1 yi · log(se i) � , (9) where y is the ground-truth label of an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' If Stu Loss Lstu is used in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 19 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 21, ρ = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' otherwise, ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 Multi-Teacher Causal Knowledge Distillation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1 Distillation Framework To make the model simultaneously retain good classifica- tion performance and localization performance, we design a multi-teacher causal distillation framework that randomly selects the knowledge of the pre-trained classification teacher and localization teacher to supervise the learning of our stu- dent as shown in Figure 4 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' First, the pre-trained classification teacher provides logit knowledge to help the student network to achieve good clas- sification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Inspired by [19], we design our log- its distillation loss function Lld between classification teacher and student as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Lld = KL( ˜Zstu, ˜Zcls tch), (10) ˜Zstu i = exp(Zstu i /T cls) �n j=1 exp(Zstu j /T cls), (11) ˜Zcls tch i = exp(Zcls tch i /T cls) �n j=1 exp(Zcls tch j /T cls), (12) where KL is the Kullback-Leibler divergence function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Zstu and Zcls tch respectively denote the output logits Z ∈ Rn of the student network and the classification teacher network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' n is the number of classes and i ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' T cls is the logits distillation temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Second, the pre-trained localization teacher brings the localization knowledge in the class activation maps to help the student network to accurately localize the whole object rather than the most discriminative part of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' We design the activation distillation loss function Lad between localization teacher and student as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Lad = MSE( ˜Astu, ˜Aloc tch), (13) ˜Astu i = exp(Astu i /T loc) �nhw j=1 exp(Astu j /T loc) , (14) ˜Aloc tch i = exp(Aloc tch i /T loc) �nhw j=1 exp(Aloc tch j /T loc) , (15) where MSE is the Mean Squared Error function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Astu and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Aloc tch respectively denote the class activation maps A ∈ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='[]8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Feifei Shao ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Yawei Luo* ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Shengjian Wu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Qiyi Li ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Fei Gao ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Yi Yang ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Jun Xiao ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Class activation maps ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Pre-trained ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='classification ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='teacher ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Pre-trained ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='localization ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='teacher ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Student ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Logits Loss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='ℒ������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Student ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Fore Loss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='ℒ ������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Diff Loss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='ℒ������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Student ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Student ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Student ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Student ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='network ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='(b) Classification teacher ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='(c) Localization teacher ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Original image logits ������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Original image logits ������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='(a) Distillation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Heatmap H ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Stu Loss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='ℒ������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='∫ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='∫ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Random ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='selection ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Threshold ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Background ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='threshold ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Area Loss ℒ������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Area Loss ℒ������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Activation Loss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='ℒ������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Original image ������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Foreground ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='image ������������fore ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Stu Loss ℒ������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Stu Loss ℒ������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Foreground image ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='logits ������������������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Background image ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='logits ������������������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Original image ������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Background ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='image ������������������������������������������������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 4: Overview of the multi-teacher causal distillation framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' (a) The classification teacher and localization teacher are applied stochastically during each distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' (b) The classification teacher adopts the student network as its model structure but adds Area Loss and Fore Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' The two orange student networks in the classification teacher are the same model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' (c) The localization teacher uses the student network as its model structure but adds Area Loss and Diff Loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' The three orange student networks in the localization teacher are the same model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Rnhw of the student network and the localization teacher network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' h and w are the height and width of the activation map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' T loc is the activation distillation temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Finally, the total loss function Ltotal of the multi-teacher causal dis- tillation framework in Figure 4 (a) is given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Ltotal =α · [(ϵ == 1) · Lld + (ϵ == 2) · Lad] + (1 − α) · Lstu, (16) where α is the distillation strength hyper-parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' ϵ ∈ {0, 1, 2} denotes the controller of the random selection in Figure 4 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2 Classification Teacher Network As shown in Figure 4 (b), to prevent the degradation of knowl- edge transfer from the model capacity gap between student and teacher [13, 34], we adopt the same network structure with the student as our classification teacher but add two extra loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Specifically, given an image I, it is first forwarded into the model to produce the original im- age logits S = {s1, s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' , sn} and localization heatmap H ∈ Rh×w, where h and w respectively denote the height and width of the original image I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Then, we obtain a binary foreground mask M fore by segmenting the heatmap H us- ing a threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Next, we generate a foreground image Ifore by projecting the M fore into the original image I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Finally, Ifore is fed into the model again to produce the foreground image logits Sfore = {sfore 1 , sfore 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' , sfore n }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' To clearly distinguish the foreground area from the back- ground area in heatmap H, we first introduce the Area Loss of [32, 58] to reduce the activation value of the foreground area and the background area in heatmap H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Then, we ap- ply a Fore Loss to activate the object areas in the image by classifying the foreground image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' As a result, Area Loss and Fore Loss terms together can suppress the background areas and highlight the foreground areas [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Area Loss Larea and Fore Loss Lfore are given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Larea = 1 h · w h � i=1 w � j=1 Hi,j, (17) Lfore = − n � i=1 yi · log(sfore i ), (18) where y is the ground-truth label of an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Finally, our classification teacher learns to minimize the Area Loss Larea, Fore Loss Lfore, and Stu Loss Lstu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Lcls tch = Lstu + µ · Lfore + η · Larea, (19) where Lcls tch is the total loss of our classification teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' µ and η are the hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 Localization Teacher Network As shown in Figure 4 (c), similar to the classification teacher network, we also use the same network structure with the student as our localization teacher but add two extra loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Given an image I, it is first forwarded into the model to produce the original image logits S = {s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' , sn} Further Improving Weakly-supervised Object Localization via Causal Knowledge Distillation 9 Vanilla student Localization teacher Input image Bohemian_Waxwing_0057_177784 Brandt_Cormorant_0091_22825 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 5: Visualization comparison between vanilla student (without knowledge distillation) and localization teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' and localization heatmap H ∈ Rh×w, where h and w re- spectively denote the height and width of the original im- age I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Then, we respectively obtain the binary foreground mask M fore and background mask M back by segmenting the heatmap H using a threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Next, we generate the foreground image Ifore and background image Iback by re- spectively projecting the M fore and M back into the origi- nal image I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Finally, the foreground image Ifore and back- ground image Iback are fed into the model again to yield the foreground image logits Sfore = {sfore 1 , sfore 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' , sfore n } and background image logits Sback = {sback 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' , sback n }, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' To make the foreground image contain the integral ob- ject as much as possible while reducing the object informa- tion in the background area, we design a difference loss (Diff Loss) Ldiff, which is given as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Ldiff = − n � i=1 yi · log(sfore i − sback i ) − n � i=1 yi · log(si + (sfore i − sback i ) 2 ), (20) where y and s are the ground-truth label of an image and the original image prediction, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Finally, our lo- calization teacher learns to minimize the Area Loss Larea, Diff Loss Ldiff, and Stu Loss Lstu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Lloc tch = Lstu + β · Ldiff + δ · Larea, (21) where Lloc tch denotes the total loss of the localization teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' β and δ are the hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Figure 5 shows the com- parison result between our vanilla student and localization teacher, which verifies our localization teacher can produce a high-quality class activation map and alleviate the discrim- inative region problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 4 Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1 Datasets and Evaluation Metrics Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' The proposed KD-CI-CAM was evaluated on two public datasets: CUB-200-2011 [51] and ILSVRC 2016 [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 1) CUB-200-2011 is an extended version of Caltech-UCSD Birds 200 (CUB-200) [54] containing 200 bird species which focuses on the study of subordinate categorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Based on the CUB-200, CUB-200-2011 adds more images for each category and labels new part localization annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' CUB- 200-2011 contains 5, 994 images in the training set and 5, 794 images in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Each image of CUB-200-2011 is an- notated by the bounding boxes, part locations, and attribute labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 2) ILSVRC 2016 is the dataset originally prepared for the ImageNet Large Scale Visual Recognition Challenge (ILSVRC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' It contains 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2 million images of 1, 000 cate- gories in the training set, 50, 000 in the validation set, and 100, 000 images in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' For both datasets, we only utilize the image-level classification labels for training, as constrained by the problem setting in WSOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Evaluation Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' We leverage the classification ac- curacy (Cls) and localization accuracy (Loc) as the eval- uation metrics for WSOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' The former includes Top-1 and Top-5 classification accuracy, while the latter includes Top- 1, Top-5, and GT-known localization accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Top-1 classi- fication accuracy denotes the accuracy of the highest predic- tion score (likewise for localization accuracy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Top-5 classi- fication accuracy denotes that if one of the five predictions with the highest score is correct, it counts as correct (like- wise for localization accuracy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' GT-known localization ac- curacy is the accuracy that only considers localization re- gardless of classification result compared to Top-1 localiza- tion accuracy [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2 Implementation Details We adopt the VGG16 [45] and InceptionV3 [48] pre-trained on the ImageNet [38] as our backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' We augment the training images with RandAugment [48] and use Adam [23] to optimize our network with β1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='9 and β2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='99 on the CUB-200-2011 [51] and ILSVRC 2016 [38] datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' If we use VGG16 as our backbone and train it on the CUB- 200-2011, we insert four non-local blocks to the backbone before every bottleneck layer excluding the first one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Oth- erwise, we insert three non-local blocks into the backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' The newly added non-local blocks are randomly initialized except for the batch normalization layers, which are initial- ized as zero [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' On the CUB-200-2011 dataset [51], on one hand, we use VGG16 [45] as the backbone and train our student model with the learning rate lr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0005, batch size bz = 6, up- date rate λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='01, epoch e = 100, distillation temperature 10 Feifei Shao Yawei Luo* Shengjian Wu Qiyi Li Fei Gao Yi Yang Jun Xiao T cls = 15 and T loc = 15, distillation strength α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' The hyper-parameters of classification and localization teachers are consistent with the student model except for µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='04, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5, and δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' At the test stage, we first resize images to 344 × 344 and then centrally crop it to 224 × 224 inspired by [52, 56, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Then, we generate the bounding box by segmenting the localization map us- ing a threshold θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' On the other hand, we use In- ceptionV3 [48] as the backbone and train our student model with the learning rate lr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0001, batch size bz = 6, up- date rate λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='01, epoch e = 100, distillation temperature T cls = 15 and T loc = 15, distillation strength α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' The hyper-parameters of classification and localization teachers are consistent with the student model except for µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2, and δ = 2e − 08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' In the testing phase, we first resize images to 500 × 500 and then centrally crop it to 299 × 299 inspired by [52, 56, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Then, we generate the bounding box by segmenting the localization map using a threshold θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' On the ILSVRC 2016 dataset [38], we use VGG16 as the backbone and train our student model with the learning rate lr = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='8e − 05, batch size bz = 48, update rate λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='001, and epoch e = 20, distillation temperature T cls = 2 and T loc = 2, distillation strength α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' We train classifi- cation and localization teachers using µ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0, η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='04, β = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0, and δ = 1e − 06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' The other hyper-parameters of teachers are consistent with the student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' At test time, we first resize images to 288 × 288 and then centrally crop it to 224 × 224 inspired by [52, 56, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Then, we generate the bounding box by segmenting the localization map using a threshold θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 Comparison with State-of-the-Art Methods We compare KD-CI-CAM with other state-of-the-art (SOTA) methods on the CUB-200-2011 [51] and ILSVRC 2016 [38] datasets shown in Table 1 and Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' To validate the robustness of our solutions, we imple- ment our approach with different backbones on the CUB- 200-2011 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' We observe that KD-CI-CAM significantly outperforms the current SOTA method under multiple evalu- ation metrics shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' More concretely, if the back- bone is VGG16 [45], KD-CI-CAM achieves 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2% Top-1 classification accuracy that is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='9% higher than the current SOTA FAM [32] and outperforms it by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='7% and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3% in the Top-1 localization accuracy and GT-known localization ac- curacy, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Besides, KD-CI-CAM reaches 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0% Top-1 localization accuracy that is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='7% higher than the cur- rent SOTA BAS [56] and outperforms it by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5% in the GT-known localization accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Compared with the GT- known localization SOTA BridgeGap [22], KD-CI-CAM is in a narrow margin that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6% lower for the GT-known lo- calization accuracy, but it brings a significant performance Table 1: Performance (%) on the CUB-200-2011 test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' IncepV3: InceptionV3 Methods Backbone Top-1 Cls Top-1 Loc GT-known Loc CAM [72]16 VGG16 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2 ACoL [67]18 VGG16 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='9 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='9 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 ADL [8]19 VGG16 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4 DANet [61]19 VGG16 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5 NL-CCAM [62]20 VGG16 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4 MEIL [31]20 VGG16 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='8 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='8 PSOL [65]20 VGG16 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 GCNet [26]20 VGG16 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='8 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2 RCAM [3]20 VGG16 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='9 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='7 MCIR [2]21 VGG16 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1 SLT-Net [16]21 VGG16 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='8 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6 SPA [35]21 VGG16 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 ORNet [58]21 VGG16 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='7 FAM [32]21 VGG16 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 PDM [33]22 VGG16 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='9 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2 BAS [56]22 VGG16 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1 BridgeGap [22]22 VGG16 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='8 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2 CREAM [59]22 VGG16 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0 KD-CI-CAM VGG16 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6 SPG [68]18 IncepV3 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6 ADL [8]19 IncepV3 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0 DANet [61]19 IncepV3 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5 PSOL [65]20 IncepV3 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5 I2C [69]20 IncepV3 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6 SLT-Net [16]21 IncepV3 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5 SPA [35]21 IncepV3 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1 FAM [32]21 IncepV3 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='7 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 PDM [33]22 IncepV3 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6 BAS [56]22 IncepV3 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2 CREAM [59]22 IncepV3 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='8 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4 KD-CI-CAM IncepV3 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='7 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 gain of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2% over BridgeGap [22] in the Top-1 localiza- tion accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' When we use InceptionV3 [48] as the back- bone, compared with the current SOTA method in terms of Top-1 localization accuracy and GT-known localization, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=', BAS [56], KD-CI-CAM respectively outperforms it by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0% and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1% in the Top-1 localization and GT-known localization accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Although compared with the current Top-1 classification SOTA FAM [32], KD-CI-CAM yields a slightly lower Top-1 classification accuracy, but it brings a significant performance gain of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6% and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0% over FAM in the Top-1 localization accuracy and GT-known localization accuracy, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' For more general scenarios as on the ILSVRC 2016 [38] which suffers less from the “entangled context” due to the huge amount of images and various backgrounds, KD-CI- CAM can also perform on par with the state of the arts, espe- cially in the Top-1 classification accuracy shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Specifically, we observe that KD-CI-CAM reaches 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5% Top-1 classification accuracy that is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1% higher than the current SOTA SLT-Net [16] and has the same performance with SLT-Net [16] in the Top-1 localization accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Com- pared with the current GT-known localization SOTA FAM, Further Improving Weakly-supervised Object Localization via Causal Knowledge Distillation 11 Table 2: Performance(%) on the ILSVRC 2016 validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Methods Backbone Top-1 Cls Top-1 Loc GT-known Loc CAM [72]16 VGG16 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='8 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0 ACoL [67]18 VGG16 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='8 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0 ADL [8]19 VGG16 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='9 NL-CCAM [62]20 VGG16 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2 MEIL [31]20 VGG16 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='8 PSOL [65]20 VGG16 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='9 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0 RCAM [3]20 VGG16 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='7 MCIR [2]21 VGG16 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 SLT-Net [16]21 VGG16 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2 SPA [35]21 VGG16 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1 ORNet [58]21 VGG16 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1 FAM [32]21 VGG16 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='9 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='7 PDM [33]22 VGG16 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='7 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 BAS [56]22 VGG16 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6 BridgeGap [22]22 VGG16 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='9 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='9 CREAM [59]22 VGG16 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 KD-CI-CAM VGG16 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 KD-CI-CAM yields a lower GT-known localization accu- racy but it brings a significant performance gain of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6% over FAM [32] in the Top-1 classification accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4 Ablation Study We conduct two groups of ablation experiments on the CUB- 200-2011 [51] and ILSVRC 2016 [38] datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' The first one is the importance of each module in the architecture, which is designed to demonstrate the effectiveness of the causal context pool and knowledge distillation in the “en- tangled context” problem and “C-L dilemma” as shown in Table 3 and Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' The other one is the importance of diverse knowledge in the multi-teacher causal distillation framework, which is reported in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Importance of each module in the architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Our proposed architecture has two components: causal context pool and knowledge distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 1) Causal Context Pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' To validate the importance of the causal context pool, we im- plement our approach with different backbones on the CUB- 200-2011 dataset [51] and ILSVRC 2016 dataset [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' In Ta- ble 3, causal context pool significantly brings 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1%, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='9% and 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6% improvement in the Top-1 localization, Top-5 lo- calization, and GT-known localization when the backbone is InceptionV3 [48] on the CUB-200-2011 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Mean- while, if the backbone is VGG16 [45], the causal context pool also brings comprehensive improvement in the Top- 1 localization, Top-5 localization, and GT-known localiza- tion on both CUB-200-2011 [51] and ILSVRC 2016 [38] datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 2) Knowledge Distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Employing knowledge distillation can comprehensively improve classification and localization accuracy at all backbones and datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' For ex- ample, knowledge distillation respectively brings extra 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1%, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5%, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5% improvement in the Top-1 classification, Top-1 localization, and GT-known localization accuracy when the backbone is VGG16 [45] on the CUB-200-2011 dataset as shown in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' We are surprised to find that knowl- edge distillation also performs well on the ILSVRC 2016 dataset that it brings extra 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4%, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2%, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='7% improve- ment in the Top-1 classification, Top-1 localization, and GT- known localization accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 3) Quantitative Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' To gain more insights into the effectiveness of the causal con- text pool and knowledge distillation in the “entangled con- text” and “C-L dilemma” problems, we plot the visualiza- tion results in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' For example, Figure 1 (a) depicts the effect of causal context pool by comparing different meth- ods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' We observe that CI-CAM and KD-CI-CAM employing causal context pool can distinguish the boundary between object and co-occurrence background clearer than vanilla CAM [72] and NL-CCAM [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Besides, Figure 1 (b) pro- vides multiple independent experiments using different train- ing hyperparameters of CI-CAM and KD-CI-CAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' We find that KD-CI-CAM with knowledge distillation is capable of simultaneously reaching the highest classification and local- ization accuracy on both VGG16 [45] and InceptionV3 [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Importance of diverse knowledge in distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' To reveal the importance of diverse knowledge in distillation, we set up multiple distillation experiments by using differ- ent teachers, which are reported in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' We observe that employing a classification teacher can bring comprehensive improvement in classification performance at all backbones and datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' And using a localization teacher also com- prehensively improves localization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Compared with only using a single classification teacher or localiza- tion teacher, employing both classification and localization teachers together in distillation can bring larger improve- ment in both classification and localization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Tak- ing VGG16 [45] as the backbone for example, on the CUB- 200-2011 dataset, we obtain 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='9% improvement in the Top-1 classification and Top-5 classification accuracy by only using the classification teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' When only using the localization teacher, it brings 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3%, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2%, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4% im- provement in the Top-1 localization, Top-5 localization, and GT-known localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' If we use both classification and lo- calization teachers together, our method respectively further improves the performance and brings 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1%, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5%, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4% improvement in the Top-1 classification, Top-1 localization, and GT-known localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' On the ILSVRC 2016 dataset, we obtain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1% improvement in the Top-1 clas- sification and Top-5 classification by only using the classi- fication teacher, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' When only using the localiza- tion teacher, we respectively obtain 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='9%, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='7%, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1% improvement in the Top-1 localization, Top-5 localization, and GT-known localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Employing both classification and localization teachers together can comprehensively im- prove the classification and localization performance more 12 Feifei Shao Yawei Luo* Shengjian Wu Qiyi Li Fei Gao Yi Yang Jun Xiao Table 3: Ablation studies using different components on the CUB-200-2011 and ILSVRC 2016 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Baseline perfor- mance is our re-implemented results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 1) Base: Baseline method (NL-CCAM), 2) ConPool: Causal context pool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 3) KD: Knowledge distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 4) IncepV3: InceptionV3 Dataset Backbone Base ConPool KD Top-1 Cls(%) Top-5 Cls(%) Top-1 Loc(%) Top-5 Loc(%) GT-known Loc(%) CUB-200-2011 test set VGG16 √ 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='7 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5 √ √ 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4) 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='9) 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4 (+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6) √ √ √ 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2 (+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1 (+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5) 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0 (+7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4) 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5 (+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4) 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6 (+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1) IncepV3 √ 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='8 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='8 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='8 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='9 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2 √ √ 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='7 (+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='9) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='8) 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='9 (+11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='8 (+12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='9) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='8 (+13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6) √ √ √ 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='7 (+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='9) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 (+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5) 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 (+13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5) 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0 (+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 (+15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1) ILSVRC 2016 val set VGG16 √ 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='8 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='9 √ √ 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0) 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4) 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6) 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='7) √ √ √ 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2) 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2 (+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6) 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1 (+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3) 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 (+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4) Table 4: Ablation experiments using diverse knowledge on the CUB-200-2011 and ILSVRC 2016 datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 1) Stu: Student (CI-CAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 2) ClsTea: Classification teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 3) LocTea: Localization teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 4) IncepV3: InceptionV3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Dataset Backbone Stu ClsTea LocTea Top-1 Cls(%) Top-5 Cls(%) Top-1 Loc(%) Top-5 Loc(%) GT-known Loc(%) CUB-200-2011 test set VGG16 √ 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1 √ √ 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4 (+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3) 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='9 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='9) 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0) 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2 (-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2) 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0 (-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1) √ √ 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1 (+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4 (+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4) 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='8 (+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3) 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6 (+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2) 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5 (+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4) √ √ √ 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2 (+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1 (+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1) 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0 (+6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5) 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5 (+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1) 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6 (+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5) IncepV3 √ 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='7 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='9 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='8 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='8 √ √ 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='8 (+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4) 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4) 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='7 (-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1) √ √ 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='7 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1) 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='9 (+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1 (+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3) 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='9 (+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1) √ √ √ 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='7 (+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='7) 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 (+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4) 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0 (+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2) 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 (+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5) ILSVRC 2016 val set VGG16 √ 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6 √ √ 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1) 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='8 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='8) 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4 (+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0) 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='9) √ √ 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1 (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3) 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='9 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='9) 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1 (+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='7) 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='7 (+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1) √ √ √ 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6 (+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2) 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2 (+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2) 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1 (+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='7) 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 (+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='7) Table 5: Analysis experiments on the CUB-200-2011 and ILSVRC 2016 datasets by using different localization teachers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 1) Stu: Student (CI-CAM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 2) ClsTea: Classification teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 3) LocTea: Localization teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 4) IncepV3: InceptionV3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Dataset Backbone Stu ClsTea Two kinds of LocTea KD-CI-CAM Top-1 Cls(%) Top-1 Loc(%) GT-known Loc(%) Top-1 Cls(%) Top-1 Loc(%) GT-known Loc(%) CUB-200-2011 test set VGG16 √ √ 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='9 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='8 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='8 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6 IncepV3 √ √ 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='9 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='0 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='1 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='7 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 ILSVRC 2016 val set VGG16 √ √ 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='6 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='4 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='7 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='2 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='3 than only using a single classification teacher or localization teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content='5 Discussion In this section, we will discuss how to select our localiza- tion teachers according to their Top-1 localization and GT- known localization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' We conduct three groups of experiments with different backbones on the CUB-200- 2011 dataset and ILSVRC 2016 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' In Table 5, there are one classification teacher and two kinds of localization teachers in every group experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' The first localization teacher has the highest Top-1 localization accuracy, and the second one has the highest GT-known localization accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Taking the CUB-200-2011 dataset for example, we train KD-CI-CAM twice using the same classification teacher and two different localization teachers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' If the backbone is VGG16, we first train the first-row KD-CI-CAM using a classifica- tion teacher and the first-row localization teacher with higher Top-1 localization accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Then, we train the second-row KD-CI-CAM using the same classification teacher and the second-row localization teacher with higher GT-known lo- calization accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' From Table 5, we find that despite the Top-1 classification and Top-1 localization of the second- row localization teacher being significantly lower than that of the first-row localization teacher, the second-row KD-CI- CAM significantly outperforms the first-row KD-CI-CAM under all the evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' To validate the generality of this phenomenon,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' we set up two repeated experiments on Further Improving Weakly-supervised Object Localization via Causal Knowledge Distillation 13 CI-CAM heat map NL-CCAM boxes NL-CCAM heat map CAM boxes CAM heat map CI-CAM boxes Image (a) CUB-200-2011 (b) ILSVRC 2016 KD-CI-CAM heat map KD-CI-CAM boxes CI-CAM heat map NL-CCAM boxes NL-CCAM heat map CAM boxes CAM heat map CI-CAM boxes Image KD-CI-CAM heat map KD-CI-CAM boxes 使用了center crop，' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' using fish Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 6: Qualitative object localization results compared with the CAM and NL-CCAM methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' The predicted bounding boxes are in green, and the ground-truth boxes are in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' The yellow arrows indicate the regions suffer from entangled contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' the ILSVRC 2016 dataset and InceptionV3 [48] backbone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Similar to the above observation, KD-CI-CAM can achieve higher accuracy by using the localization teacher with higher GT-known localization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' As a consequence, we advocate that the selection of localization teacher should be based on its GT-known localization performance rather than the Top-1 localization performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 5 Conclusions In this paper, we target the “entangled context” and “C-L dilemma” problems in the WSOL task, which remain un- noticed and unexplored by existing efforts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' To this end, we propose a principled framework for solving both “entan- gled context” and “C-L dilemma” problems in one go.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Con- cretely, we first address the “entangled context” via causal intervention by analyzing the causal relationship between image features, context, and image labels, and cutting off the effect from confounder context to image features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' In terms of the model network, we propose a causal context pool to accumulate all contexts of each class, and then re-project the fused contexts to the feature maps of convolutional lay- ers to make the feature boundary clearer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Second, we de- sign a multi-teacher casual distillation framework as shown 14 Feifei Shao Yawei Luo* Shengjian Wu Qiyi Li Fei Gao Yi Yang Jun Xiao in Figure 4 (a) for solving the “C-L dilemma” problem by balancing the absorption of classification knowledge and lo- calization knowledge during model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' To our knowl- edge, we have made a very early attempt to apprehend and approach the “entangled context” and “C-L dilemma” prob- lems for WSOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' Extensive experiments have demonstrated that the “entangled context” and “C-L dilemma” are practi- cal issues within the WSOL task and our proposed methods are effective towards them as shown in Figure 1 and Fig- ure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' 6 Acknowledge This work was supported by the National Key Research & Development Project of China (2021ZD0110700), Zhejiang Innovation Foundation (2019R52002), the National Natu- ral Science Foundation of China (U19B2043, 61976185), China Postdoctoral Science Foundation (2022T150567), and the Fundamental Research Funds for the Central Universi- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAzT4oBgHgl3EQfIPvm/content/2301.01060v1.pdf'} +page_content=' H Alaskar, A Hussain, B Almaslukh, T Vaiyapuri, Z Sbai, and Arun Kumar Dubey.' metadata={'source': 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Detectability of Red Edges in +Exoplanet Reflection Spectra +Jonathan Gomez Barrientos +,1, 2 Ryan J. MacDonald +,1, 3, 4 Nikole K. Lewis +,1 and Lisa Kaltenegger +1 +1Department of Astronomy and Carl Sagan Institute, Cornell University, 122 Sciences Drive, Ithaca, NY 14853, USA +2Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA 91125, USA +3Department of Astronomy, University of Michigan, Ann Arbor, MI 48109, USA +4NHFP Sagan Fellow +ABSTRACT +Reflection spectroscopy holds great promise for characterizing the atmospheres and surfaces of po- +tentially habitable terrestrial exoplanets. The surface of the modern Earth exhibits a sharp albedo +change near 750 nm caused by vegetation — the red edge — which would leave a strong spectral sig- +nature if present on an exoplanet. However, the retrieval of wavelength-dependent surface properties +from reflection spectra has seen relatively little study. Here, we propose a new surface albedo param- +eterization capable of retrieving the wavelength location of a priori unknown ‘edge-like’ features. We +demonstrate that a wavelength-dependent surface albedo model achieves higher accuracy in retrieving +atmospheric composition. Wavelength-dependent surfaces are also generally preferred over a uniform +albedo model when retrieving simulated reflection spectra for a modern Earth analog, even for mod- +erate signal-to-noise ratios (S/N = 10) and Earth-like clouds. Further, the location of the modern +Earth’s red edge can be robustly and precisely constrained (within 70 nm for S/N = 10). Our results +suggest that future space-based direct imaging missions have the potential to infer surface compositions +for rocky exoplanets, including spectral edges similar to those caused by life on the modern Earth. +Keywords: planets and satellites: atmospheres — planets and satellites: terrestrial planets +1. INTRODUCTION +One of the most profound discoveries of the exoplanet +era is the realization that rocky planets in the classical +habitable zone are prolific around nearby stars (e.g., Pe- +tigura et al. 2013; Bryson et al. 2021). Remote observa- +tions of such rocky planets hold the potential to deepen +our understanding of the physical, chemical, climate, +and biological processes shaping these worlds (see e.g., +Kaltenegger 2017; Wordsworth & Kreidberg 2021, for +recent reviews). Spectroscopic observations ultimately +provide our clearest window into the atmospheric and +surface properties of rocky exoplanets. +The near-term focus for rocky exoplanet atmospheric +characterization lies with transiting planets orbiting +small stars, such as M dwarfs (e.g. Scalo et al. 2007; +Barstow & Irwin 2016; Morley et al. 2017; Fauchez et al. +Corresponding author: Jonathan Gomez Barrientos +jdg276@cornell.edu +2021) or white dwarfs (e.g., Kaltenegger & MacDonald +et al. 2020, Lin et al. 2022). Transmission spectroscopy +— the variation of planetary radius with wavelength — +is currently the most common technique used to diag- +nose the composition, temperature profile, and aerosol +properties of exoplanet atmospheres (e.g., Brown 2001; +Sing et al. 2016; Kreidberg 2018; MacDonald & Lewis +2022). Existing transmission spectra from the Hubble +Space Telescope and ground-based facilities have ruled +out clear H2-dominated atmospheres for several rocky +exoplanets (e.g., de Wit et al. 2016, 2018; Wakeford +et al. 2019; Diamond-Lowe et al. 2020; Libby-Roberts +et al. 2021; Garcia et al. 2022). The recently launched +JWST will observe terrestrial exoplanets transiting M +dwarfs, which should lead to detections of molecules +such as CO2, CH4, and H2O (e.g., Krissansen-Totton +et al. 2018; Lustig-Yaeger et al. 2019; Lin et al. 2021). +However, terrestrial exoplanets in the habitable zone of +G-type stars like our Sun are not optimal targets for +transmission spectroscopy (due to their low planet-star +arXiv:2301.01775v1 [astro-ph.EP] 4 Jan 2023 + +ID2 +Gomez Barrientos et al. +radius ratio, infrequent transits, and atmospheric refrac- +tion; see e.g. B´etr´emieux & Kaltenegger 2014). +Directly-detected reflected stellar light is a promis- +ing avenue to characterize rocky exoplanets orbiting G- +type stars. +Detecting reflected light from an Earth- +like exoplanet around a star like our Sun requires sen- +sitivity to planet-star contrast ratios at visible wave- +lengths of Fp/F∗ ∼ 10−10, where Fp and F∗ are, re- +spectively, the fluxes from the planet and star as ob- +served at Earth. The Roman Space Telescope (sched- +uled for launch in 2026) will offer important progress +towards this goal, demonstrating space-based coronog- +raphy by imaging cool giant planets with Fp/F∗ ∼ 10−8 +to 10−9 (Kasdin et al. 2020). +Recently, the Astro +2020 Decadal Survey recommended the development of +a large IR/Optical/UV space-based telescope — notion- +ally for launch in the 2040s — to characterize Earth-like +exoplanets around Sun-like stars (Decadal Survey on As- +tronomy & Astrophysics 2020). Such a mission would +be designed to detect atmospheric and surface biosigna- +tures for a population of Earth-sized exoplanets. +Biosignatures are remotely detectable features indica- +tive of a biological process (for recent reviews, see +Kaltenegger 2017; Schwieterman et al. 2018; Fujii et al. +2018). For exoplanets, biosignatures include the simul- +taneous detection of a pair of oxidizing and reducing +gases (e.g. +O2 / O3 and CH4) or temporal variabil- +ity (e.g, Keeling et al. 1976, Ford et al. 2001; Meadows +2006, 2008; Cowan et al. 2012; Fujii et al. 2017). An- +other important class of biosignatures — and the focus +of this study — are surface biosignatures, here defined +as remotely detectable spectroscopic features caused by +the presence of biology on a planetary surface (e.g., Des +Marais et al. 2002; Schwieterman et al. 2015; Hegde et al. +2015; Fonseca Coelho et al. 2022). +The red edge is a proposed surface biosignature +caused by a biology-induced change in the wavelength- +dependent surface albedo. On the modern Earth, where +plants cover ∼ 60% of the land area, the red edge can be +detected in both resolved reflection spectra (Sagan et al. +1993) and disc-averaged Moonshine (e.g. Des Marais +et al. 2002; Woolf et al. 2002; Turnbull et al. 2006). +This photosynthetic red edge has changed in strength +throughout Earth’s history (e.g., from changing sur- +face coverage, types of biota), but should have been +detectable for the last 1 billion years (O’Malley-James +& Kaltenegger 2018; O’Malley-James & Kaltenegger +2019a). +The origin of Earth’s red edge is chlorophyll +reflecting more light redwards of ∼ 750 nm. For exoplan- +ets, many studies have suggested that a similar change +in the surface albedo at a characteristic wavelength +could constitute a biosignature analogous to Earth’s red +edge (e.g., Seager et al. 2005; Takizawa et al. 2017; +O’Malley-James & Kaltenegger 2018; O’Malley-James +& Kaltenegger 2019a). Reflected light spectra of an exo- +Earth would consequently display a sharp contrast ra- +tio increase if its surface featured a signature like the +red edge. This study proposes a method to retrieve the +wavelength location of albedo changes, like the red edge, +from reflection spectra of rocky exoplanets. +Spectroscopic retrieval is a method commonly used to +infer planetary properties (e.g., atmospheric composi- +tion, temperature, and clouds) from observed spectra. +Bayesian retrieval techniques compare model spectra +for a wide range of possible planet properties (typically +≳ 105) to a set of observations, thereby obtaining prob- +ability distributions for the planet properties (e.g., Ben- +neke & Seager 2012; MacDonald & Madhusudhan 2017; +Molli`ere et al. 2019). Reflected light exoplanet retrieval +techniques were initially developed for directly imaged +cool giant exoplanets, largely in preparation for the Ro- +man Space Telescope (Lupu et al. 2016; Nayak et al. +2017; Lacy et al. 2019; Damiano & Hu 2020; Damiano +et al. 2020; Carri´on-Gonz´alez et al. 2020, 2021; Mukher- +jee et al. 2021). Subsequent studies have extended re- +flected light retrievals to directly imaged sub-Neptunes +and terrestrial planets (Feng et al. 2018; Damiano & +Hu 2021, 2022; Robinson & Salvador 2022; Wang et al. +2022), which would be observable with a future large +IR/Optical/UV space-based telescope. +Terrestrial planet reflection spectral retrievals must +additionally consider surface reflection. Most previous +studies have assumed a uniform-in-wavelength surface +albedo (Feng et al. 2018; Damiano & Hu 2022; Robinson +& Salvador 2022). Consequently, the spectral imprint +of a wavelength-dependent surface, including the red +edge, has seen little investigation. Recently, Wang et al. +(2022) found that retrievals using a three-albedo model +can outperform a single-albedo model — demonstrat- +ing that reflection spectra can constrain wavelength- +dependent surface albedos (see also Brandt & Spiegel +2014). Wang et al. (2022)’s albedo retrieval technique +considered three fixed wavelength bands at visible wave- +lengths (blue, green, and red). Here, we introduce a gen- +eralized albedo retrieval technique designed to identify +sharp albedo changes at a priori unknown wavelengths +analogous to Earth’s red edge. +In this study, +we demonstrate that wavelength- +dependent +surface +albedos +can +be +retrieved +from +moderate-quality reflection spectra of Earth-like ex- +oplanets. +In particular, the wavelength location of +Earth’s vegetative red edge can be precisely constrained +by a future large IR/Optical/UV space-based telescope. +In what follows, we first introduce our modeling and re- + +Retrieving Red Edges from Exoplanet Reflection Spectra +3 +trieval methodology in Section 2. We demonstrate that +our parametric albedo prescription provides an excel- +lent fit to reflection spectra for an Earth-like surface in +Section 3. We explore the sensitivity of surface albedo +retrievals to data quality and clouds in Section 4, be- +fore investigating constraints on other atmospheric and +planetary properties in Section 5. Finally, in Section 6, +we summarize our results and discuss their implications. +2. REFLECTION SPECTRA MODELING AND +RETRIEVAL FRAMEWORK +This paper investigates the surface, bulk planetary, +and atmospheric properties that can be retrieved from +observed reflection spectra of an exo-Earth. We begin +in Section 2.1 by presenting a self-consistent 1D model +for an Earth-like exoplanet orbiting a Sun-like star and +describe the computation of its reflection spectrum. We +then outline the generation of synthetic reflection spec- +tral observations and our Bayesian retrieval method for +Earth-like exoplanet reflection spectra in Section 2.2. +2.1. Reflection Spectra for an Exo-Earth +2.1.1. Atmospheric Model +We generate an atmospheric model resembling the +modern Earth using Exo-Prime2 (see e.g., Kaltenegger +et al. 2010; Madden & Kaltenegger 2020; Kasting & Ack- +erman 1986; Pavlov et al. 2000; Pavlov & Kasting 2002; +Segura et al. 2005, 2007) – a 1D radiative-convective +terrestrial atmosphere code. Exo-Prime2 couples 1D cli- +mate and photochemistry models to compute the verti- +cal temperature structure and atmospheric mixing ra- +tio profiles for a planet, assuming an incident stellar +spectrum and planetary outgassing rates. Exo-Prime2 +also includes feedback from wavelength-dependent sur- +face albedos and clouds. For Earth-like clouds, we use +the MODIS 20 µm cloud albedo model (King et al. 1997; +Rossow & Schiffer 1999), which provides a reasonable +average for many clouds of different droplet size. The +application of Exo-Prime2 to model Earth-like plan- +ets around different host stars and through geological +time has been extensively described in the literature +(e.g., Kaltenegger et al. 2010; Rugheimer et al. 2013; +Rugheimer & Kaltenegger 2018; Madden & Kaltenegger +2020; Lin et al. 2022; Kaltenegger & Lin 2021). +The +resulting pressure-temperature (P-T) and mixing ratio +profiles computed by Exo-Prime2 are shown in Figure 1. +2.1.2. Surface Model +We model a representative Earth-like surface us- +ing wavelength-dependent albedos from the USGS and +ASTER spectral libraries (Baldridge et al. 2009; Kokaly +et al. 2017; Clark et al. 2007). We create an average +present-day Earth surface albedo from 8 raw albedos of +snow, water, coast, sand, trees, grass, basalt, and granite +(after Kaltenegger et al. 2007). We assume an Earth- +like surface consisting of 70% ocean, 28% land, and 2% +coast. The land surface consists of 30% grass, 30% trees, +9% granite, 9% basalt, 15% snow, and 7% sand. We use +the surface-fraction weighted albedo (see Figure 2, bot- +tom panel) in our 1D radiative transfer models. +2.1.3. Reflection Spectra Computation +A distant observer directly imaging an exoplanet mea- +sures the wavelength-dependent planet-star flux ratio. +At wavelengths where reflected light dominates over +thermal emission, the flux ratio can be expressed as +Fp(α, λ) +Fs(λ) += Ag(λ) Φ(α, λ) +�Rp +d +�2 +(1) +where Ag is the planet’s geometric albedo spectrum, Φ is +the phase function, α is the orbital phase, Rp is the plan- +etary radius, and d is the planet-star orbital distance. +The geometric albedo is traditionally defined as the ra- +tio of the observed flux from the planet at full phase to +that from a perfectly reflecting Lambert disk. The phase +function encodes the dilution of the planetary brightness +for phase angles without full illumination (Φ = 1 when +α = 0). While the geometric albedo encodes informa- +tion about an atmosphere’s composition, temperature, +cloud properties, and surface reflection, the phase func- +tion is controlled by both the stellar illumination and +atmospheric scattering. +We generate model reflection spectra for an Earth-like +planet around a Sun-like star using the open-source ra- +diative transfer code PICASO Batalha et al. (2019). To +compute Fp/Fs from the geometric albedo we provide +PICASO with Rp and d (fixed to 1 AU) and assume ob- +servations at full phase (Φ = 1) unless otherwise noted. +We note that observations will more typically occur at +non-zero phase, which would dampen the resultant re- +flection spectra. However, in this proof of concept study, +we choose to focus on full phase to reduce the complexity +of the radiative transfer calculations required within the +retrieval framework. We set the planetary reference ra- +dius such that r(P = 1 bar) = R⊕, the surface at 1 bar, +and the surface gravity to 9.81ms−2. +For the stellar +spectrum, we used PICASO to interpolate the Castelli +& Kurucz (2003) grid for a Sun-analog star with Teff = +5780 K, log g = 4.437, and [Fe/H] = 0.0122. +For the radiative transfer calculation, we provide +PICASO with the P-T profile, mixing ratio profiles, +and the wavelength-dependent surface albedo from our +Earth-like Exo-Prime2 model. +Our reflection spectra +computations span the near-UV to near-IR, ranging + +4 +Gomez Barrientos et al. +200 +220 +240 +260 +280 +T (K) +-6 +-5 +-4 +-3 +-2 +-1 +0 +logPressure(bar) +7.1 +6.8 +6.5 +6.2 +5.9 +5.6 +logCH4 +7 +6 +5 +4 +3 +2 +logH2O +8 +7 +6 +5 +logO3 +9.5 +8.5 +7.5 +6.5 +logN2O +-6 +-5 +-4 +-3 +-2 +-1 +0 +logPressure(bar) +3.6 +3.5 +3.4 +3.3 +logCO2 +0.17 +0.19 +0.21 +O2 +0.76 +0.77 +0.78 +0.79 +N2 +0 +20 +40 +60 +80 +100 +Altitude (km) +0 +20 +40 +60 +80 +100 +Altitude (km) +Figure 1. Pressure-temperature profile and volume mixing ratio profiles for a model of the modern Earth. The main spectrally +relevant gases from 0.3–2.5 µm are shown, alongside the bulk N2 fraction. +from 0.3–2.5 µm. +We consider molecular line opacity +for H2O, O2, O3, CH4, CO2, and N2O (see Batalha et al. +2019, for details on the opacity database), alongside +Rayleigh scattering from N2 and O2. For computational +efficiency, we downsampled PICASO’s molecular cross +sections by 10× (from R=10,000). We tested different +resampling factors and find that 10× downsampling pro- +vides a reliable balance between speed and accuracy. +Our PICASO model accounting for Earth-like clouds +assumes +optical +properties +consistent +with +water. +Specifically, we use an asymmetry factor of 0.85 and +a single scattering albedo of 1.0 (after Feng et al. +2018). +We place the cloud base (in log10 bars) at +log pb = −0.23, set its vertical extent (also in log10 +bars) to log dp = −0.53, and the cloud optical depth +to log τ = −1.0. We selected these values for the cloud +vertical extent by calibrating the continuum flux from +0.4–1 µm of our 1D models to reproduce the reflection +spectrum of Earth from Robinson et al. (2011). We note +that assuming a constant cloud albedo increases the re- +flected flux beyond 1 µm compared to Robinson et al. +(2011), but does not significantly influence our analysis. +2.1.4. Impact of the Red Edge on Reflection Spectra +Figure 2 shows example reflection spectra for a cloud- +free exo-Earth, both with and without a wavelength- +dependent surface albedo. As expected, the red edge has +a pronounced effect on the reflection spectrum. The red +edge causes a marked increase in Fp/F∗ for wavelengths +immediately following 0.75 µm. As we will see in Sec- +tion 3, this sudden change is a distinct feature enabling +the spectroscopic detectability of the red edge. We also +note that the red edge occurs near strong O3 and O2 +features, which underscores the importance of account- +ing for a wavelength-dependent surface albedo when +searching for atmospheric biosignature gases. With our +‘ground-truth’ model described, we proceed to outline +our retrieval framework. +2.2. Retrieval Framework +We have developed a Bayesian retrieval wrapper +around the PICASO radiative transfer code. We em- +ploy this retrieval framework in subsequent sections to +demonstrate that information on the red edge can be +reliably retrieved from reflection spectra of exo-Earths. +Here, we describe the simulated data used in our re- +trievals and our retrieval configuration. +2.2.1. Simulated Data & Noise Model +Our aim is to investigate the retrievability of the red +edge as a function of data quality, rather than for a spe- +cific future mission architecture. Consequently, we gen- +erated several synthetic datasets, for both the cloud-free +and cloudy models described in Section 2.1.3, spanning +signal-to-noise ratios of S/N = 5, 10, 15, and 20 (at +a reference wavelength of 0.55 µm) and spectral resolu- +tions of R = 70 and 140. We account for wavelength- +dependent noise for the simulated observations using a + +Retrieving Red Edges from Exoplanet Reflection Spectra +5 +0.5 +1.0 +1.5 +2.0 +2.5 +0 +1 +2 +3 +4 +5 +Fp/Fs × 10−10 +Red edge +H2O +O3 +O3 +O2 +H2O +H2O +H2O +CH4 +CO2 +Constant Albedo +Earth-like Albedo +0.5 +1.0 +1.5 +2.0 +2.5 +Wavelength( m) +0.00 +0.05 +0.10 +0.15 +0.20 +Albedo +Figure 2. Impact of the red edge on reflection spectra. Top: model reflection spectra for a cloud-free Earth-like planet orbiting +a Sun-like star. A realistic wavelength-dependent surface albedo (green) produces a sharp increase in the flux ratio around +0.75 µm compared to a constant surface albedo (purple). Prominent molecular absorption features are labeled. Bottom: the +corresponding surface albedos. The location of the vegetative red edge is highlighted in both panels (red shading). For clarity, +both models are shown at a spectral resolution of R = 70. +constant spectral resolution noise model scaling relation +as done in (Robinson et al. 2016; Feng et al. 2018) +S/N(λ) ∝ q(λ) T (λ) Ag(λ) Φ(α, λ) B(λ) λ +(2) +where q is the detector quantum efficiency, T is the +throughput, and B is a blackbody representing the par- +ent star (see Feng et al. 2018). We adopt functions for +q and T from the Python package coronagraph1, which +is an open source noise simulator for coronagraph-based +1 https://github.com/jlustigy/coronagraph.git +observations of directly imaged exoplanets (e.g., Robin- +son et al. 2016; Lustig-Yaeger et al. 2019). For the black- +body, we use Teff = 5780 K. +When generating each simulated dataset, we do not +randomize the placement of each data point by sam- +pling from a Gaussian distribution. +Rather, the data +are centered on the (true) planet-to-star flux ratio — +corresponding to the model after binning down to the +data resolution — and assigned error bars according to +our noise model at the desired S/N. We note that run- +ning retrievals on a dataset with Gaussian noise can bias +the retrieval results, especially for low spectral resolu- + +6 +Gomez Barrientos et al. +tion and S/N (see Feng et al. 2018). However, running +retrievals with Gaussian scatter still allows for spectral +features to be recovered (see e.g., Lin et al. 2021, Ap- +pendix A). To avoid biasing our retrieval results to a spe- +cific random noise draw, we run ‘scatter-free’ retrievals, +which produce posterior distributions equivalent to the +ensemble average over many individual noise instances. +We include an example retrieval with Gaussian scatter +in Appendix A. +2.2.2. Retrieval Configuration +Bayesian retrieval codes repeatedly call a parameter- +ized radiative transfer forward model to identify the +range of bulk planetary, atmospheric, and surface prop- +erties consistent with a given dataset. +Our retrieval +framework employs PICASO (Batalha et al. 2019) as +the radiative transfer forward model and the MultiNest +(Feroz & Hobson 2008; Feroz et al. 2009, 2019) wrap- +per PyMultiNest Buchner et al. (2014) for the sampling +algorithm used to explore the parameter space. +We parameterize the atmospheric and surface prop- +erties using a simplified prescription that captures the +salient features shaping terrestrial exoplanet reflection +spectra. +We parameterize the P-T profile with an +isotherm. We assume H2O, O2, O3, CH4, CO2, and N2O +are the main spectrally-active gases with sufficient abun- +dances to shape the spectrum for Earth-like planets, +with each gas ascribed a single free parameter for the +uniform-in-altitude volume mixing ratio. +We also as- +sume the primary atmospheric gas is N2, with its mix- +ing ratio determined by the condition that mixing ra- +tios must sum to one. We prescribe three further free +parameters for the planetary radius and gravity (evalu- +ated at 1 mbar) and the surface pressure. These choices +are similar to those made by other reflected-light re- +trieval studies (e.g., Feng et al. 2018; Damiano & Hu +2022; Robinson & Salvador 2022; Wang et al. 2022). +Since our simulated observations incorporate an +Earth-like wavelength-dependent surface albedo, we +propose a new parametric treatment for wavelength- +dependent surface albedos +As(λ) = +� +� +� +� +� +� +� +As,1 +λ < λ1 +As,2 +λ1 ≤ λ ≤ λ2 +As,3 +λ > λ2 +(3) +where As,1, As,2, and As,3 define the surface albedo in +three distinct wavelength regions, λ1 marks the transi- +tion from As,1 → As,2, and λ2 marks the transition from +As,2 → As,3. This surface albedo prescription thus has +five free parameters. To avoid discontinuities at λ1 and +λ2, we compute this function on a wavelength grid at R +Table 1. Free parameters included in our PICASO retrievals. +Parameter +Description +Reference Value +Prior Range +log O2 +Oxygen mixing ratio +-0.678 +[-10,0] +log O3 +Ozone mixing ratio +-6.25 +[-10,-1] +log H2O +Water vapor mixing ratio +-2.72 +[-10,-1] +log CO2 +Carbon dioxide mixing ratio +-3.44 +[-10,-1] +log CH4 +Methane mixing ratio +-5.77 +[-10,-1] +log N2O +Nitrous oxide mixing ratio +-6.55 +[-10,-1] +log P0 +Surface pressure +0.0 +[-2,2] +Rp +Planet radius at 1 mbar +1.007 +[0.5,2.0] +g +Gravity at 1 mbar +9.66 +[1.0,25] +T +Temperature +289 +[100,800] +λ1 +Albedo transition point +0.72 +[0.3,2.5] +λ2 +Albedo transition point +1.40 +[0.3,2.5] +As,1 +Surface Albedo +0.09 +[0,1] +As,2 +Surface Albedo +0.15 +[0,1] +As,3 +Surface Albedo +0.06 +[0,1] +log pb +Cloud-base pressure +-0.23 +[-2,2] +log dp +Cloud width +-0.53 +[-2,2] +log τ +Cloud optical depth +-1.0 +[-2,2] +Note—The reference values for each parameter correspond to either ‘ground +truth’ values from the input model (e.g., planet radius and cloud properties; +see Section 2.1) or representative average values (e.g., mixing ratios and albedo +parameters; see Section 2.2.2). All priors are uniform distributions. += 1,000 and convolve it with a Gaussian with a standard +deviation of 28 wavelength grid spaces (corresponding +to 28 nm at 1 µm). Our albedo parameterization thus +resembles a smoothed double-step function (similar to +the function used by Taylor et al. 2021 to parameter- +ize the single-scattering albedo of clouds in giant planet +nightside emission spectra). +We shall demonstrate in +subsequent sections that the proposed parameterization +is sufficiently flexible to capture both the strong wave- +length dependence of the red edge and a possible sec- +ondary reflectance edge in the infrared (see Figure 2). +For retrievals including clouds, we add three further +parameters: the cloud base pressure, its vertical pressure +extent, and optical depth. Following Feng et al. (2018), +we assume water-like clouds with a fixed asymmetry pa- +rameter (0.85) and single scattering albedo (1.0). +In +total, the most complex retrievals we consider thus have +a total of 18 free parameters (summarized in Table 1). +We validated our retrieval framework against simulated +data from the Robinson et al. (2011) model (see Ap- +pendix B). +Our retrieval analysis covers multiple model and data +scenarios. First, in Section 3, we evaluate the retriev- +ability of albedo changes for our cloud-free model, since +this model has the strongest spectral red edge. +We +initially ran four retrievals on the simulated data at +(S/N)ref = 5, 10, 15, 20 and R = 70, where we parame- + +Retrieving Red Edges from Exoplanet Reflection Spectra +7 +terize the planet’s wavelength-dependent surface albedo +with Equation 3. +Then, we ran a similar set of re- +trievals with a constant-in-wavelength surface albedo. +Doing so enables us to perform Bayesian model compar- +isons between the wavelength-dependent surface and the +constant-in-wavelength surface models (e.g., Benneke & +Seager 2013; Trotta 2017). We also ran a retrieval with +R = 140 at (S/N)ref = 10 to investigate the impact of +retrieving data at a higher spectral resolution. Lastly, +we ran a retrieval at R = 70 and (S/N)ref = 10 for a +more realistic dataset including a cloud deck to inves- +tigate how clouds impact the retrieval results. All our +MultiNest retrievals use 2,000 live points, which typi- +cally involve the computation of 106 models. +We summarize the prior range for each retrieval free +parameter in Table 1. We generally allow generous prior +ranges, encompassing a wide range of physically plausi- +ble values, with all priors being uniform distributions. +As in Feng et al. (2018), we allow for oxygen-rich at- +mospheres by extending its prior range to 100% (but +rejecting any parameter combinations where the sum +of the non-N2 mixing ratios exceed unity). Our prior +range for the planet radius and gravity terminates at +2.0 R⊕ and 25.0 ms−2, respectively, since this study fo- +cuses on Earth-like planets. We note that PICASO re- +quires that the pressure corresponding to Rp must be +less than the highest pressure in the atmospheric pres- +sure grid (i.e. the surface pressure). We circumvent this +issue by defining the planet radius and gravity parame- +ters at 1 mbar, such that the surface pressure prior range +(10−2–102 bar) is always deeper than 1 mbar. +We also include ‘reference values’ for each parameter +in Table 1 — corresponding closest to the original input +Exo-Prime2 model (see Section 2.1) — for comparison +to the retrieval results. Since the input gas mixing ra- +tios depend on height, while the retrievals assume uni- +form mixing ratios, we set the reference values as the +average of the true mixing ratio profile from the sur- +face to 25 km altitude. Our reference temperature is the +planet’s surface temperature. For the albedo parame- +ters, our reference values are determined by averaging +the surface albedo over 0.3–0.72 µm, 0.72–1.4 µm, and +1.4–2.5 µm. The reference planet radius and gravity cor- +respond to the true values from the Exo-Prime2 model +(scaled to 1 mbar). Similarly, the surface pressure and +cloud parameter reference values correspond exactly to +the original model inputs (see Section 2.1.3). +We now turn to present the results of our retrievals +including wavelength-dependent surface albedos. +3. THE NECESSITY FOR WAVELENGTH- +DEPENDENT SURFACE ALBEDOS IN +REFLECTED-LIGHT RETRIEVALS +Here, we demonstrate that reflection spectra of ter- +restrial exoplanets contain recoverable information on +wavelength-dependent surfaces. We show that not only +can data commensurate with future direct imaging mis- +sions constrain wavelength-dependent surface albedos, +but that assuming a constant surface albedo may result +in biased atmospheric inferences. +3.1. Can a Uniform Albedo Fit the Earth’s Red Edge? +We first assess whether a retrieval model assuming a +constant-in-wavelength surface albedo can adequately fit +the reflection spectrum of an exoplanet with a realistic +Earth-like surface. We have already seen in Section 2.1.4 +and Figure 2 that the Earth’s red edge induces a sharp +change in Fp/F∗ around 750 nm, so here we quantify +whether such a spectral signature is detectable and its +impact on atmospheric retrievals. +In Figure 3, we demonstrate that a uniform albedo +model often struggles to capture the spectral morphol- +ogy of an Earth-like exoplanet. +Our ‘ground truth’ +model is the cloud-free scenario described in Section 2.1, +which produces the strongest red edge, while the simu- +lated data (here, (S/N)ref = 10) and retrieval configu- +ration are detailed in Section 2.2. We see that the re- +trieved spectrum for the uniform albedo model begins +to deviate from the simulated data for optical wave- +lengths longer than 0.65 µm. In the optical and near-IR, +where the S/N is highest, the uniform albedo model is +often discrepant with the data to 2σ. The root cause +of this model-data mismatch is that the uniform albedo +model has a roughly constant continuum Fp/F∗ from +0.6–0.9 µm (outside O2 absorption features), which can- +not reproduce the sharp spectral continuum change as- +sociated with the vegetation red edge near 0.72 µm. +In contrast, our wavelength-dependent surface albedo +parameterization well matches the reflection spectrum of +our Earth-like exoplanet. Equation 3 allows our retrieval +code to reproduce both the large increase in Fp/F∗ at +0.72 µm caused by the vegetation red edge and the gen- +eral morphology of the spectrum in the visible, and near- +IR. Statistically, our retrieval including a wavelength- +dependent albedo is favored over the uniform model with +a Bayes factor of ln B = 30.4 (equivalent to 8.1 σ using +the relations in Benneke & Seager 2012), which would be +considered a conclusive detection on the Jeffrey’s scale +of Bayesian model comparison (e.g. Trotta 2017). +We illustrate why the uniform albedo model strug- +gles to fit our data by comparing the retrieved sur- +face albedos in the bottom panel of Figure 3. +While + +8 +Gomez Barrientos et al. +0.5 +1.0 +1.5 +2.0 +2.5 +0 +1 +2 +3 +4 +5 +6 +Fp/Fs × 10−10 +Non-uniform Model (R = 70) +Uniform Model (R = 70) +1σ confidence +2σ confidence +1σ confidence +2σ confidence +Data (S/N = 10) +0.5 +1.0 +1.5 +2.0 +2.5 +Wavelength( m) +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +Albedo +Retrieved Albedo +True Albedo +Retrieved Albedo +1σ confidence +2σ confidence +1σ confidence +2σ confidence +Figure 3. Top: reflection spectra retrieval results assuming a wavelength-dependent surface albedo (purple contours) and +a uniform surface albedo (orange contours). The simulated data (R = 70 and (S/N)ref = 10) corresponds to an Earth-like +exoplanet orbiting a Sun-like star with an Earth-like wavelength-dependent surface. +Bottom: the retrieved surface albedo +profiles corresponding to the two models in the top panel. The true Earth-like surface albedo used to generate the simulated +data is overlaid for comparison (black curve). +the uniform albedo model correctly captures the sur- +face albedo in the near-IR wavelengths beyond 1.4 µm, +it significantly underestimates the true surface albedo +from 0.75–1.35 µm. However, our proposed parameter- +ization demonstrates that one can retrieve wavelength- +dependent surface properties — in particular the loca- +tion of the vegetation red edge — at even a moderate +signal-to-noise ratio ((S/N)ref = 10). +The retrieved +surface albedo profile also correctly infers a decrease +in the albedo for near-IR wavelengths beyond 1.3 µm. +While our wavelength-dependent surface retrieval some- +what overestimates the magnitude of the albedo (likely + +Retrieving Red Edges from Exoplanet Reflection Spectra +9 +Figure 4. +Posterior probability distribution for the re- +trieved surface albedo parameters in Equation 3 (for sim- +ulated data at R = 70 and (S/N)ref = 10). +The green +lines mark the ground truth references values (see Table 1). +The orange contours show the 1σ, 2σ, and 3σ confidence +regions for each retrieved parameter. The blue lines in the +histograms show the median (solid line) and marginalized 1σ +confidence region (dashed lines) for each parameter. +due to other complexities not captured in the model, +such as variable atmospheric abundances with height), it +correctly captures the general shape of the wavelength- +dependent albedo profile and lies within 2σ of the true +Earth-like surface albedo. +We further show the posteriors of our retrieved albedo +parameters in Figure 4. All five parameters are well- +constrained by the data, demonstrating that reflec- +tion spectra are highly sensitive to the wavelength- +dependence of the surface albedo. +In particular, the +parameter encoding the wavelength location of the red +edge, λ1, is retrieved to a remarkable precision of 8 nm. +Similarly, the retrieval identifies a secondary albedo +change near 1.4 µm, encoded by λ2, with a precision of +≈ 200 nm (though the long posterior tail to higher val- +ues indicates this is harder to constrain than λ1). The +three albedo parameters are slightly overestimated, as +noted above, but are consistent within 2 σ of the refer- +ence values (see Table 1). Overall, Figure 4 shows that +Equation 3 offers a parametrization capable of captur- +ing the key wavelength-dependent features of a realistic +Earth-like surface. +3 +2 +1 +0 +Probability denstiy +logO2 +7.0 +6.5 +6.0 +5.5 +5.0 +Probability denstiy +logO3 +4 +3 +2 +1 +logH2O +Non-uniform Albedo +Uniform Albedo +Figure 5. Impact on the retrieved molecular abundances +of O2, O3, and H2O from assuming a uniform surface albedo. +Accounting for a wavelength-dependent surface (green his- +tograms) results in good agreement with the ground truth +reference values (black lines, see Table 1), while a retrieval +assuming a uniform albedo (orange) can bias these abun- +dances by an order of magnitude. +3.2. Biases from Assuming a Uniform Albedo +The assumption of a uniform surface albedo can bias +inferred properties of an exoplanet. +Since a retrieval +code employs every available means to minimize model- +data residuals, it can attempt to compensate for the non- +inclusion of a wavelength-dependent surface by modi- +fying the retrieved abundances of chemical species in +the atmosphere (since their cross sections are also wave- +length dependent). In Figure 5, we demonstrate that +one consequence from assuming a uniform surface albedo +is biased abundance inferences for several key molecules +in our model. Specifically, we find that the volume mix- +ing ratios of O3 and H2O are over-estimated by an order +of magnitude and the bulk atmospheric gas would be +identified as O2 rather than N2. +This finding under- +scores an important point: accurate abundance infer- +ences for atmospheric gases can depend on the inclusion +of a wavelength-dependent surface albedo in reflected +light retrieval frameworks. Since Figure 5 corresponds +to the moderate case of R = 70 and (S/N)ref = 10, +wavelength-dependent surface spectral properties will +be an important consideration for future direct imaging +missions for exoplanets, especially for missions focused +on Earth-like exoplanets. + +入1 = 0.723+0.008 +0.007 +0.012 +0.107 += 0.114+0.054 +-0.03510 +1.5 +0 +2 +9 +.70 +12 +16 +24 +7 +O: +O: +: +: +Q: +Q: +0. +入1 +As,1 +入2 +As,310 +Gomez Barrientos et al. +4. FACTORS INFLUENCING RETRIEVALS OF +WAVELENGTH-DEPENDENT +SURFACE ALBEDOS +Having +established +the +need +for +a +wavelength- +dependent surface albedo model in reflected-light re- +trievals, we next explore how these results depend on +data quality (S/N in Section 4.1 and R in Section 4.2) +and the inclusion of clouds (Section 4.3). +4.1. Sensitivity to S/N +Figure 6 shows our retrieved spectra and surface +albedo profiles for (S/N)ref = 5, 10, 15, and 20. We +see that even at (S/N)ref = 5, the retrieval correctly +identifies a sharp rise in the surface albedo near 0.7 µm +— consistent with the wavelength of the vegetation red +edge on modern Earth. The uncertainty in the retrieved +wavelength of this feature is remarkably small (λ1 deter- +mined to ≈ 15 nm). This suggests that sudden changes +in surface albedo are an effect of first order importance +even for low signal-to-noise observations. With a dou- +bling to (S/N)ref = 10, we see further improvements in +the retrieved surface albedo profile: (i) the uncertainty +in the location of the sharp rise in surface albedo is +halved (λ1 determined to ≈ 8 nm); (ii) a hint emerges of +a secondary albedo change near 1.4 µm (λ2 determined +to ≈ 200 nm); and (iii) the true surface albedo profile is +correctly captured throughout most of the wavelength +range to within 2σ. +For (S/N)ref = 15, the retrieval +becomes more confident about the existence of a sec- +ondary albedo edge (λ2 determined to ≈ 70 nm). Fi- +nally, at (S/N)ref = 20 the retrieved model attains even +better overall agreement with the true albedo model. +We also find that the tendency to overestimate the re- +trieved albedo (see Section 3.1) becomes less prevalent +for higher S/N. +Table 2 quantifies the preference for our wavelength- +dependent albedo model (Equation 3) over a uniform +albedo model. For (S/N)ref = 5, our Bayesian model +comparison finds moderate evidence for a non-uniform +surface albedo (2.7σ). A slight increase to (S/N)ref = 10 +suffices to conclusively detect at least one discontinuity +(8.1σ). Further increases in S/N can help to detect a +non-uniform surface albedo also for cloudy atmospheres, +where the effect is smaller because clouds block part of +the light from the underlying surface from view (see Sec- +tion 4.3). Since our retrievals thus far have only consid- +ered data at R = 70, we next explore variable spectral +resolution for a fixed signal-to-noise ratio. +4.2. The Role of Spectral Resolution +Figure 7 shows how the retrieved surface albedo pro- +file changes with spectral resolution. +Specifically, we +Table +2. +Predicted detection significances for an Earth-like +wavelength-dependent surface albedo as a function of S/N. +Model +(S/N)ref = 5 +(S/N)ref = 10 +(S/N)ref = 15 +(S/N)ref = 20 +Clear +2.7σ +8.1σ +12.7σ +17.2σ +Cloudy +— +2.9σ +5.6σ +— +Note—(S/N)ref is defined at 0.55 µm (see Section 2.2.1). All data is at R = 70 +from 0.3–2.5 µm. Cloudy spectra retrievals at (S/N)ref = 5 and (S/N)ref = +20 were not computed, so ‘—’ is used for their detection significances. +illustrate the expected improvement from doubling the +spectral resolution from R = 70 to R = 140 (hence dou- +bling the number of datapoints from 0.3–2.5 µm). We +see that the retrieved albedo from the higher resolu- +tion data is in better agreement with the true albedo, +especially at longer wavelengths where the errors bars +are largest. Further, the uncertainty on the retrieved +albedo parameters decrease (As,1 and As,2 improve by +≈ 25%; As,3 improves by ≈ 50%). We further note that +the at R = 140 the retrieval of the second, smaller edge +around 1.4 µm (λ2) improves (by ≈ 68%), resulting in +a retrieved albedo shape more consistent with the true +Earth-like surface. +4.3. The Impact of Clouds +Figure 8 shows how the inclusion of clouds affects +the retrieved spectrum and surface albedo profile. Our +cloud properties for this demonstration were chosen to +resemble the continuum flux from the model in Robin- +son et al. (2011) (see Section 2.1.1). We see that while +the data is well fit, the retrieved albedo profile is gen- +erally overestimated when clouds are included (see also +Wang et al. 2022 for a discussion on cloud-surface de- +generacies). +Nevertheless, the wavelength of the red +edge is still reliably retrieved and well constrained even +in the presence of clouds (λ1 determined to 17 nm). +The retrieved albedo also shows a slight decrease at +longer wavelengths, but the secondary albedo change +near 1.4 µm is not well constrained. Overall, the pres- +ence of a cloud deck can lead one to infer an artificially +brighter surface outside the 1σ uncertainty region of the +retrieved albedo profile. +Clouds increase the minimum signal-to-noise ratio re- +quired to detect a wavelength-dependent surface fea- +ture. For example, Table 2 demonstrates that the de- +tection significance for a non-uniform surface albedo at +(S/N)ref = 10 drops from 8.1σ (cloud-free) to 2.9σ +(including clouds). Such lower significances arise from +cloud-surface degeneracies broadening albedo uncertain- +ties (see Figure 8). However, we still find a detection of a + +Retrieving Red Edges from Exoplanet Reflection Spectra +11 +0.5 +1.0 +1.5 +2.0 +2.5 +0 +1 +2 +3 +4 +5 +6 +Fp/Fs × 10−10 +Median Model (R = 70) +1σ confidence +2σ confidence +Binned Model +Data (S/N = 5) +0.5 +1.0 +1.5 +2.0 +2.5 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +Albedo +Retrieved Albedo +True Albedo +0.5 +1.0 +1.5 +2.0 +2.5 +0 +1 +2 +3 +4 +5 +6 +Fp/Fs × 10−10 +Median Model (R = 70) +1σ confidence +2σ confidence +Binned Model +Data (S/N = 10) +0.5 +1.0 +1.5 +2.0 +2.5 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +Albedo +Retrieved Albedo +True Albedo +0.5 +1.0 +1.5 +2.0 +2.5 +0 +1 +2 +3 +4 +5 +6 +Fp/Fs × 10−10 +Median Model (R = 70) +1σ confidence +2σ confidence +Binned Model +Data (S/N = 15) +0.5 +1.0 +1.5 +2.0 +2.5 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +Albedo +Retrieved Albedo +True Albedo +0.5 +1.0 +1.5 +2.0 +2.5 +Wavelength( m) +0 +1 +2 +3 +4 +5 +6 +Fp/Fs × 10−10 +Median Model (R = 70) +1σ confidence +2σ confidence +Binned Model +Data (S/N = 20) +0.5 +1.0 +1.5 +2.0 +2.5 +Wavelength( m) +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +Albedo +Retrieved Albedo +True Albedo +Figure 6. Retrieved reflection spectra and surface albedo profiles as a function of signal-to-noise ratio. Left panels: comparison +of the median retrieved spectrum (blue line), and its corresponding 1σ and 2σ confidence regions (purple contours), with +simulated data for an Earth-like exoplanet (cyan errors) at (S/N)ref = 5, 10, 15, and 20. The median model binned to the +resolution of the data (gold diamonds) well fit the data. Right panels: corresponding retrieved surface albedo profiles (blue line +and purple contours) inferred from each dataset compared to the true Earth-like surface model (black line). +wavelength-dependent surface albedo for (S/N)ref = 15 +(5.6σ). These results show that, while clouds can com- +plicate the inference of wavelength-dependent surface +features, it is still possible to identify non-uniform sur- +face albedos for Earth-like cloud coverage. + +12 +Gomez Barrientos et al. +0.5 +1.0 +1.5 +2.0 +2.5 +Wavelength( m) +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +Albedo +Retrieved Albedo (R = 70) +True Albedo +Retrieved Albedo (R = 140) +1σ confidence +2σ confidence +1σ confidence +2σ confidence +Figure 7. +Impact of spectral resolution on the retrieved +surface albedos of modern Earth seen as an exoplanet. The +retrieved albedos for two distinct datasets are overlaid: (i) +R = 70 and (S/N)ref = 10 (purple contours); and (ii) R = +140 and (S/N)ref = 10 (orange contours). The true model +Earth surface used to generate both simulated datasets is +included for comparison (black line). +0.5 +1.0 +1.5 +2.0 +2.5 +0 +1 +2 +3 +4 +5 +6 +7 +Fp/Fs × 10−10 +Retrieved Model (R = 70) +1σ confidence +2σ confidence +Binned Model +Data (S/N = 10) +0.5 +1.0 +1.5 +2.0 +2.5 +Wavelength( m) +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +Albedo +Retrieved Albedo +True Albedo +1σ confidence +2σ confidence +Figure 8. Top: retrieved reflection spectrum of a cloudy +Earth-like exoplanet with a wavelength-dependent surface +albedo (for simulated data at R = 70 and (S/N)ref = 10). +The median retrieved spectrum binned to the resolution of +the data (gold diamonds) generally agrees with the simulated +observations (cyan errors) within the retrieval confidence in- +tervals (purple contours). Bottom: corresponding retrieved +surface albedo profile (purple contours) compared to the true +surface albedo (black line). +5. FULL RETRIEVAL RESULTS FOR PLANETARY +AND ATMOSPHERIC PROPERTIES +For completeness, here we show our full retrieval re- +sults for other planetary and atmospheric properties. +Table 3 summarizes the retrieved values of all 15 free +parameters for our cloud-free scenario, along with their +1σ uncertainties, as a function of S/N. We also show the +posterior distributions for each parameter in Figure 9. +5.1. Planetary Properties +Figure 9 (top left) shows that the reliable inference of +bulk planetary properties requires moderate signal-to- +noise ratios for cloud-free models. At (S/N)ref = 5, the +1 mbar radius is underestimated and the surface pressure +slightly overestimated. The temperature is correctly re- +trieved, albeit with broad uncertainty (≈ 100 K). We +find that (S/N)ref = 10 is the minimum to reliably re- +trieve these parameters. In particular, for the tempera- +ture a well-defined peak appears around 300 K and the +uncertainty shrinks to ≈ 70 K. The surface pressure and +planetary radius are less biased for (S/N)ref ≥ 10, with +the reference values correctly retrieved within 2σ. For +the 1 mbar gravity, we find only a lower limit for all our +signal-to-noise ratios. +5.2. Molecular Abundances +Figure 9 (top right) shows that the abundances of +gases with strong absorption features in the optical and +near-infrared (O3, O2, and H2O) are generally well con- +strained. The O3 abundance can always be constrained +better than a factor of 2 (0.3 dex), even for (S/N)ref = 5. +The ease of constraining O3 is driven by its strong +absorption at optical and near-UV wavelengths. +The +O2 posterior is the broadest due to the smaller num- +ber of data points spanning its narrow absorption fea- +tures. Nevertheless, O2 can be constrained to 0.3 dex for +(S/N)ref = 20. We note that the O3, O2, and H2O abun- +dances are slightly underestimated for (S/N)ref = 5, but +are reliably retrieved for (S/N)ref ≥ 10. +Our retrievals are unable to detect gases with only +weak absorption features in the modeled wavelength +range. We can place an upper limit on the CH4 abun- +dance for (S/N)ref = 5, but CO2 and N2O require +(S/N)ref = 15 for upper limits. +We find a tentative +hint of CO2 at (S/N)ref = 20 centered on the reference +value, but the posterior tail to lower abundances indi- +cates a non-detection of CO2 absorption. Constraints +on gases such as CO2 and CH4 for a modern Earth- +like atmosphere at low resolution and low signal-to-noise +would benefit from observations of thermal emission in +the mid-infrared (e.g. Des Marais et al. 2002; Kalteneg- +ger et al. 2007; Konrad et al. 2021). + +Retrieving Red Edges from Exoplanet Reflection Spectra +13 +0.5 +0.0 +0.5 +1.0 +1.5 +logP0 (bar) +Probability Denstiy +Planetary Parameters +Molecular Abundances +Surface Albedo Parameters +0 +5 +10 +15 +20 +25 +g(ms−2) +0.6 +0.8 +1.0 +1.2 +Rp (RE) +Probability Denstiy +0.70 +0.72 +0.74 +0.76 +λ1 ( m) +Probability Denstiy +0.06 +0.08 +0.10 +0.12 +0.14 +0.16 +As, 1 +0.1 +0.2 +0.3 +0.4 +As, 2 +1.1 +1.2 +1.3 +1.4 +1.5 +1.6 +1.7 +λ2 ( m) +0.0 +0.1 +0.2 +0.3 +As, 3 +4 +3 +2 +1 +0 +logO2 +7.5 +7.0 +6.5 +6.0 +logO3 +4.0 +3.5 +3.0 +2.5 +2.0 +logH2O +7 +5 +3 +1 +logCO2 +9 +7 +5 +3 +logCH4 +9 +7 +5 +3 +1 +logN2O +100 +200 +300 +400 +500 +T(K) +S/N=20 +S/N=15 +S/N=10 +S/N=5 +Figure 9. Retrieval results for bulk planetary, atmospheric, and surface parameters for a cloud-free Earth-like planet. Four +different retrieval results are shown as a function of S/N (shading and histogram outlines). The reference values for the ground +truth model are annotated (black dashed lines). +Table 3. Retrieval results for a clear atmosphere at R=70 with reference +values and 1σ uncertainties. +Parameter +Reference +S/N=5 +S/N=10 +S/N=15 +S/N=20 +log O2 +-0.678 +−1.87+0.85 +−1.02 +−1.29+0.52 +−0.57 +−1.14+0.37 +−0.42 +−1.06+0.28 +−0.32 +log O3 +-6.25 +−6.64+0.27 +−0.27 +−6.44+0.18 +−0.18 +−6.40+0.14 +−0.14 +−6.37+0.10 +−0.11 +log H2O +-2.72 +−3.09+0.48 +−0.43 +−2.73+0.31 +−0.35 +−2.66+0.23 +−0.28 +−2.61+0.17 +−0.23 +log CO2 +-3.44 +−4.52+2.19 +−3.49 +−4.40+2.11 +−3.69 +−4.71+1.81 +−3.56 +−4.00+1.09 +−3.61 +log CH4 +-5.77 +−6.64+1.71 +−2.10 +−6.96+1.81 +−2.02 +−7.28+1.72 +−1.85 +−7.19+1.62 +−1.89 +log N2O +-6.55 +−4.41+2.16 +−3.55 +−5.90+2.44 +−2.73 +−6.55+2.21 +−2.34 +−6.59+2.06 +−2.28 +log P0 +0.0 +0.61+0.34 +−0.39 +0.40+0.27 +−0.26 +0.39+0.21 +−0.23 +0.38+0.19 +−0.19 +Rp +1.007 +0.76+0.12 +−0.07 +0.86+0.10 +−0.08 +0.88+0.08 +−0.07 +0.90+0.06 +−0.05 +g +9.66 +13.07+7.57 +−7.14 +13.51+7.31 +−6.52 +13.67+6.92 +−5.89 +13.42+6.73 +−4.82 +T +289 +321+96 +−97 +321+72 +−63 +310+47 +−42 +305+36 +−32 +λ1 +0.72 +0.723+0.015 +−0.014 +0.723+0.008 +−0.007 +0.723+0.005 +−0.005 +0.723+0.004 +−0.004 +λ2 +1.40 +1.86+0.46 +−0.48 +1.42+0.33 +−0.11 +1.37+0.08 +−0.06 +1.36+0.07 +−0.04 +As,1 +0.09 +0.104+0.027 +−0.024 +0.101+0.012 +−0.012 +0.100+0.009 +−0.009 +0.099+0.007 +−0.007 +As,2 +0.15 +0.25+0.05 +−0.06 +0.21+0.04 +−0.04 +0.20+0.03 +−0.03 +0.19+0.02 +−0.02 +As,3 +0.06 +0.19+0.25 +−0.11 +0.11+0.05 +−0.03 +0.10+0.03 +−0.02 +0.10+0.02 +−0.02 +Note—S/N is defined at 0.55 µm (see Section 2.2.1). +5.3. Surface Properties +Figure 9 (bottom) highlights trends in the retrieved +surface albedo parameters. As discussed in Sections 3 +and 4, our main results are: i) including wavelength +dependent surface albedo in retrievals can improve the +accuracy of atmospheric inferences; and ii) even for +low to moderate signal-to-noise ratios one can constrain +the wavelength-dependent surface albedo for an Earth- +analog planet. We highlight here that the λ1 posterior +demonstrates that a sharp change occurs in the surface +albedo around 0.72 µm, even for (S/N)ref = 5, which +is consistent with the modern Earth’s red edge. This +albedo transition is remarkably well constrained, as in- +dicated by the narrow 1σ intervals in Table 2. Similarly, +the posteriors for λ2 at (S/N)ref ≥ 10 indicate that there +is another sharp feature in the wavelength-dependent +surface albedo around 1.4 µm. Compared with the pos- +teriors for the bulk planetary properties and molecular +abundances, these results suggest that surface albedo +changes are one of the most reliable features to detect +in reflection spectra of Earth-like planets. + +14 +Gomez Barrientos et al. +3 +2 +1 +0 +Probability denstiy +logO2 +7.5 +7.0 +6.5 +6.0 +5.5 +Probability denstiy +logO3 +5 +4 +3 +2 +1 +logH2O +Clear +Cloudy +Figure 10. Retrieved molecular abundances from reflection +spectra of an Earth-like exoplanet with a clear (green) and +cloudy (orange) atmosphere. The simulated data used has +R = 70 and (S/N)ref = 10). +The ground truth reference +values are overlaid (black lines, see Table 1). +5.4. The Influence of Clouds +While Figure 9 corresponded to cloud-free models, +clouds can also increase the uncertainty on the re- +trieved atmospheric composition. Figure 10 shows that +a cloud deck increases the abundance uncertainties for +detectable species. The cloud parameters broaden the +1σ constraint for the oxygen, ozone, and water vapor +abundances (from 0.55 dex to 0.61 dex for O2; from +0.18 dex to 0.30 dex for O3, and from 0.33 dex to 0.50 dex +for H2O). These effects can be attributed to the degener- +acy that emerges between the location of the cloud-base +and the gas mixing ratios (see Appendix C). Despite the +broader distributions, the O2, O3, and H2O abundances +are still retrieved to within 1σ of their reference values +when clouds are included in our model. +Finally, Figure 11 shows the cloud parameter con- +straints for R = 70 and (S/N)ref = 10 (corresponding +to the retrieved spectrum in Figure 8). The retrieved +optical depth (log τ), pressure extent (log dp), and base +pressure (log pc) of the cloud deck are all correctly re- +trieved within 1σ. The posteriors for log τ and log dp are +broad due to the degenerate nature of these parameters. +However, the bounded constraint on log pc demonstrates +that our retrieval technique correctly identifies the pres- +ence of a cloud deck as a necessary model component +distinct from the wavelength-dependent surface albedo. +Figure 11. Retrieved cloud parameters for simulated data +at R = 70 and (S/N)ref = 10. The retrieved median values +(blue solid lines) agree with the reference values (green lines) +within the 1σ confidence regions (dashed lines). +6. SUMMARY AND DISCUSSION +In this study, we investigated the potential to retrieve +wavelength-dependent surface properties from reflection +spectra of rocky exoplanets and the influence of such sur- +faces on the retrieval of molecular abundances and bulk +planetary properties. We accomplished this by develop- +ing and implementing a Bayesian retrieval approach to +infer wavelength-dependent surface properties from sim- +ulated observations of a self-consistent modern Earth- +like planet. +Our retrieval analysis demonstrated that +it is possible to infer wavelength-dependent surface in- +formation at moderate signal-to-noise ratios. Our key +results from this work include the following: +1. Reflected-light retrievals of terrestrial exoplanets +should account for wavelength-dependent surface +albedos to achieve unbiased results. If one assumes +a constant surface albedo, some optical and near- +infrared data can be poorly fit and the retrieved +mixing ratios can be biased. +2. We introduced a five-parameter prescription to +account for sharp ‘edge-like’ changes in the sur- +face albedo at a priori unknown wavelengths. We +demonstrated that this parameterization can re- +cover realistic surface albedo profiles from reflec- +tion spectra of a modern Earth analog. + +log Tc +0.312 +F0.75. +0.722 +ogdp +0.640 +0.491 +6 +0.0 +0.8 +log Tc +log dp +log pcRetrieving Red Edges from Exoplanet Reflection Spectra +15 +3. Wavelength-dependent surface albedos can be +readily retrieved from cloud-free reflection spec- +tra. +Even with S/N = 5, one can infer evi- +dence of Earth-like wavelength-dependent surface +features (2.7σ). The easiest feature to detect for +our modern Earth analog is a sharp change in +the surface albedo at visible wavelengths (i.e. the +modern Earth’s red edge). Improved data quality +(S/N ≥ 10) allows a secondary albedo change in +the infrared to be constrained as well. +4. Clouds can lower the detection significance of +wavelength-dependent surface features. Neverthe- +less, one can still infer modern Earth-like sur- +face features in the presence of clouds at S/N = +10 (2.9σ). +Also, the wavelength of the modern +Earth’s red edge can still be reliably retrieved. +5. Higher spectral resolution improves constraints on +wavelength-dependent surface features for cloud- +free modern Earth-like models. Specifically, data +at R = 140 outperforms R = 70 data in identifying +surface features at infrared wavelengths. +6. The mixing ratios of gases with strong absorption +features in the optical and near-infrared — namely +O3, O2, and H2O — can be precisely retrieved (< +0.6 dex) with moderate quality data (S/N = 10) +when wavelength dependent surface albedos are +included in retrievals. Gases with weaker infrared +absorption, such as CH4, CO2, and N2O, are +largely unconstrained by reflection spectra (upper +limits only). +7. Several other planetary properties (e.g. planet ra- +dius, temperature, and cloud properties) can be +retrieved from reflection spectra. +However, the +planetary gravity can not be determined from re- +flection spectra alone. +We proceed to discuss the implications of our findings. +6.1. Retrievability of Diverse Surface Compositions +Wavelength-dependent surface albedos significantly +impact the reflection spectra of directly-imaged terres- +trial exoplanets (e.g. Figure 2). Our retrieval analysis +demonstrates that future direct imaging missions could +find evidence of wavelength-dependent surface features +and constrain the shape of surface albedo profiles. Our +results complement and expand on the recent study by +Wang et al. (2022), by offering a novel parameteriza- +tion to retrieve changes in the surface albedo at a priori +unknown wavelengths. +In particular, our demonstra- +tion that sharp features like the modern Earth’s veg- +etation red edge can be reliably retrieved (to 150 nm +0.5 +1.0 +1.5 +2.0 +2.5 +Wavelength( m) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Albedo +Sand +Seawater +Granite +Basalt +Trees +Snow +Figure 12. +Wavelength-dependent albedos of materials +commonly found on Earth’s surface. +for cloud-free models at S/N = 5) is very promising +for a proposed future large IR/Optical/UV space-based +telescope (e.g. Decadal Survey on Astronomy & Astro- +physics 2020). While we find that clouds can result in +overestimated surface albedos (in agreement with results +from Feng et al. 2018, Robinson & Salvador 2022, and +Damiano & Hu 2022), the red edge’s wavelength loca- +tion can nonetheless be correctly retrieved for modern +Earth-like planets even for Earth-like cloud coverage. +More generally, detecting wavelength-dependent sur- +face albedos from reflection spectra offers the opportu- +nity to constrain the surface composition of rocky exo- +planets. Other materials like sand, basalt, and granite, +which cover substantial regions of Earth’s surface, have +unique albedo profiles (see Figure 12) and shape the re- +flection spectra of Earth-like exoplanets (see e.g. Mad- +den & Kaltenegger 2020). These profiles could poten- +tially be extracted from spectra of rocky worlds whose +surfaces are dominated by these materials (Pham & +Kaltenegger 2021, 2022). +However, for retrieval pur- +poses the flexibility of our surface albedo parameteriza- +tion (Equation 3) would need to be tested for these sur- +face compositions. Our parameterization was inspired +by the modern Earth’s surface albedo, hence it is able to +locate sharp albedo changes such as the red edge. Future +work should investigate the flexibility of our parameter- +ization for other surfaces, such as oceans or deserts, to +determine if a generalized parameterization is necessary. +An additional caveat is that our retrievals have fo- +cused on zero orbital phase (i.e. full illumination), while +future direct imaging observations will be constrained to +higher phase angle. The left panel of Figure 13 shows +the impact of higher phase angles through PICASO cal- +culations at orbital phases of 0◦, 30◦, 60◦, and 90◦ for +the model in Figure 2. We see that orbital phase acts to +scale down Fp/F∗, while preserving the relative ampli- +tude of the red edge relative to the surrounding contin- + +16 +Gomez Barrientos et al. +0.5 +1.0 +1.5 +2.0 +2.5 +Wavelength( m) +0 +1 +2 +3 +4 +5 +Fp/Fs × 10−10 +α = 0o +α = 30o +α = 60o +α = 90o +Figure 13. Left: Impact of different orbital phase angles on reflection spectra of an Earth-like exoplanet. The sequence of +models have the same atmospheric and surface properties as Figure 2, but with partial illumination. Right: retrieved reflection +spectrum of a cloud-free Earth-like exoplanet at orbital phase of 60◦ with a wavelength-dependent surface albedo (for simulated +data at R = 70 and (S/N)ref = 20). The inset shows the posterior distribution of the retrieved red edge wavelength. +uum. The right panel of Figure 13 shows the impact of +a non-zero orbital phase on a retrieval of our cloud-free +Earth-like exoplanet model. Due to the computational +requirements of non-zero orbital phase retrievals, for this +demonstration we consider only an orbital phase of 60◦ +and observations with (S/N)ref = 20 and R = 70. We +see that the location of the vegetation red edge can still +be constrained for non-zero phases, but the uncertainty +becomes about 50% larger due to the lower flux ratio. +Future retrieval studies could investigate the impact of a +wide range of non-zero orbital phases (e.g. Nayak et al. +2017), or multi-phase observations (Damiano et al. 2020; +Carri´on-Gonz´alez et al. 2021), on wavelength-dependent +surface albedo constraints. +Nevertheless, our results +show great promise for the detectability and character- +ization of rocky exoplanet surfaces from reflection spec- +tra. +6.2. Opportunities for Detecting Surface Biosignatures +The potential habitability of Earth-like exoplanets or- +biting Sun-like stars could be assessed by constraining +wavelength-dependent surface properties. Future direct- +imaging missions will focus on characterizing these at- +mospheres to search for biosignature gas pairs like O3 +combined with CH4. Our results suggest that such mis- +sions can also retrieve surface spectral features, and that +not accounting for wavelength-dependent surfaces can +bias the retrieved abundances of biosignature gas pairs. +We stress that including wavelength-dependent surface +albedos in retrievals is an opportunity, since it enables an +expanded mission science case including searches for sur- +face biosignatures. The vegetation red edge is one candi- +date, but its universality remains uncertain. Exoplanets +could have reflectance edges at different wavelengths (see +e.g. Kiang et al. 2007) or photosynthetic organisms that +do not show red edge features (see e.g. Cockell et al. +2009). Some minerals also exhibit sharp spectral fea- +tures near optical wavelengths (e.g., Seager et al. 2005; +O’Malley-James & Kaltenegger 2018). +Thus, any de- +tection of reflectance edges would need to be carefully +placed in context with other signatures of habitability +before attributing a biological origin. +The promising detectability of the modern Earth’s red +edge also suggests that other surface biosignatures would +benefit from retrieval studies. One such alternative sur- +face biosignature is biofluorescence. +On Earth, coral +and other organisms absorb harmful shortwave radia- +tion and re-emit it at longer wavelengths as a protec- +tion mechanism. Like the vegetation red edge, biofluo- +rescence can dramatically increase a planet’s brightness +at specific wavelengths (O’Malley-James & Kaltenegger +2019b, 2018). Biofluorescence could therefore manifest +as a time-dependent spectral edge that may be retriev- +able from reflection spectra of Earth-like exoplanets. +6.3. Constraining Atmospheric Properties of Earth-like +Exoplanets via Reflected Light +One of the primary science goals of future space- +based observatories with direct imaging capabilities will +be to characterize the atmospheres of Earth-like exo- +planets orbiting Sun-like stars. +Our results indicate +that a large space-based observatory with the capabil- +ity to achieve Fp/F∗ ∼ 10−10 could precisely constrain +the abundances of several biosignature pair gases on +modern Earth analogs. +Assuming a cloud-free atmo- +sphere, we showed that the O2, O3, and H2O abun- +dances can be constrained within 0.6 dex for S/N = 10 +and R = 70. +Crucially, the O3 abundance can be +constrained to within a factor of 2 (0.3 dex) even for +S/N = 5. +However, it will be more challenging for +these missions to detect biosignature gases with weak +absorption features. For instance, we could only place + +0.006Retrieving Red Edges from Exoplanet Reflection Spectra +17 +an upper limit on the CH4 mixing ratio for a modern +Earth-like analog. Furthermore, clouds can also impact +our ability to constrain and detect some biosignatures. +Clouds broaden the uncertainties in the retrieved molec- +ular abundances of O2, O3, and H2O, which makes it +more challenging to constrain their abundances. +Our retrieval results suggest that meaningful infor- +mation can be extracted from reflected light spectra of +a modern Earth analog, even with a lower SNR than +indicated by previous work, if the available data covers +an expanded wavelength range. Feng et al. (2018) found +that S/N = 15 is generally a prerequisite to constrain +the abundances of O2, O3, and H2O. In comparison, +we find that these gases can be precisely constrained for +S/N ≥ 10. +These differences are mainly attributable +to the wavelength range of the simulated data — we +use 0.3–2.5 µm, while Feng et al. (2018) considered 0.4– +1.0 µm. The longer wavelength coverage in our retrievals +decreased the minimum S/N necessary to constrain H2O +because of the three additional water features at 1.1, 1.4, +and 1.9 µm. Similarly, the short wavelength coverage of +O3 absorption lowered the S/N necessary for constrain- +ing the O3 abundance. However, the difference between +the minimum signal-to-noise ratios for retrieving the O2 +abundance are: i) due to our differing noise models (we +assumed an agnostic detector efficiency while Feng et al. +(2018) based their model on Roman Space Telescope- +like detectors); and ii) due to the wavelength dependent +surface features we added to the retrieval process. +6.4. Characterizing Exoplanets Orbiting Sun-like Stars +Ultimately, a key driver of exoplanet science is the +characterization of potentially habitable planets around +Sun-like stars. The Astro 2020 Decadal Survey (Decadal +Survey on Astronomy & Astrophysics 2020) specifically +highlights the goal of searching for atmospheric biosig- +natures on Earth-like exoplanets orbiting Sun-like stars. +Our findings indicate that a future direct imaging mis- +sion observing reflected light could also detect surface +biosignatures — including the vegetation red edge — via +the retrieval of a wavelength-dependent surface albedo. +Our odds of detecting life in the solar neighborhood can +only be enhanced by considering all the ways life shapes +its host planet. For a spectral edge encoded in light from +a distant star may, one day, illuminate the surface of a +world not too dissimilar to our own. +We thank Natasha Batalha and Zifan Lin for helpful +discussions. We also thank the anonymous referee for +a helpful report. J.G.B. is supported by Cornell Uni- +versity’s Ronald E. McNair Post-baccalaureate Achieve- +ment Program and the Carl Sagan Institute. +1 +2 +3 +4 +5 +Software: +PICASO (Batalha et al. 2019) , corona- +graph (Lustig-Yaeger et al. 2019) +DATA AVAILABILITY +The +realistic +Earth-like +surface +albedo +and +the +raw albedo files used in this work are available at +https://doi.org/10.5281/zenodo.6977238. +APPENDIX +A. RETRIEVALS WITH GAUSSIAN SCATTER +Here, we show a retrieved reflection spectrum from synthetic observations with Gaussian scatter. Figure 14 shows +that the retrieved spectra captures the overall spectral morphology of the input model and that the location of the +vegetation red edge is still constrained. +B. RETRIEVAL MODEL VALIDATION +Here, we validate our retrieval framework using the model Earth spectrum from Robinson et al. (2011), which has +been validated against Earthshine data. The simulated data was generated by binning the model spectrum from its +native resolution to R=70. We then used our noise model to simulate data at (S/N)ref = 10. Figure 15 shows that our +retrieval model reproduces the general spectral morphology of the Robinson et al. (2011) model. As with the cloudy +1D simulated data, the retrieved albedo is brighter than the model surface albedo. Nevertheless, our retrieval results +indicate that the location of the red edge can still be constrained at (S/N)ref = 10. +C. FULL POSTERIOR DISTRIBUTIONS +Here, we include full posterior distributions from representative retrievals of our modern Earth analog. Figure 16 +shows the posterior distribution for a cloud-free retrieval at (S/N)ref = 10 and R = 70 including a wavelength- +dependent surface albedo. Similarly, Figure 17 shows the posterior for the cloudy scenario discussed in Section 4.3. + +18 +Gomez Barrientos et al. +Figure 14. 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R., Gillon, M., et al. 2016, Nature, +537, 69, doi: 10.1038/nature18641 + +20 +Gomez Barrientos et al. +logP0 = 0.404+0.274 +−0.261 +5 +10 +15 +20 +g +g = 13.506+7.305 +−6.519 +0.75 +0.90 +1.05 +1.20 +Rp +Rp = 0.860+0.101 +−0.080 +0.70 +0.72 +0.74 +0.76 +λ1 +λ1 = 0.723+0.008 +−0.007 +0.04 +0.08 +0.12 +0.16 +As, 1 +As, 1 = 0.101+0.012 +−0.012 +0.16 +0.24 +0.32 +As, 2 +As, 2 = 0.205+0.041 +−0.038 +1.0 +1.5 +2.0 +λ2 +λ2 = 1.420+0.332 +−0.107 +0.2 +0.4 +0.6 +0.8 +As, 3 +As, 3 = 0.114+0.054 +−0.035 +3.2 +2.4 +1.6 +0.8 +logO2 +logO2 = −1.291+0.518 +−0.575 +7.2 +6.8 +6.4 +6.0 +logO3 +logO3 = −6.435+0.182 +−0.184 +3.6 +3.0 +2.4 +1.8 +logH2O +logH2O = −2.729+0.312 +−0.352 +8 +6 +4 +2 +logCO2 +logCO2 = −4.402+2.114 +−3.693 +9.0 +7.5 +6.0 +4.5 +logCH4 +logCH4 = −6.962+1.808 +−2.017 +8 +6 +4 +2 +logN2O +logN2O = −5.904+2.443 +−2.733 +0.5 +0.0 +0.5 +1.0 +logP0 +200 +300 +400 +T +5 +10 +15 +20 +g +0.75 +0.90 +1.05 +1.20 +Rp +0.70 +0.72 +0.74 +0.76 +λ1 +0.04 +0.08 +0.12 +0.16 +As, 1 +0.16 +0.24 +0.32 +As, 2 +1.0 +1.5 +2.0 +λ2 +0.2 +0.4 +0.6 +0.8 +As, 3 +3.2 +2.4 +1.6 +0.8 +logO2 +7.2 +6.8 +6.4 +6.0 +logO3 +3.6 +3.0 +2.4 +1.8 +logH2O +8 +6 +4 +2 +logCO2 +9.0 +7.5 +6.0 +4.5 +logCH4 +8 +6 +4 +2 +logN2O +200 +300 +400 +T +T = 321+72 +−63 +Figure 16. The full posterior distribution for the cloud-free simulated data at R = 70 and (S/N)ref = 10. The retrieved +median (solid blue lines) and 16th and 84th percentiles (dashed blue lines) for each parameter are overlaid for comparison with +the ground truth reference values (green lines). Note that the mixing ratio reference values represent the average value from +the surface to 25 km altitude. Similarly, the surface albedo parameter reference values represent the average albedo over a +wavelength range. + +Retrieving Red Edges from Exoplanet Reflection Spectra +21 +logP0 = 0.577+0.341 +−0.369 +5 +10 +15 +20 +g +g = 12.517+7.836 +−7.033 +0.60 +0.75 +0.90 +1.05 +Rp +Rp = 0.740+0.089 +−0.063 +0.5 +1.0 +1.5 +2.0 +λ1 +λ1 = 0.717+0.017 +−0.017 +0.2 +0.4 +0.6 +0.8 +As, 1 +As, 1 = 0.159+0.136 +−0.092 +0.2 +0.4 +0.6 +0.8 +As, 2 +As, 2 = 0.406+0.150 +−0.138 +1.0 +1.5 +2.0 +λ2 +λ2 = 1.687+0.520 +−0.346 +0.2 +0.4 +0.6 +0.8 +As, 3 +As, 3 = 0.241+0.234 +−0.151 +4.5 +3.0 +1.5 +logO2 +logO2 = −0.949+0.551 +−0.662 +7.2 +6.6 +6.0 +5.4 +logO3 +logO3 = −6.345+0.298 +−0.312 +6.0 +4.5 +3.0 +1.5 +logH2O +logH2O = −3.119+0.466 +−0.538 +8 +6 +4 +2 +logCO2 +logCO2 = −3.905+1.514 +−3.561 +9.0 +7.5 +6.0 +4.5 +logCH4 +logCH4 = −6.779+1.681 +−2.044 +8 +6 +4 +2 +logN2O +logN2O = −5.929+2.281 +−2.576 +200 +300 +400 +T +T = 329+90 +−89 +1.6 +0.8 +0.0 +0.8 +logτc +logτc = −0.312+0.751 +−0.722 +1 +0 +1 +logdp +logdp = −0.493+0.595 +−0.640 +0.0 +0.8 +1.6 +logP0 +1 +0 +1 +logpc +5 +10 +15 +20 +g +0.60 +0.75 +0.90 +1.05 +Rp +0.5 +1.0 +1.5 +2.0 +λ1 +0.2 +0.4 +0.6 +0.8 +As, 1 +0.2 +0.4 +0.6 +0.8 +As, 2 +1.0 +1.5 +2.0 +λ2 +0.2 +0.4 +0.6 +0.8 +As, 3 +4.5 +3.0 +1.5 +logO2 +7.2 +6.6 +6.0 +5.4 +logO3 +6.0 +4.5 +3.0 +1.5 +logH2O +8 +6 +4 +2 +logCO2 +9.0 +7.5 +6.0 +4.5 +logCH4 +8 +6 +4 +2 +logN2O +200 +300 +400 +T +1.6 +0.8 +0.0 +0.8 +logτc +1 +0 +1 +logdp +1 +0 +1 +logpc +logpc = −0.643+0.487 +−0.491 +Figure 17. The full posterior distribution for the cloudy simulated data at R = 70 and (S/N)ref = 10. The retrieved median +(solid blue lines) and 16th and 84th percentiles (dashed blue lines) for each parameter are overlaid for comparison with the +ground truth reference values (green lines). + +22 +Gomez Barrientos et al. +de Wit, J., Wakeford, H. R., Lewis, N. K., et al. 2018, +Nature Astronomy, 2, 214, +doi: 10.1038/s41550-017-0374-z +Decadal Survey on Astronomy, & Astrophysics. 2020, +Pathways to Discovery in Astronomy and Astrophysics +for the 2020s, doi: 10.17226/26141 +Des Marais, D. J., Harwit, M. O., Jucks, K. W., et al. 2002, +Astrobiology, 2, 153, doi: 10.1089/15311070260192246 +Diamond-Lowe, H., Charbonneau, D., Malik, M., Kempton, +E. M. R., & Beletsky, Y. 2020, AJ, 160, 188, +doi: 10.3847/1538-3881/abaf4f +Fauchez, T. J., Villanueva, G. L., Sergeev, D. E., et al. +2021, arXiv e-prints, arXiv:2109.11460. +https://arxiv.org/abs/2109.11460 +Feng, Y. K., Robinson, T. D., Fortney, J. 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W. +2002, ApJ, 574, 430, doi: 10.1086/340929 +Wordsworth, R., & Kreidberg, L. 2021, arXiv e-prints, +arXiv:2112.04663. https://arxiv.org/abs/2112.04663 + diff --git a/P9AzT4oBgHgl3EQfzv7f/content/tmp_files/load_file.txt b/P9AzT4oBgHgl3EQfzv7f/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..445a7fef48824b89f9a1fbc59ea11ed9a8b02ab4 --- /dev/null +++ b/P9AzT4oBgHgl3EQfzv7f/content/tmp_files/load_file.txt @@ -0,0 +1,2240 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf,len=2239 +page_content='Draft version January 6, 2023 Typeset using LATEX twocolumn style in AASTeX631 In Search of the Edge: A Bayesian Exploration of the Detectability of Red Edges in Exoplanet Reflection Spectra Jonathan Gomez Barrientos ,1, 2 Ryan J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' MacDonald ,1, 3, 4 Nikole K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Lewis ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1 and Lisa Kaltenegger 1 1Department of Astronomy and Carl Sagan Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Cornell University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 122 Sciences Drive,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Ithaca,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' NY 14853,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' USA 2Division of Geological and Planetary Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' California Institute of Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Pasadena,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' CA 91125,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' USA 3Department of Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' University of Michigan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Ann Arbor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' MI 48109,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' USA 4NHFP Sagan Fellow ABSTRACT Reflection spectroscopy holds great promise for characterizing the atmospheres and surfaces of po- tentially habitable terrestrial exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The surface of the modern Earth exhibits a sharp albedo change near 750 nm caused by vegetation — the red edge — which would leave a strong spectral sig- nature if present on an exoplanet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' However, the retrieval of wavelength-dependent surface properties from reflection spectra has seen relatively little study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Here, we propose a new surface albedo param- eterization capable of retrieving the wavelength location of a priori unknown ‘edge-like’ features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We demonstrate that a wavelength-dependent surface albedo model achieves higher accuracy in retrieving atmospheric composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Wavelength-dependent surfaces are also generally preferred over a uniform albedo model when retrieving simulated reflection spectra for a modern Earth analog, even for mod- erate signal-to-noise ratios (S/N = 10) and Earth-like clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Further, the location of the modern Earth’s red edge can be robustly and precisely constrained (within 70 nm for S/N = 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Our results suggest that future space-based direct imaging missions have the potential to infer surface compositions for rocky exoplanets, including spectral edges similar to those caused by life on the modern Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Keywords: planets and satellites: atmospheres — planets and satellites: terrestrial planets 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' INTRODUCTION One of the most profound discoveries of the exoplanet era is the realization that rocky planets in the classical habitable zone are prolific around nearby stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', Pe- tigura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Bryson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Remote observa- tions of such rocky planets hold the potential to deepen our understanding of the physical, chemical, climate, and biological processes shaping these worlds (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', Kaltenegger 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Wordsworth & Kreidberg 2021, for recent reviews).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Spectroscopic observations ultimately provide our clearest window into the atmospheric and surface properties of rocky exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The near-term focus for rocky exoplanet atmospheric characterization lies with transiting planets orbiting small stars, such as M dwarfs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Scalo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Barstow & Irwin 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Morley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Fauchez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Corresponding author: Jonathan Gomez Barrientos jdg276@cornell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='edu 2021) or white dwarfs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', Kaltenegger & MacDonald et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2020, Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Transmission spectroscopy — the variation of planetary radius with wavelength — is currently the most common technique used to diag- nose the composition, temperature profile, and aerosol properties of exoplanet atmospheres (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', Brown 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Sing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Kreidberg 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' MacDonald & Lewis 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Existing transmission spectra from the Hubble Space Telescope and ground-based facilities have ruled out clear H2-dominated atmospheres for several rocky exoplanets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', de Wit et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2016, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Wakeford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Diamond-Lowe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Libby-Roberts et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Garcia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The recently launched JWST will observe terrestrial exoplanets transiting M dwarfs, which should lead to detections of molecules such as CO2, CH4, and H2O (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', Krissansen-Totton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Lustig-Yaeger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' However, terrestrial exoplanets in the habitable zone of G-type stars like our Sun are not optimal targets for transmission spectroscopy (due to their low planet-star arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='01775v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='EP] 4 Jan 2023 ID2 Gomez Barrientos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' radius ratio, infrequent transits, and atmospheric refrac- tion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' B´etr´emieux & Kaltenegger 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Directly-detected reflected stellar light is a promis- ing avenue to characterize rocky exoplanets orbiting G- type stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Detecting reflected light from an Earth- like exoplanet around a star like our Sun requires sen- sitivity to planet-star contrast ratios at visible wave- lengths of Fp/F∗ ∼ 10−10, where Fp and F∗ are, re- spectively, the fluxes from the planet and star as ob- served at Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The Roman Space Telescope (sched- uled for launch in 2026) will offer important progress towards this goal, demonstrating space-based coronog- raphy by imaging cool giant planets with Fp/F∗ ∼ 10−8 to 10−9 (Kasdin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Recently, the Astro 2020 Decadal Survey recommended the development of a large IR/Optical/UV space-based telescope — notion- ally for launch in the 2040s — to characterize Earth-like exoplanets around Sun-like stars (Decadal Survey on As- tronomy & Astrophysics 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Such a mission would be designed to detect atmospheric and surface biosigna- tures for a population of Earth-sized exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Biosignatures are remotely detectable features indica- tive of a biological process (for recent reviews, see Kaltenegger 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Schwieterman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Fujii et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' For exoplanets, biosignatures include the simul- taneous detection of a pair of oxidizing and reducing gases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' O2 / O3 and CH4) or temporal variabil- ity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='g, Keeling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 1976, Ford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Meadows 2006, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Cowan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Fujii et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' An- other important class of biosignatures — and the focus of this study — are surface biosignatures, here defined as remotely detectable spectroscopic features caused by the presence of biology on a planetary surface (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', Des Marais et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Schwieterman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Hegde et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Fonseca Coelho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The red edge is a proposed surface biosignature caused by a biology-induced change in the wavelength- dependent surface albedo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' On the modern Earth, where plants cover ∼ 60% of the land area, the red edge can be detected in both resolved reflection spectra (Sagan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 1993) and disc-averaged Moonshine (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Des Marais et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Woolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Turnbull et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' This photosynthetic red edge has changed in strength throughout Earth’s history (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', from changing sur- face coverage, types of biota), but should have been detectable for the last 1 billion years (O’Malley-James & Kaltenegger 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' O’Malley-James & Kaltenegger 2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The origin of Earth’s red edge is chlorophyll reflecting more light redwards of ∼ 750 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' For exoplan- ets, many studies have suggested that a similar change in the surface albedo at a characteristic wavelength could constitute a biosignature analogous to Earth’s red edge (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', Seager et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Takizawa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' O’Malley-James & Kaltenegger 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' O’Malley-James & Kaltenegger 2019a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Reflected light spectra of an exo- Earth would consequently display a sharp contrast ra- tio increase if its surface featured a signature like the red edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' This study proposes a method to retrieve the wavelength location of albedo changes, like the red edge, from reflection spectra of rocky exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Spectroscopic retrieval is a method commonly used to infer planetary properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', atmospheric composi- tion, temperature, and clouds) from observed spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Bayesian retrieval techniques compare model spectra for a wide range of possible planet properties (typically ≳ 105) to a set of observations, thereby obtaining prob- ability distributions for the planet properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', Ben- neke & Seager 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' MacDonald & Madhusudhan 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Molli`ere et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Reflected light exoplanet retrieval techniques were initially developed for directly imaged cool giant exoplanets, largely in preparation for the Ro- man Space Telescope (Lupu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Nayak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Lacy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Damiano & Hu 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Damiano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Carri´on-Gonz´alez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2020, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Mukher- jee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Subsequent studies have extended re- flected light retrievals to directly imaged sub-Neptunes and terrestrial planets (Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Damiano & Hu 2021, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Robinson & Salvador 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2022), which would be observable with a future large IR/Optical/UV space-based telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Terrestrial planet reflection spectral retrievals must additionally consider surface reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Most previous studies have assumed a uniform-in-wavelength surface albedo (Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Damiano & Hu 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Robinson & Salvador 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Consequently, the spectral imprint of a wavelength-dependent surface, including the red edge, has seen little investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Recently, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' (2022) found that retrievals using a three-albedo model can outperform a single-albedo model — demonstrat- ing that reflection spectra can constrain wavelength- dependent surface albedos (see also Brandt & Spiegel 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' (2022)’s albedo retrieval technique considered three fixed wavelength bands at visible wave- lengths (blue, green, and red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Here, we introduce a gen- eralized albedo retrieval technique designed to identify sharp albedo changes at a priori unknown wavelengths analogous to Earth’s red edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' In this study, we demonstrate that wavelength- dependent surface albedos can be retrieved from moderate-quality reflection spectra of Earth-like ex- oplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' In particular, the wavelength location of Earth’s vegetative red edge can be precisely constrained by a future large IR/Optical/UV space-based telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' In what follows, we first introduce our modeling and re- Retrieving Red Edges from Exoplanet Reflection Spectra 3 trieval methodology in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We demonstrate that our parametric albedo prescription provides an excel- lent fit to reflection spectra for an Earth-like surface in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We explore the sensitivity of surface albedo retrievals to data quality and clouds in Section 4, be- fore investigating constraints on other atmospheric and planetary properties in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Finally, in Section 6, we summarize our results and discuss their implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' REFLECTION SPECTRA MODELING AND RETRIEVAL FRAMEWORK This paper investigates the surface, bulk planetary, and atmospheric properties that can be retrieved from observed reflection spectra of an exo-Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We begin in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1 by presenting a self-consistent 1D model for an Earth-like exoplanet orbiting a Sun-like star and describe the computation of its reflection spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We then outline the generation of synthetic reflection spec- tral observations and our Bayesian retrieval method for Earth-like exoplanet reflection spectra in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Reflection Spectra for an Exo-Earth 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Atmospheric Model We generate an atmospheric model resembling the modern Earth using Exo-Prime2 (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', Kaltenegger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Madden & Kaltenegger 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Kasting & Ack- erman 1986;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Pavlov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Pavlov & Kasting 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Segura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2005, 2007) – a 1D radiative-convective terrestrial atmosphere code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Exo-Prime2 couples 1D cli- mate and photochemistry models to compute the verti- cal temperature structure and atmospheric mixing ra- tio profiles for a planet, assuming an incident stellar spectrum and planetary outgassing rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Exo-Prime2 also includes feedback from wavelength-dependent sur- face albedos and clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' For Earth-like clouds, we use the MODIS 20 µm cloud albedo model (King et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Rossow & Schiffer 1999), which provides a reasonable average for many clouds of different droplet size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The application of Exo-Prime2 to model Earth-like plan- ets around different host stars and through geological time has been extensively described in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', Kaltenegger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Rugheimer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Rugheimer & Kaltenegger 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Madden & Kaltenegger 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Kaltenegger & Lin 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The resulting pressure-temperature (P-T) and mixing ratio profiles computed by Exo-Prime2 are shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Surface Model We model a representative Earth-like surface us- ing wavelength-dependent albedos from the USGS and ASTER spectral libraries (Baldridge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Kokaly et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We create an average present-day Earth surface albedo from 8 raw albedos of snow, water, coast, sand, trees, grass, basalt, and granite (after Kaltenegger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We assume an Earth- like surface consisting of 70% ocean, 28% land, and 2% coast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The land surface consists of 30% grass, 30% trees, 9% granite, 9% basalt, 15% snow, and 7% sand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We use the surface-fraction weighted albedo (see Figure 2, bot- tom panel) in our 1D radiative transfer models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Reflection Spectra Computation A distant observer directly imaging an exoplanet mea- sures the wavelength-dependent planet-star flux ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' At wavelengths where reflected light dominates over thermal emission, the flux ratio can be expressed as Fp(α, λ) Fs(λ) = Ag(λ) Φ(α, λ) �Rp d �2 (1) where Ag is the planet’s geometric albedo spectrum, Φ is the phase function, α is the orbital phase, Rp is the plan- etary radius, and d is the planet-star orbital distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The geometric albedo is traditionally defined as the ra- tio of the observed flux from the planet at full phase to that from a perfectly reflecting Lambert disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The phase function encodes the dilution of the planetary brightness for phase angles without full illumination (Φ = 1 when α = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' While the geometric albedo encodes informa- tion about an atmosphere’s composition, temperature, cloud properties, and surface reflection, the phase func- tion is controlled by both the stellar illumination and atmospheric scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We generate model reflection spectra for an Earth-like planet around a Sun-like star using the open-source ra- diative transfer code PICASO Batalha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' To compute Fp/Fs from the geometric albedo we provide PICASO with Rp and d (fixed to 1 AU) and assume ob- servations at full phase (Φ = 1) unless otherwise noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We note that observations will more typically occur at non-zero phase, which would dampen the resultant re- flection spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' However, in this proof of concept study, we choose to focus on full phase to reduce the complexity of the radiative transfer calculations required within the retrieval framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We set the planetary reference ra- dius such that r(P = 1 bar) = R⊕, the surface at 1 bar, and the surface gravity to 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='81ms−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' For the stellar spectrum, we used PICASO to interpolate the Castelli & Kurucz (2003) grid for a Sun-analog star with Teff = 5780 K, log g = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='437, and [Fe/H] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' For the radiative transfer calculation, we provide PICASO with the P-T profile, mixing ratio profiles, and the wavelength-dependent surface albedo from our Earth-like Exo-Prime2 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Our reflection spectra computations span the near-UV to near-IR, ranging 4 Gomez Barrientos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 200 220 240 260 280 T (K) 6 5 4 3 2 1 0 logPressure(bar) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='6 logCH4 7 6 5 4 3 2 logH2O 8 7 6 5 logO3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 logN2O 6 5 4 3 2 1 0 logPressure(bar) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='3 logCO2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='21 O2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='79 N2 0 20 40 60 80 100 Altitude (km) 0 20 40 60 80 100 Altitude (km) Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Pressure-temperature profile and volume mixing ratio profiles for a model of the modern Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The main spectrally relevant gases from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='3–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 µm are shown, alongside the bulk N2 fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='3–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We consider molecular line opacity for H2O, O2, O3, CH4, CO2, and N2O (see Batalha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2019, for details on the opacity database), alongside Rayleigh scattering from N2 and O2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' For computational efficiency, we downsampled PICASO’s molecular cross sections by 10× (from R=10,000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We tested different resampling factors and find that 10× downsampling pro- vides a reliable balance between speed and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Our PICASO model accounting for Earth-like clouds assumes optical properties consistent with water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Specifically, we use an asymmetry factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='85 and a single scattering albedo of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 (after Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We place the cloud base (in log10 bars) at log pb = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='23, set its vertical extent (also in log10 bars) to log dp = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='53, and the cloud optical depth to log τ = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We selected these values for the cloud vertical extent by calibrating the continuum flux from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4–1 µm of our 1D models to reproduce the reflection spectrum of Earth from Robinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We note that assuming a constant cloud albedo increases the re- flected flux beyond 1 µm compared to Robinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' (2011), but does not significantly influence our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Impact of the Red Edge on Reflection Spectra Figure 2 shows example reflection spectra for a cloud- free exo-Earth, both with and without a wavelength- dependent surface albedo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' As expected, the red edge has a pronounced effect on the reflection spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The red edge causes a marked increase in Fp/F∗ for wavelengths immediately following 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='75 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' As we will see in Sec- tion 3, this sudden change is a distinct feature enabling the spectroscopic detectability of the red edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We also note that the red edge occurs near strong O3 and O2 features, which underscores the importance of account- ing for a wavelength-dependent surface albedo when searching for atmospheric biosignature gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' With our ‘ground-truth’ model described, we proceed to outline our retrieval framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Retrieval Framework We have developed a Bayesian retrieval wrapper around the PICASO radiative transfer code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We em- ploy this retrieval framework in subsequent sections to demonstrate that information on the red edge can be reliably retrieved from reflection spectra of exo-Earths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Here, we describe the simulated data used in our re- trievals and our retrieval configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Simulated Data & Noise Model Our aim is to investigate the retrievability of the red edge as a function of data quality, rather than for a spe- cific future mission architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Consequently, we gen- erated several synthetic datasets, for both the cloud-free and cloudy models described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='3, spanning signal-to-noise ratios of S/N = 5, 10, 15, and 20 (at a reference wavelength of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='55 µm) and spectral resolu- tions of R = 70 and 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We account for wavelength- dependent noise for the simulated observations using a Retrieving Red Edges from Exoplanet Reflection Spectra 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 0 1 2 3 4 5 Fp/Fs × 10−10 Red edge H2O O3 O3 O2 H2O H2O H2O CH4 CO2 Constant Albedo Earth-like Albedo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 Wavelength( m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='20 Albedo Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Impact of the red edge on reflection spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Top: model reflection spectra for a cloud-free Earth-like planet orbiting a Sun-like star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' A realistic wavelength-dependent surface albedo (green) produces a sharp increase in the flux ratio around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='75 µm compared to a constant surface albedo (purple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Prominent molecular absorption features are labeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Bottom: the corresponding surface albedos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The location of the vegetative red edge is highlighted in both panels (red shading).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' For clarity, both models are shown at a spectral resolution of R = 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' constant spectral resolution noise model scaling relation as done in (Robinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2018) S/N(λ) ∝ q(λ) T (λ) Ag(λ) Φ(α, λ) B(λ) λ (2) where q is the detector quantum efficiency, T is the throughput, and B is a blackbody representing the par- ent star (see Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We adopt functions for q and T from the Python package coronagraph1, which is an open source noise simulator for coronagraph-based 1 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='com/jlustigy/coronagraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='git observations of directly imaged exoplanets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', Robin- son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Lustig-Yaeger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' For the black- body, we use Teff = 5780 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' When generating each simulated dataset, we do not randomize the placement of each data point by sam- pling from a Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Rather, the data are centered on the (true) planet-to-star flux ratio — corresponding to the model after binning down to the data resolution — and assigned error bars according to our noise model at the desired S/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We note that run- ning retrievals on a dataset with Gaussian noise can bias the retrieval results, especially for low spectral resolu- 6 Gomez Barrientos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' tion and S/N (see Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' However, running retrievals with Gaussian scatter still allows for spectral features to be recovered (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2021, Ap- pendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' To avoid biasing our retrieval results to a spe- cific random noise draw, we run ‘scatter-free’ retrievals, which produce posterior distributions equivalent to the ensemble average over many individual noise instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We include an example retrieval with Gaussian scatter in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Retrieval Configuration Bayesian retrieval codes repeatedly call a parameter- ized radiative transfer forward model to identify the range of bulk planetary, atmospheric, and surface prop- erties consistent with a given dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Our retrieval framework employs PICASO (Batalha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2019) as the radiative transfer forward model and the MultiNest (Feroz & Hobson 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Feroz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2009, 2019) wrap- per PyMultiNest Buchner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' (2014) for the sampling algorithm used to explore the parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We parameterize the atmospheric and surface prop- erties using a simplified prescription that captures the salient features shaping terrestrial exoplanet reflection spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We parameterize the P-T profile with an isotherm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We assume H2O, O2, O3, CH4, CO2, and N2O are the main spectrally-active gases with sufficient abun- dances to shape the spectrum for Earth-like planets, with each gas ascribed a single free parameter for the uniform-in-altitude volume mixing ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We also as- sume the primary atmospheric gas is N2, with its mix- ing ratio determined by the condition that mixing ra- tios must sum to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We prescribe three further free parameters for the planetary radius and gravity (evalu- ated at 1 mbar) and the surface pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' These choices are similar to those made by other reflected-light re- trieval studies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Damiano & Hu 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Robinson & Salvador 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Since our simulated observations incorporate an Earth-like wavelength-dependent surface albedo, we propose a new parametric treatment for wavelength- dependent surface albedos As(λ) = � � � � � � � As,1 λ < λ1 As,2 λ1 ≤ λ ≤ λ2 As,3 λ > λ2 (3) where As,1, As,2, and As,3 define the surface albedo in three distinct wavelength regions, λ1 marks the transi- tion from As,1 → As,2, and λ2 marks the transition from As,2 → As,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' This surface albedo prescription thus has five free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' To avoid discontinuities at λ1 and λ2, we compute this function on a wavelength grid at R Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Free parameters included in our PICASO retrievals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Parameter Description Reference Value Prior Range log O2 Oxygen mixing ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='678 [-10,0] log O3 Ozone mixing ratio 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='25 [-10,-1] log H2O Water vapor mixing ratio 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='72 [-10,-1] log CO2 Carbon dioxide mixing ratio 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='44 [-10,-1] log CH4 Methane mixing ratio 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='77 [-10,-1] log N2O Nitrous oxide mixing ratio 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='55 [-10,-1] log P0 Surface pressure 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 [-2,2] Rp Planet radius at 1 mbar 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='007 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0] g Gravity at 1 mbar 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='66 [1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0,25] T Temperature 289 [100,800] λ1 Albedo transition point 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='72 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='3,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5] λ2 Albedo transition point 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='40 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='3,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5] As,1 Surface Albedo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='09 [0,1] As,2 Surface Albedo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='15 [0,1] As,3 Surface Albedo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='06 [0,1] log pb Cloud-base pressure 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='23 [-2,2] log dp Cloud width 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='53 [-2,2] log τ Cloud optical depth 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 [-2,2] Note—The reference values for each parameter correspond to either ‘ground truth’ values from the input model (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', planet radius and cloud properties;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1) or representative average values (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', mixing ratios and albedo parameters;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' All priors are uniform distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' = 1,000 and convolve it with a Gaussian with a standard deviation of 28 wavelength grid spaces (corresponding to 28 nm at 1 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Our albedo parameterization thus resembles a smoothed double-step function (similar to the function used by Taylor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2021 to parameter- ize the single-scattering albedo of clouds in giant planet nightside emission spectra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We shall demonstrate in subsequent sections that the proposed parameterization is sufficiently flexible to capture both the strong wave- length dependence of the red edge and a possible sec- ondary reflectance edge in the infrared (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' For retrievals including clouds, we add three further parameters: the cloud base pressure, its vertical pressure extent, and optical depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Following Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' (2018), we assume water-like clouds with a fixed asymmetry pa- rameter (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='85) and single scattering albedo (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' In total, the most complex retrievals we consider thus have a total of 18 free parameters (summarized in Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We validated our retrieval framework against simulated data from the Robinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' (2011) model (see Ap- pendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Our retrieval analysis covers multiple model and data scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' First, in Section 3, we evaluate the retriev- ability of albedo changes for our cloud-free model, since this model has the strongest spectral red edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We initially ran four retrievals on the simulated data at (S/N)ref = 5, 10, 15, 20 and R = 70, where we parame- Retrieving Red Edges from Exoplanet Reflection Spectra 7 terize the planet’s wavelength-dependent surface albedo with Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Then, we ran a similar set of re- trievals with a constant-in-wavelength surface albedo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Doing so enables us to perform Bayesian model compar- isons between the wavelength-dependent surface and the constant-in-wavelength surface models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', Benneke & Seager 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Trotta 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We also ran a retrieval with R = 140 at (S/N)ref = 10 to investigate the impact of retrieving data at a higher spectral resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Lastly, we ran a retrieval at R = 70 and (S/N)ref = 10 for a more realistic dataset including a cloud deck to inves- tigate how clouds impact the retrieval results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' All our MultiNest retrievals use 2,000 live points, which typi- cally involve the computation of 106 models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We summarize the prior range for each retrieval free parameter in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We generally allow generous prior ranges, encompassing a wide range of physically plausi- ble values, with all priors being uniform distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' As in Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' (2018), we allow for oxygen-rich at- mospheres by extending its prior range to 100% (but rejecting any parameter combinations where the sum of the non-N2 mixing ratios exceed unity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Our prior range for the planet radius and gravity terminates at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 R⊕ and 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 ms−2, respectively, since this study fo- cuses on Earth-like planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We note that PICASO re- quires that the pressure corresponding to Rp must be less than the highest pressure in the atmospheric pres- sure grid (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' the surface pressure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We circumvent this issue by defining the planet radius and gravity parame- ters at 1 mbar, such that the surface pressure prior range (10−2–102 bar) is always deeper than 1 mbar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We also include ‘reference values’ for each parameter in Table 1 — corresponding closest to the original input Exo-Prime2 model (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1) — for comparison to the retrieval results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Since the input gas mixing ra- tios depend on height, while the retrievals assume uni- form mixing ratios, we set the reference values as the average of the true mixing ratio profile from the sur- face to 25 km altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Our reference temperature is the planet’s surface temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' For the albedo parame- ters, our reference values are determined by averaging the surface albedo over 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='3–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='72 µm, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='72–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4 µm, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The reference planet radius and gravity cor- respond to the true values from the Exo-Prime2 model (scaled to 1 mbar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Similarly, the surface pressure and cloud parameter reference values correspond exactly to the original model inputs (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We now turn to present the results of our retrievals including wavelength-dependent surface albedos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' THE NECESSITY FOR WAVELENGTH- DEPENDENT SURFACE ALBEDOS IN REFLECTED-LIGHT RETRIEVALS Here, we demonstrate that reflection spectra of ter- restrial exoplanets contain recoverable information on wavelength-dependent surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We show that not only can data commensurate with future direct imaging mis- sions constrain wavelength-dependent surface albedos, but that assuming a constant surface albedo may result in biased atmospheric inferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Can a Uniform Albedo Fit the Earth’s Red Edge?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We first assess whether a retrieval model assuming a constant-in-wavelength surface albedo can adequately fit the reflection spectrum of an exoplanet with a realistic Earth-like surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We have already seen in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4 and Figure 2 that the Earth’s red edge induces a sharp change in Fp/F∗ around 750 nm, so here we quantify whether such a spectral signature is detectable and its impact on atmospheric retrievals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' In Figure 3, we demonstrate that a uniform albedo model often struggles to capture the spectral morphol- ogy of an Earth-like exoplanet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Our ‘ground truth’ model is the cloud-free scenario described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1, which produces the strongest red edge, while the simu- lated data (here, (S/N)ref = 10) and retrieval configu- ration are detailed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We see that the re- trieved spectrum for the uniform albedo model begins to deviate from the simulated data for optical wave- lengths longer than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='65 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' In the optical and near-IR, where the S/N is highest, the uniform albedo model is often discrepant with the data to 2σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The root cause of this model-data mismatch is that the uniform albedo model has a roughly constant continuum Fp/F∗ from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='6–0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='9 µm (outside O2 absorption features), which can- not reproduce the sharp spectral continuum change as- sociated with the vegetation red edge near 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='72 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' In contrast, our wavelength-dependent surface albedo parameterization well matches the reflection spectrum of our Earth-like exoplanet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Equation 3 allows our retrieval code to reproduce both the large increase in Fp/F∗ at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='72 µm caused by the vegetation red edge and the gen- eral morphology of the spectrum in the visible, and near- IR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Statistically, our retrieval including a wavelength- dependent albedo is favored over the uniform model with a Bayes factor of ln B = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4 (equivalent to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1 σ using the relations in Benneke & Seager 2012), which would be considered a conclusive detection on the Jeffrey’s scale of Bayesian model comparison (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Trotta 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We illustrate why the uniform albedo model strug- gles to fit our data by comparing the retrieved sur- face albedos in the bottom panel of Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' While 8 Gomez Barrientos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 0 1 2 3 4 5 6 Fp/Fs × 10−10 Non-uniform Model (R = 70) Uniform Model (R = 70) 1σ confidence 2σ confidence 1σ confidence 2σ confidence Data (S/N = 10) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 Wavelength( m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='30 Albedo Retrieved Albedo True Albedo Retrieved Albedo 1σ confidence 2σ confidence 1σ confidence 2σ confidence Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Top: reflection spectra retrieval results assuming a wavelength-dependent surface albedo (purple contours) and a uniform surface albedo (orange contours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The simulated data (R = 70 and (S/N)ref = 10) corresponds to an Earth-like exoplanet orbiting a Sun-like star with an Earth-like wavelength-dependent surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Bottom: the retrieved surface albedo profiles corresponding to the two models in the top panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The true Earth-like surface albedo used to generate the simulated data is overlaid for comparison (black curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' the uniform albedo model correctly captures the sur- face albedo in the near-IR wavelengths beyond 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4 µm, it significantly underestimates the true surface albedo from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='75–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='35 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' However, our proposed parameter- ization demonstrates that one can retrieve wavelength- dependent surface properties — in particular the loca- tion of the vegetation red edge — at even a moderate signal-to-noise ratio ((S/N)ref = 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The retrieved surface albedo profile also correctly infers a decrease in the albedo for near-IR wavelengths beyond 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='3 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' While our wavelength-dependent surface retrieval some- what overestimates the magnitude of the albedo (likely Retrieving Red Edges from Exoplanet Reflection Spectra 9 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Posterior probability distribution for the re- trieved surface albedo parameters in Equation 3 (for sim- ulated data at R = 70 and (S/N)ref = 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The green lines mark the ground truth references values (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The orange contours show the 1σ, 2σ, and 3σ confidence regions for each retrieved parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The blue lines in the histograms show the median (solid line) and marginalized 1σ confidence region (dashed lines) for each parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' due to other complexities not captured in the model, such as variable atmospheric abundances with height), it correctly captures the general shape of the wavelength- dependent albedo profile and lies within 2σ of the true Earth-like surface albedo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We further show the posteriors of our retrieved albedo parameters in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' All five parameters are well- constrained by the data, demonstrating that reflec- tion spectra are highly sensitive to the wavelength- dependence of the surface albedo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' In particular, the parameter encoding the wavelength location of the red edge, λ1, is retrieved to a remarkable precision of 8 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Similarly, the retrieval identifies a secondary albedo change near 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4 µm, encoded by λ2, with a precision of ≈ 200 nm (though the long posterior tail to higher val- ues indicates this is harder to constrain than λ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The three albedo parameters are slightly overestimated, as noted above, but are consistent within 2 σ of the refer- ence values (see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Overall, Figure 4 shows that Equation 3 offers a parametrization capable of captur- ing the key wavelength-dependent features of a realistic Earth-like surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 3 2 1 0 Probability denstiy logO2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 Probability denstiy logO3 4 3 2 1 logH2O Non-uniform Albedo Uniform Albedo Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Impact on the retrieved molecular abundances of O2, O3, and H2O from assuming a uniform surface albedo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Accounting for a wavelength-dependent surface (green his- tograms) results in good agreement with the ground truth reference values (black lines, see Table 1), while a retrieval assuming a uniform albedo (orange) can bias these abun- dances by an order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Biases from Assuming a Uniform Albedo The assumption of a uniform surface albedo can bias inferred properties of an exoplanet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Since a retrieval code employs every available means to minimize model- data residuals, it can attempt to compensate for the non- inclusion of a wavelength-dependent surface by modi- fying the retrieved abundances of chemical species in the atmosphere (since their cross sections are also wave- length dependent).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' In Figure 5, we demonstrate that one consequence from assuming a uniform surface albedo is biased abundance inferences for several key molecules in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Specifically, we find that the volume mix- ing ratios of O3 and H2O are over-estimated by an order of magnitude and the bulk atmospheric gas would be identified as O2 rather than N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' This finding under- scores an important point: accurate abundance infer- ences for atmospheric gases can depend on the inclusion of a wavelength-dependent surface albedo in reflected light retrieval frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Since Figure 5 corresponds to the moderate case of R = 70 and (S/N)ref = 10, wavelength-dependent surface spectral properties will be an important consideration for future direct imaging missions for exoplanets, especially for missions focused on Earth-like exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 入1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='723+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='107 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='114+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='054 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='03510 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 0 2 9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='70 12 16 24 7 O: O: : : Q: Q: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 入1 As,1 入2 As,310 Gomez Barrientos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' FACTORS INFLUENCING RETRIEVALS OF WAVELENGTH-DEPENDENT SURFACE ALBEDOS Having established the need for a wavelength- dependent surface albedo model in reflected-light re- trievals, we next explore how these results depend on data quality (S/N in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1 and R in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2) and the inclusion of clouds (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Sensitivity to S/N Figure 6 shows our retrieved spectra and surface albedo profiles for (S/N)ref = 5, 10, 15, and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We see that even at (S/N)ref = 5, the retrieval correctly identifies a sharp rise in the surface albedo near 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='7 µm — consistent with the wavelength of the vegetation red edge on modern Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The uncertainty in the retrieved wavelength of this feature is remarkably small (λ1 deter- mined to ≈ 15 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' This suggests that sudden changes in surface albedo are an effect of first order importance even for low signal-to-noise observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' With a dou- bling to (S/N)ref = 10, we see further improvements in the retrieved surface albedo profile: (i) the uncertainty in the location of the sharp rise in surface albedo is halved (λ1 determined to ≈ 8 nm);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' (ii) a hint emerges of a secondary albedo change near 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4 µm (λ2 determined to ≈ 200 nm);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' and (iii) the true surface albedo profile is correctly captured throughout most of the wavelength range to within 2σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' For (S/N)ref = 15, the retrieval becomes more confident about the existence of a sec- ondary albedo edge (λ2 determined to ≈ 70 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Fi- nally, at (S/N)ref = 20 the retrieved model attains even better overall agreement with the true albedo model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We also find that the tendency to overestimate the re- trieved albedo (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1) becomes less prevalent for higher S/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Table 2 quantifies the preference for our wavelength- dependent albedo model (Equation 3) over a uniform albedo model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' For (S/N)ref = 5, our Bayesian model comparison finds moderate evidence for a non-uniform surface albedo (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='7σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' A slight increase to (S/N)ref = 10 suffices to conclusively detect at least one discontinuity (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Further increases in S/N can help to detect a non-uniform surface albedo also for cloudy atmospheres, where the effect is smaller because clouds block part of the light from the underlying surface from view (see Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Since our retrievals thus far have only consid- ered data at R = 70, we next explore variable spectral resolution for a fixed signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The Role of Spectral Resolution Figure 7 shows how the retrieved surface albedo pro- file changes with spectral resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Specifically, we Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Predicted detection significances for an Earth-like wavelength-dependent surface albedo as a function of S/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Model (S/N)ref = 5 (S/N)ref = 10 (S/N)ref = 15 (S/N)ref = 20 Clear 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='7σ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1σ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='7σ 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2σ Cloudy — 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='9σ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='6σ — Note—(S/N)ref is defined at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='55 µm (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' All data is at R = 70 from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='3–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Cloudy spectra retrievals at (S/N)ref = 5 and (S/N)ref = 20 were not computed, so ‘—’ is used for their detection significances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' illustrate the expected improvement from doubling the spectral resolution from R = 70 to R = 140 (hence dou- bling the number of datapoints from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='3–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We see that the retrieved albedo from the higher resolu- tion data is in better agreement with the true albedo, especially at longer wavelengths where the errors bars are largest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Further, the uncertainty on the retrieved albedo parameters decrease (As,1 and As,2 improve by ≈ 25%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' As,3 improves by ≈ 50%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We further note that the at R = 140 the retrieval of the second, smaller edge around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4 µm (λ2) improves (by ≈ 68%), resulting in a retrieved albedo shape more consistent with the true Earth-like surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The Impact of Clouds Figure 8 shows how the inclusion of clouds affects the retrieved spectrum and surface albedo profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Our cloud properties for this demonstration were chosen to resemble the continuum flux from the model in Robin- son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' (2011) (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We see that while the data is well fit, the retrieved albedo profile is gen- erally overestimated when clouds are included (see also Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2022 for a discussion on cloud-surface de- generacies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Nevertheless, the wavelength of the red edge is still reliably retrieved and well constrained even in the presence of clouds (λ1 determined to 17 nm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The retrieved albedo also shows a slight decrease at longer wavelengths, but the secondary albedo change near 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4 µm is not well constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Overall, the pres- ence of a cloud deck can lead one to infer an artificially brighter surface outside the 1σ uncertainty region of the retrieved albedo profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Clouds increase the minimum signal-to-noise ratio re- quired to detect a wavelength-dependent surface fea- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' For example, Table 2 demonstrates that the de- tection significance for a non-uniform surface albedo at (S/N)ref = 10 drops from 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1σ (cloud-free) to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='9σ (including clouds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Such lower significances arise from cloud-surface degeneracies broadening albedo uncertain- ties (see Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' However, we still find a detection of a Retrieving Red Edges from Exoplanet Reflection Spectra 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 0 1 2 3 4 5 6 Fp/Fs × 10−10 Median Model (R = 70) 1σ confidence 2σ confidence Binned Model Data (S/N = 5) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='30 Albedo Retrieved Albedo True Albedo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 0 1 2 3 4 5 6 Fp/Fs × 10−10 Median Model (R = 70) 1σ confidence 2σ confidence Binned Model Data (S/N = 10) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='30 Albedo Retrieved Albedo True Albedo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 0 1 2 3 4 5 6 Fp/Fs × 10−10 Median Model (R = 70) 1σ confidence 2σ confidence Binned Model Data (S/N = 15) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='30 Albedo Retrieved Albedo True Albedo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 Wavelength( m) 0 1 2 3 4 5 6 Fp/Fs × 10−10 Median Model (R = 70) 1σ confidence 2σ confidence Binned Model Data (S/N = 20) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 Wavelength( m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='30 Albedo Retrieved Albedo True Albedo Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Retrieved reflection spectra and surface albedo profiles as a function of signal-to-noise ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Left panels: comparison of the median retrieved spectrum (blue line), and its corresponding 1σ and 2σ confidence regions (purple contours), with simulated data for an Earth-like exoplanet (cyan errors) at (S/N)ref = 5, 10, 15, and 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The median model binned to the resolution of the data (gold diamonds) well fit the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Right panels: corresponding retrieved surface albedo profiles (blue line and purple contours) inferred from each dataset compared to the true Earth-like surface model (black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' wavelength-dependent surface albedo for (S/N)ref = 15 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='6σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' These results show that, while clouds can com- plicate the inference of wavelength-dependent surface features, it is still possible to identify non-uniform sur- face albedos for Earth-like cloud coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 12 Gomez Barrientos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 Wavelength( m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='30 Albedo Retrieved Albedo (R = 70) True Albedo Retrieved Albedo (R = 140) 1σ confidence 2σ confidence 1σ confidence 2σ confidence Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Impact of spectral resolution on the retrieved surface albedos of modern Earth seen as an exoplanet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The retrieved albedos for two distinct datasets are overlaid: (i) R = 70 and (S/N)ref = 10 (purple contours);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' and (ii) R = 140 and (S/N)ref = 10 (orange contours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The true model Earth surface used to generate both simulated datasets is included for comparison (black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 0 1 2 3 4 5 6 7 Fp/Fs × 10−10 Retrieved Model (R = 70) 1σ confidence 2σ confidence Binned Model Data (S/N = 10) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 Wavelength( m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='9 Albedo Retrieved Albedo True Albedo 1σ confidence 2σ confidence Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Top: retrieved reflection spectrum of a cloudy Earth-like exoplanet with a wavelength-dependent surface albedo (for simulated data at R = 70 and (S/N)ref = 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The median retrieved spectrum binned to the resolution of the data (gold diamonds) generally agrees with the simulated observations (cyan errors) within the retrieval confidence in- tervals (purple contours).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Bottom: corresponding retrieved surface albedo profile (purple contours) compared to the true surface albedo (black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' FULL RETRIEVAL RESULTS FOR PLANETARY AND ATMOSPHERIC PROPERTIES For completeness, here we show our full retrieval re- sults for other planetary and atmospheric properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Table 3 summarizes the retrieved values of all 15 free parameters for our cloud-free scenario, along with their 1σ uncertainties, as a function of S/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We also show the posterior distributions for each parameter in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Planetary Properties Figure 9 (top left) shows that the reliable inference of bulk planetary properties requires moderate signal-to- noise ratios for cloud-free models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' At (S/N)ref = 5, the 1 mbar radius is underestimated and the surface pressure slightly overestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The temperature is correctly re- trieved, albeit with broad uncertainty (≈ 100 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We find that (S/N)ref = 10 is the minimum to reliably re- trieve these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' In particular, for the tempera- ture a well-defined peak appears around 300 K and the uncertainty shrinks to ≈ 70 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The surface pressure and planetary radius are less biased for (S/N)ref ≥ 10, with the reference values correctly retrieved within 2σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' For the 1 mbar gravity, we find only a lower limit for all our signal-to-noise ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Molecular Abundances Figure 9 (top right) shows that the abundances of gases with strong absorption features in the optical and near-infrared (O3, O2, and H2O) are generally well con- strained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The O3 abundance can always be constrained better than a factor of 2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='3 dex), even for (S/N)ref = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The ease of constraining O3 is driven by its strong absorption at optical and near-UV wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The O2 posterior is the broadest due to the smaller num- ber of data points spanning its narrow absorption fea- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Nevertheless, O2 can be constrained to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='3 dex for (S/N)ref = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We note that the O3, O2, and H2O abun- dances are slightly underestimated for (S/N)ref = 5, but are reliably retrieved for (S/N)ref ≥ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Our retrievals are unable to detect gases with only weak absorption features in the modeled wavelength range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We can place an upper limit on the CH4 abun- dance for (S/N)ref = 5, but CO2 and N2O require (S/N)ref = 15 for upper limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We find a tentative hint of CO2 at (S/N)ref = 20 centered on the reference value, but the posterior tail to lower abundances indi- cates a non-detection of CO2 absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Constraints on gases such as CO2 and CH4 for a modern Earth- like atmosphere at low resolution and low signal-to-noise would benefit from observations of thermal emission in the mid-infrared (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Des Marais et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Kalteneg- ger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Konrad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Retrieving Red Edges from Exoplanet Reflection Spectra 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 logP0 (bar) Probability Denstiy Planetary Parameters Molecular Abundances Surface Albedo Parameters 0 5 10 15 20 25 g(ms−2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2 Rp (RE) Probability Denstiy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='76 λ1 ( m) Probability Denstiy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='16 As, 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4 As, 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='7 λ2 ( m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='3 As, 3 4 3 2 1 0 logO2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 logO3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 logH2O 7 5 3 1 logCO2 9 7 5 3 logCH4 9 7 5 3 1 logN2O 100 200 300 400 500 T(K) S/N=20 S/N=15 S/N=10 S/N=5 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Retrieval results for bulk planetary, atmospheric, and surface parameters for a cloud-free Earth-like planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Four different retrieval results are shown as a function of S/N (shading and histogram outlines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The reference values for the ground truth model are annotated (black dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Retrieval results for a clear atmosphere at R=70 with reference values and 1σ uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Parameter Reference S/N=5 S/N=10 S/N=15 S/N=20 log O2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='678 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='87+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='85 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='02 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='29+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='52 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='57 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='14+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='37 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='42 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='06+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='28 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='32 log O3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='25 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='64+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='27 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='27 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='44+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='18 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='18 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='40+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='14 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='14 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='37+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='10 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='11 log H2O 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='72 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='09+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='48 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='43 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='73+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='31 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='35 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='66+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='23 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='28 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='61+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='17 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='23 log CO2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='44 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='52+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='19 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='49 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='40+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='11 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='69 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='71+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='81 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='56 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='00+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='09 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='61 log CH4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='77 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='64+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='71 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='10 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='96+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='81 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='02 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='28+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='72 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='85 −7.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='10+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='03 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='10+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='02 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='02 Note—S/N is defined at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='55 µm (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Surface Properties Figure 9 (bottom) highlights trends in the retrieved surface albedo parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' As discussed in Sections 3 and 4, our main results are: i) including wavelength dependent surface albedo in retrievals can improve the accuracy of atmospheric inferences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' and ii) even for low to moderate signal-to-noise ratios one can constrain the wavelength-dependent surface albedo for an Earth- analog planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We highlight here that the λ1 posterior demonstrates that a sharp change occurs in the surface albedo around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='72 µm, even for (S/N)ref = 5, which is consistent with the modern Earth’s red edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' This albedo transition is remarkably well constrained, as in- dicated by the narrow 1σ intervals in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Similarly, the posteriors for λ2 at (S/N)ref ≥ 10 indicate that there is another sharp feature in the wavelength-dependent surface albedo around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Compared with the pos- teriors for the bulk planetary properties and molecular abundances, these results suggest that surface albedo changes are one of the most reliable features to detect in reflection spectra of Earth-like planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 14 Gomez Barrientos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 3 2 1 0 Probability denstiy logO2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 Probability denstiy logO3 5 4 3 2 1 logH2O Clear Cloudy Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Retrieved molecular abundances from reflection spectra of an Earth-like exoplanet with a clear (green) and cloudy (orange) atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The simulated data used has R = 70 and (S/N)ref = 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The ground truth reference values are overlaid (black lines, see Table 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The Influence of Clouds While Figure 9 corresponded to cloud-free models, clouds can also increase the uncertainty on the re- trieved atmospheric composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Figure 10 shows that a cloud deck increases the abundance uncertainties for detectable species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The cloud parameters broaden the 1σ constraint for the oxygen, ozone, and water vapor abundances (from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='55 dex to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='61 dex for O2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='18 dex to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='30 dex for O3, and from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='33 dex to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='50 dex for H2O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' These effects can be attributed to the degener- acy that emerges between the location of the cloud-base and the gas mixing ratios (see Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Despite the broader distributions, the O2, O3, and H2O abundances are still retrieved to within 1σ of their reference values when clouds are included in our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Finally, Figure 11 shows the cloud parameter con- straints for R = 70 and (S/N)ref = 10 (corresponding to the retrieved spectrum in Figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The retrieved optical depth (log τ), pressure extent (log dp), and base pressure (log pc) of the cloud deck are all correctly re- trieved within 1σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The posteriors for log τ and log dp are broad due to the degenerate nature of these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' However, the bounded constraint on log pc demonstrates that our retrieval technique correctly identifies the pres- ence of a cloud deck as a necessary model component distinct from the wavelength-dependent surface albedo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Retrieved cloud parameters for simulated data at R = 70 and (S/N)ref = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The retrieved median values (blue solid lines) agree with the reference values (green lines) within the 1σ confidence regions (dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' SUMMARY AND DISCUSSION In this study, we investigated the potential to retrieve wavelength-dependent surface properties from reflection spectra of rocky exoplanets and the influence of such sur- faces on the retrieval of molecular abundances and bulk planetary properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We accomplished this by develop- ing and implementing a Bayesian retrieval approach to infer wavelength-dependent surface properties from sim- ulated observations of a self-consistent modern Earth- like planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Our retrieval analysis demonstrated that it is possible to infer wavelength-dependent surface in- formation at moderate signal-to-noise ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Our key results from this work include the following: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Reflected-light retrievals of terrestrial exoplanets should account for wavelength-dependent surface albedos to achieve unbiased results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' If one assumes a constant surface albedo, some optical and near- infrared data can be poorly fit and the retrieved mixing ratios can be biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We introduced a five-parameter prescription to account for sharp ‘edge-like’ changes in the sur- face albedo at a priori unknown wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We demonstrated that this parameterization can re- cover realistic surface albedo profiles from reflec- tion spectra of a modern Earth analog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' log Tc 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='312 F0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='722 ogdp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='640 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='491 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='8 log Tc log dp log pcRetrieving Red Edges from Exoplanet Reflection Spectra 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Wavelength-dependent surface albedos can be readily retrieved from cloud-free reflection spec- tra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Even with S/N = 5, one can infer evi- dence of Earth-like wavelength-dependent surface features (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='7σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The easiest feature to detect for our modern Earth analog is a sharp change in the surface albedo at visible wavelengths (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' the modern Earth’s red edge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Improved data quality (S/N ≥ 10) allows a secondary albedo change in the infrared to be constrained as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Clouds can lower the detection significance of wavelength-dependent surface features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Neverthe- less, one can still infer modern Earth-like sur- face features in the presence of clouds at S/N = 10 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='9σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Also, the wavelength of the modern Earth’s red edge can still be reliably retrieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Higher spectral resolution improves constraints on wavelength-dependent surface features for cloud- free modern Earth-like models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Specifically, data at R = 140 outperforms R = 70 data in identifying surface features at infrared wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The mixing ratios of gases with strong absorption features in the optical and near-infrared — namely O3, O2, and H2O — can be precisely retrieved (< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='6 dex) with moderate quality data (S/N = 10) when wavelength dependent surface albedos are included in retrievals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Gases with weaker infrared absorption, such as CH4, CO2, and N2O, are largely unconstrained by reflection spectra (upper limits only).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Several other planetary properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' planet ra- dius, temperature, and cloud properties) can be retrieved from reflection spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' However, the planetary gravity can not be determined from re- flection spectra alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We proceed to discuss the implications of our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Retrievability of Diverse Surface Compositions Wavelength-dependent surface albedos significantly impact the reflection spectra of directly-imaged terres- trial exoplanets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Our retrieval analysis demonstrates that future direct imaging missions could find evidence of wavelength-dependent surface features and constrain the shape of surface albedo profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Our results complement and expand on the recent study by Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' (2022), by offering a novel parameteriza- tion to retrieve changes in the surface albedo at a priori unknown wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' In particular, our demonstra- tion that sharp features like the modern Earth’s veg- etation red edge can be reliably retrieved (to 150 nm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 Wavelength( m) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 Albedo Sand Seawater Granite Basalt Trees Snow Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Wavelength-dependent albedos of materials commonly found on Earth’s surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' for cloud-free models at S/N = 5) is very promising for a proposed future large IR/Optical/UV space-based telescope (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Decadal Survey on Astronomy & Astro- physics 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' While we find that clouds can result in overestimated surface albedos (in agreement with results from Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2018, Robinson & Salvador 2022, and Damiano & Hu 2022), the red edge’s wavelength loca- tion can nonetheless be correctly retrieved for modern Earth-like planets even for Earth-like cloud coverage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' More generally, detecting wavelength-dependent sur- face albedos from reflection spectra offers the opportu- nity to constrain the surface composition of rocky exo- planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Other materials like sand, basalt, and granite, which cover substantial regions of Earth’s surface, have unique albedo profiles (see Figure 12) and shape the re- flection spectra of Earth-like exoplanets (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Mad- den & Kaltenegger 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' These profiles could poten- tially be extracted from spectra of rocky worlds whose surfaces are dominated by these materials (Pham & Kaltenegger 2021, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' However, for retrieval pur- poses the flexibility of our surface albedo parameteriza- tion (Equation 3) would need to be tested for these sur- face compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Our parameterization was inspired by the modern Earth’s surface albedo, hence it is able to locate sharp albedo changes such as the red edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Future work should investigate the flexibility of our parameter- ization for other surfaces, such as oceans or deserts, to determine if a generalized parameterization is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' An additional caveat is that our retrievals have fo- cused on zero orbital phase (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' full illumination), while future direct imaging observations will be constrained to higher phase angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The left panel of Figure 13 shows the impact of higher phase angles through PICASO cal- culations at orbital phases of 0◦, 30◦, 60◦, and 90◦ for the model in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We see that orbital phase acts to scale down Fp/F∗, while preserving the relative ampli- tude of the red edge relative to the surrounding contin- 16 Gomez Barrientos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 Wavelength( m) 0 1 2 3 4 5 Fp/Fs × 10−10 α = 0o α = 30o α = 60o α = 90o Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Left: Impact of different orbital phase angles on reflection spectra of an Earth-like exoplanet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The sequence of models have the same atmospheric and surface properties as Figure 2, but with partial illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Right: retrieved reflection spectrum of a cloud-free Earth-like exoplanet at orbital phase of 60◦ with a wavelength-dependent surface albedo (for simulated data at R = 70 and (S/N)ref = 20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The inset shows the posterior distribution of the retrieved red edge wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' uum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The right panel of Figure 13 shows the impact of a non-zero orbital phase on a retrieval of our cloud-free Earth-like exoplanet model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Due to the computational requirements of non-zero orbital phase retrievals, for this demonstration we consider only an orbital phase of 60◦ and observations with (S/N)ref = 20 and R = 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We see that the location of the vegetation red edge can still be constrained for non-zero phases, but the uncertainty becomes about 50% larger due to the lower flux ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Future retrieval studies could investigate the impact of a wide range of non-zero orbital phases (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Nayak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2017), or multi-phase observations (Damiano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Carri´on-Gonz´alez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2021), on wavelength-dependent surface albedo constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Nevertheless, our results show great promise for the detectability and character- ization of rocky exoplanet surfaces from reflection spec- tra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Opportunities for Detecting Surface Biosignatures The potential habitability of Earth-like exoplanets or- biting Sun-like stars could be assessed by constraining wavelength-dependent surface properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Future direct- imaging missions will focus on characterizing these at- mospheres to search for biosignature gas pairs like O3 combined with CH4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Our results suggest that such mis- sions can also retrieve surface spectral features, and that not accounting for wavelength-dependent surfaces can bias the retrieved abundances of biosignature gas pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We stress that including wavelength-dependent surface albedos in retrievals is an opportunity, since it enables an expanded mission science case including searches for sur- face biosignatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The vegetation red edge is one candi- date, but its universality remains uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Exoplanets could have reflectance edges at different wavelengths (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Kiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2007) or photosynthetic organisms that do not show red edge features (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Cockell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Some minerals also exhibit sharp spectral fea- tures near optical wavelengths (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', Seager et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' O’Malley-James & Kaltenegger 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Thus, any de- tection of reflectance edges would need to be carefully placed in context with other signatures of habitability before attributing a biological origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The promising detectability of the modern Earth’s red edge also suggests that other surface biosignatures would benefit from retrieval studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' One such alternative sur- face biosignature is biofluorescence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' On Earth, coral and other organisms absorb harmful shortwave radia- tion and re-emit it at longer wavelengths as a protec- tion mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Like the vegetation red edge, biofluo- rescence can dramatically increase a planet’s brightness at specific wavelengths (O’Malley-James & Kaltenegger 2019b, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Biofluorescence could therefore manifest as a time-dependent spectral edge that may be retriev- able from reflection spectra of Earth-like exoplanets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Constraining Atmospheric Properties of Earth-like Exoplanets via Reflected Light One of the primary science goals of future space- based observatories with direct imaging capabilities will be to characterize the atmospheres of Earth-like exo- planets orbiting Sun-like stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Our results indicate that a large space-based observatory with the capabil- ity to achieve Fp/F∗ ∼ 10−10 could precisely constrain the abundances of several biosignature pair gases on modern Earth analogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Assuming a cloud-free atmo- sphere, we showed that the O2, O3, and H2O abun- dances can be constrained within 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='6 dex for S/N = 10 and R = 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Crucially, the O3 abundance can be constrained to within a factor of 2 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='3 dex) even for S/N = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' However, it will be more challenging for these missions to detect biosignature gases with weak absorption features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' For instance, we could only place 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='006Retrieving Red Edges from Exoplanet Reflection Spectra 17 an upper limit on the CH4 mixing ratio for a modern Earth-like analog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Furthermore, clouds can also impact our ability to constrain and detect some biosignatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Clouds broaden the uncertainties in the retrieved molec- ular abundances of O2, O3, and H2O, which makes it more challenging to constrain their abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Our retrieval results suggest that meaningful infor- mation can be extracted from reflected light spectra of a modern Earth analog, even with a lower SNR than indicated by previous work, if the available data covers an expanded wavelength range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' (2018) found that S/N = 15 is generally a prerequisite to constrain the abundances of O2, O3, and H2O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' In comparison, we find that these gases can be precisely constrained for S/N ≥ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' These differences are mainly attributable to the wavelength range of the simulated data — we use 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='3–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 µm, while Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' (2018) considered 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4– 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The longer wavelength coverage in our retrievals decreased the minimum S/N necessary to constrain H2O because of the three additional water features at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4, and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='9 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Similarly, the short wavelength coverage of O3 absorption lowered the S/N necessary for constrain- ing the O3 abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' However, the difference between the minimum signal-to-noise ratios for retrieving the O2 abundance are: i) due to our differing noise models (we assumed an agnostic detector efficiency while Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' (2018) based their model on Roman Space Telescope- like detectors);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' and ii) due to the wavelength dependent surface features we added to the retrieval process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Characterizing Exoplanets Orbiting Sun-like Stars Ultimately, a key driver of exoplanet science is the characterization of potentially habitable planets around Sun-like stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The Astro 2020 Decadal Survey (Decadal Survey on Astronomy & Astrophysics 2020) specifically highlights the goal of searching for atmospheric biosig- natures on Earth-like exoplanets orbiting Sun-like stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Our findings indicate that a future direct imaging mis- sion observing reflected light could also detect surface biosignatures — including the vegetation red edge — via the retrieval of a wavelength-dependent surface albedo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Our odds of detecting life in the solar neighborhood can only be enhanced by considering all the ways life shapes its host planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' For a spectral edge encoded in light from a distant star may, one day, illuminate the surface of a world not too dissimilar to our own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We thank Natasha Batalha and Zifan Lin for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We also thank the anonymous referee for a helpful report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' is supported by Cornell Uni- versity’s Ronald E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' McNair Post-baccalaureate Achieve- ment Program and the Carl Sagan Institute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 1 2 3 4 5 Software: PICASO (Batalha et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2019) , corona- graph (Lustig-Yaeger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2019) DATA AVAILABILITY The realistic Earth-like surface albedo and the raw albedo files used in this work are available at https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='6977238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' RETRIEVALS WITH GAUSSIAN SCATTER Here, we show a retrieved reflection spectrum from synthetic observations with Gaussian scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Figure 14 shows that the retrieved spectra captures the overall spectral morphology of the input model and that the location of the vegetation red edge is still constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' RETRIEVAL MODEL VALIDATION Here, we validate our retrieval framework using the model Earth spectrum from Robinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' (2011), which has been validated against Earthshine data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The simulated data was generated by binning the model spectrum from its native resolution to R=70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' We then used our noise model to simulate data at (S/N)ref = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Figure 15 shows that our retrieval model reproduces the general spectral morphology of the Robinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' (2011) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' As with the cloudy 1D simulated data, the retrieved albedo is brighter than the model surface albedo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Nevertheless, our retrieval results indicate that the location of the red edge can still be constrained at (S/N)ref = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' FULL POSTERIOR DISTRIBUTIONS Here, we include full posterior distributions from representative retrievals of our modern Earth analog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Figure 16 shows the posterior distribution for a cloud-free retrieval at (S/N)ref = 10 and R = 70 including a wavelength- dependent surface albedo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Similarly, Figure 17 shows the posterior for the cloudy scenario discussed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 18 Gomez Barrientos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Top: retrieved reflection spectrum of a cloud-free Earth-like exoplanet with a wavelength-dependent surface albedo (for simulated data with gaussian scatter at R = 70 and (S/N)ref = 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The inset shows the posterior distribution of the retrieved red edge wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Bottom: The corresponding retrieved surface albedo profile (purple contours) compared to the true surface albedo (black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' REFERENCES Baldridge, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', Hook, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', Grove, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' I.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='40 Retrieved Albedo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='35 True Albedo l confidence 2o confidence 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='25 bedo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 Wavelength (μm)Retrieving Red Edges from Exoplanet Reflection Spectra 19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 1.' metadata={'source': 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data of the Robinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' (2011) model at R = 70 and (S/N)ref = 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Bottom: The corresponding retrieved surface albedo profile (purple contours) compared to a realistic Earth-like surface albedo (black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Carri´on-Gonz´alez, ´O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', Garc´ıa Mu˜noz, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', Cabrera, J.' metadata={'source': 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B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', Abbot, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', & Voigt, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2012, ApJL, 752, L3, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1088/2041-8205/752/1/L3 Damiano, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', & Hu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2020, AJ, 159, 175, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='3847/1538-3881/ab79a5 —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2021, AJ, 162, 200, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='3847/1538-3881/ac224d —.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2022, arXiv e-prints, arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='13816.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='org/abs/2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='13816 Damiano, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', Hu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', & Hildebrandt, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2020, AJ, 160, 206, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='3847/1538-3881/abb76a de Wit, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', Wakeford, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', Gillon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2016, Nature, 537, 69, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1038/nature18641 20 Gomez Barrientos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' logP0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='404+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='274 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='261 5 10 15 20 g g = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='506+7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='305 −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='519 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='20 Rp Rp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='860+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='101 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='080 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='76 λ1 λ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='723+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='008 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='16 As, 1 As, 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='101+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='012 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='32 As, 2 As, 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='205+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='041 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='038 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 λ2 λ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='420+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='332 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='107 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='8 As, 3 As, 3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='114+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='054 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='035 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='8 logO2 logO2 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='291+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='518 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='575 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 logO3 logO3 = −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='435+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='182 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='184 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='8 logH2O logH2O = −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='729+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='312 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='352 8 6 4 2 logCO2 logCO2 = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='402+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='114 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='693 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 logCH4 logCH4 = −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='962+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='808 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='017 8 6 4 2 logN2O logN2O = −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='904+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='443 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='733 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 logP0 200 300 400 T 5 10 15 20 g 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='20 Rp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='70 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='76 λ1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='16 As, 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='32 As, 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 λ2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='8 As, 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='8 logO2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 logO3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='8 logH2O 8 6 4 2 logCO2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 logCH4 8 6 4 2 logN2O 200 300 400 T T = 321+72 −63 Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The full posterior distribution for the cloud-free simulated data at R = 70 and (S/N)ref = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The retrieved median (solid blue lines) and 16th and 84th percentiles (dashed blue lines) for each parameter are overlaid for comparison with the ground truth reference values (green lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Note that the mixing ratio reference values represent the average value from the surface to 25 km altitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Similarly, the surface albedo parameter reference values represent the average albedo over a wavelength range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' Retrieving Red Edges from Exoplanet Reflection Spectra 21 logP0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='577+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='341 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='369 5 10 15 20 g g = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='517+7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='836 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='05 Rp Rp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='740+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='089 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 λ1 λ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='717+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='017 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='8 As, 1 As, 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='159+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='136 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='092 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='8 As, 2 As, 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='406+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='150 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='138 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 λ2 λ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='687+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='520 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='346 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='8 As, 3 As, 3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='241+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='234 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='151 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 logO2 logO2 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='949+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='551 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='662 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4 logO3 logO3 = −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='345+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='298 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='312 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 logH2O logH2O = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='119+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='466 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='538 8 6 4 2 logCO2 logCO2 = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='905+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='514 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='561 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 logCH4 logCH4 = −6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='779+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='681 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='044 8 6 4 2 logN2O logN2O = −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='929+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='281 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='576 200 300 400 T T = 329+90 −89 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='8 logτc logτc = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='312+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='751 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='722 1 0 1 logdp logdp = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='493+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='595 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='640 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='6 logP0 1 0 1 logpc 5 10 15 20 g 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='90 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='05 Rp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 λ1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='8 As, 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='8 As, 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 λ2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='8 As, 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 logO2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='4 logO3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 logH2O 8 6 4 2 logCO2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='5 logCH4 8 6 4 2 logN2O 200 300 400 T 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='8 logτc 1 0 1 logdp 1 0 1 logpc logpc = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='643+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='487 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='491 Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The full posterior distribution for the cloudy simulated data at R = 70 and (S/N)ref = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' The retrieved median (solid blue lines) and 16th and 84th percentiles (dashed blue lines) for each parameter are overlaid for comparison with the ground truth reference values (green lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 22 Gomez Barrientos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' de Wit, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', Wakeford, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', Lewis, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2018, Nature Astronomy, 2, 214, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='1038/s41550-017-0374-z Decadal Survey on Astronomy, & Astrophysics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' 2020, Pathways to Discovery in Astronomy and Astrophysics for the 2020s, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content='17226/26141 Des Marais, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} +page_content=', Harwit, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9AzT4oBgHgl3EQfzv7f/content/2301.01775v1.pdf'} diff --git a/P9FRT4oBgHgl3EQfJTdB/content/tmp_files/2301.13495v1.pdf.txt b/P9FRT4oBgHgl3EQfJTdB/content/tmp_files/2301.13495v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9b24e7a85d4b0e65ea0d05528170f7793e7609d2 --- /dev/null +++ b/P9FRT4oBgHgl3EQfJTdB/content/tmp_files/2301.13495v1.pdf.txt @@ -0,0 +1,4551 @@ +arXiv:2301.13495v1 [math.MG] 31 Jan 2023 +Dimension-free estimates on distances between subsets +of volume ε inside a unit-volume body +Abdulamin Ismailov1 +Alexei Kanel-Belov +Fyodor Ivlev +Abstract +Average distance between two points in a unit-volume body K ⊂ +Rn tends to infinity as n → ∞. However, for two small subsets of +volume ε > 0 the situation is different. For unit-volume cubes and +euclidean balls the largest distance is of order +√ +− ln ε, for simplexes +and hyperoctahedrons – of order − ln ε, for ℓp balls with p ∈ [1; 2] +– of order (− ln ε) +1 +p . These estimates are not dependent on the di- +mensionality n. The goal of the paper is to study this phenomenon. +Isoperimetric inequalities will play a key role in our approach. +Contents +1 +Introduction. +3 +2 +Preliminaries. +7 +3 +Euclidean balls. +9 +4 +Unit cubes. +17 +5 +Simplexes and ℓp-balls. +23 +5.1 +Simplexes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +23 +5.2 +ℓp-balls. +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +40 +6 +Lower bounds. +43 +7 +Discrete isoperimetric problem. +54 +8 +Conclusions. +57 +A Asymptotic behavior of Φ−1. +61 +1E-mail: nameundefinednamovich@gmail.com +1 + +B Function x(− log x)1− 1 +p is increasing on (0; 1 +2]. +63 +C Functions Vn(x) and Sn(x) in limit. +63 +D Average distance. +70 +2 + +1 +Introduction. +In high dimensions we observe a variety of different phenomena. +For ex- +ample, Vladimir Igorevich Arnold liked to ask his students the following +question: «What percent of the overall mass is occupied by the pulp of the +100-dimensional watermelon of diameter 1 meter, if the crust is of width 1 +centimeter?» The answer is approximately 1−e−1. This question in a simple +way demonstrates the concentration of measure phenomenon: how most of +the mass of a body could lie inside a thin shell. Here is another example, the +volume of a euclidean ball of radius 2023 tends to 0 as the dimensionality +goes to infinity. More generally, we have the isodiametric inequality, which +suggests that in high dimensions the diameter of a unit-volume body shall +become arbitrarily large. +The goal of this paper is to achieve a better understanding how things +work in high-dimensional spaces by studying the following phenomenon: two +points in a unit-volume convex body could be at an arbitrarily large distance +from each other; consider, for example, the unit cubes (0; 1)n – as n tends to ++∞ the diameter equal to √n also tends to infinity, similarly, the average +distance would be of order at least √n(see Appendix D); even the distance +between a point and a subset of some fixed volume ε < 1 could be arbitrarily +big, but it turns out that the distance between two subsets of some fixed +volume ε > 0 in the unit cube is bounded above by some constant dependent +on ε but not on the dimension n. What about convex bodies other than the +unit cubes? +Consider a family of unit-volume bounded convex bodies Kn. For each +Kn it makes sense to consider the supremum of all possible distances between +two subsets of some fixed volume ε ∈ (0; 1 +2). Denote this value by dn(ε). +By Φ we mean the function +Φ(a) = +� a +−∞ +e−πx2dx +Function Φ−1(ε) is asymptotically equivalent to +− 1 +√π +√ +− ln ε +as ε tends to 0(see Appendix A). +3 + +Theorem 6.2. When Kn are the unit-volume euclidean balls +lim +n→∞ dn(ε) = −2 1 +√eΦ−1(ε) +Theorems 4.3 and 6.3. When Kn are the unit cubes we have +−2 +�π +6 Φ−1(ε) ≤ lim inf +n→∞ dn(ε) ≤ lim sup +n→∞ dn(ε) ≤ −2Φ−1(ε) +Theorems 5.5 and 6.4. When Kn are the unit-volume simplexes we have +− +√ +2 +e ln(2ε) ≤ lim inf +n→∞ dn(ε) ≤ lim sup +n→∞ dn(ε) ≤ −c ln ε +for some universal constant c > 0 independent of n and ε. +Theorems 5.8 and 6.5. Fix some p ∈ [1; 2]. When Kn are the unit-volume +ℓp balls +−2Ψ−1 +p (ε) ≤ lim inf +n→∞ dn(ε) ≤ lim sup +n→∞ dn(ε) ≤ Cp(− ln ε) +1 +p, +where Cp is some universal constant determined by p, and function −2Ψ−1 +p (ε)(see +Appendix C) is asymptotically equivalent to +1 +e +1 +pΓ +� +1 + 1 +p +�(− ln ε) +1 +p +as ε → 0. +A version of our problem, in which the euclidean distance is replaced by +the Manhattan distance, can be approached by discretization. +Theorem 7.3. If by dn(ε) we denote the largest Manhattan distance between +two bodies of volume ε ∈ (0; 1 +2) in the unit cube [0; 1]n, then +lim +n→∞ +dn(ε) +√n = −2 +�π +6Φ−1(ε) +We also establish a sort of a general lower bound, showing that in a way +euclidean balls are optimal in regard to our problem. +4 + +Theorem 6.6. When Kn are unit-volume centrally symmetric bounded con- +vex bodies +−2 1 +√eΦ−1(ε) ≤ lim inf +n→∞ dn(ε) +Lower bounds on our problem could be derived simply by considering +some hyperplane cuts. But how can we bound above the distance between +two subsets A and B in a unit-volume convex body? +Well, first, we observe that if both A and B are of volume at least 1 +2, then +the distance between them is zero(see Lemma 2.1). That is why we assume +that both A and B are of some volume ε ∈ (0; 1 +2). Next we introduce the +concept of a δ-enlargement of a body defined as the set of all points at a +distance at most δ from our body, i. e. +Aδ = {x ∈ X | ∃y ∈ A: d(x, y) ≤ δ} +What happens if we replace A with its δ-enlargement for a small enough +value of δ? Roughly speaking, a layer of width δ will be added to our body. +The volume of this layer Aδ \ A could be approximated as δ · S(A), where +S(A) is the surface area of the body A. So the volume of A increases by +approximately δ·S(A), but the distance between bodies A and B will decrease +exactly by δ after we enlarge A(or might become zero). To estimate the +distance between A and B we will be slowly enlarging them simultaneously +until both bodies would be of volume 1 +2 at least, at which point the distance +between them is already zero. The double of the amount of time it took both +bodies to reach volume 1 +2 would be an upper bound on the distance between +them. +This was just a rough description of how we approach the problem. To +make this idea work we are going to need more. We have not said anything +about how our bodies may look like, at this point they could be arbitrary +subsets of volume ε, which may present a problem, since we plan to rely +on concepts such as surface area. In part, these issues might be mitigated +by the following observation: after enlarging both A and B by a little δ +distances and volumes would not change much, but smoothness properties +might improve. +Anyway, throughout this whole text we assume that the +bodies we are dealing with are as smooth as needed. +Now consider the process of a slow enlargement of a body A at its very +beginning. Instead of talking about the approximate volume of the layer +Aδ \ A it would be better to take the right derivative at the point δ = 0. +5 + +What we will get is called the Minkowski–Steiner formula for the free surface +area +µ+(A) = lim +δ→0+ +µ(Aδ) − µ(A) +δ +Thus it is vital to our approach to be able to estimate this surface area. But +we are only aware of the initial volumes of A and B, which leads us to the +isoperimetric problem: given the information about the initial volume of a +body, find a lower bound on its surface area. +Euclidean balls have really good symmetry properties. Symmetrization +techniques could be applied. Isoperimetric regions inside the euclidean balls +have been completely classified([16, Theorems 1 and 5]). This allows us to +get tight enough estimates that lead to the proof of Theorem 6.2. +Unit cubes, however, are not as good as euclidean balls in that regard. +That is why instead of dealing with the interior of the unit cube we perform +a transfer(Lemma 4.1) to a different space, where the situation with the +isoperimetric problem is better, and by doing so derive the lower bounds on +the initial space([16, Theorem 7]). +At last, to derive lower bounds in the case of simplex new ideas and +methods would need to be introduced. Here we repeat the approach from an +article by Sasha Sodin [19], where an isoperimetric inequality for ℓp balls with +p ∈ [1; 2] was proven. In particular, in case p = 1 we get hyperoctahedrons. +Theorem 5.8 is an immediate consequence of the isoperimetric inequality +established in article [19]. +Even though our method does provide asymptotically correct estimates, +we should not expect it to lead to exact constants. We bound the growth of +µ(Aδ) below by considering the isoperimetric problem for volume µ(Aδ), but +that might lead to suboptimal estimates, since as δ varies Aδ does not have +to look like an optimal isoperimetric region. +(a) region A +(b) region Aδ +(c) optimal region of volume µ(Aδ) +6 + +2 +Preliminaries. +Assume that we are working in the space X with metric d and probability +measure µ, i. e. µ(X) = 1. In this section we are going to introduce some +basic concepts related to our problem. +Definition 1. The distance between a pair of non-empty subsets A, B ⊆ X +is the infimum of distances between points from A and B +dist(A, B) = +inf +x∈A,y∈B d(x, y) +Definition 2. A point x ∈ X belongs to the δ-enlargement of a subset A ⊆ X +if it is at distance at most δ from some point of A +Aδ = {x ∈ X | ∃y ∈ A: d(x, y) ≤ δ} +We want to know how far apart from each other two subsets A, B ⊂ X +of measure µ(A) = µ(B) = ε > 0 could be. To that end, note that, if their +δ-enlargements intersect, then the distance is bounded above by 2δ +Aδ ∩ Bδ ̸= ∅ ⇒ dist(A, B) ≤ 2δ +Our problem is concerned with the case of X being an open convex +bounded subset of Rn of unit volume, d being the euclidean metric, and +µ being the Lebesgue measure. In that case the following lemma holds. +Lemma 2.1. If A and B are two subsets of X with µ(A) + µ(B) ≥ 1, then +they are at a distance 0 from each other +dist(A, B) = 0 +Proof. Assume the contrary. Let the distance between A and B be a positive +number +r = dist(A, B) > 0 +This would mean that our subsets do not intersect, and thus +µ(A) + µ(B) = µ(A ∩ B) + µ(A ∪ B) = µ(A ∪ B) = 1 +Pick a pair of points a ∈ A and b ∈ B at a distance less than 2r +d(a, b) < 2r +7 + +By c denote the midpoint of the segment between a and b. The distance +from c to both subsets A and B is strictly less than r, so the point c does +not belong to any of our subsets. Now consider a δ-neighborhood of c that +lies inside X with δ < r − d(a,b) +2 +. Clearly, it could not intersect neither A nor +B, and at the same time it has a non-zero measure, so +µ(X \ (A ∪ B)) > 0, +which leads to contradiction. +To ensure the nonemptiness of the intersection of Aδ and Bδ the following +condition would suffice +µ(Aδ) ≥ 1 +2 and µ(Bδ) ≥ 1 +2 +Thus we are interested in the growth of µ(Aε) considered as a function of ε, +since that might lead to an upper bound on δ and consequently on dist(A, B). +The derivative of µ(Aε) at ε = 0 gives us +Definition 3. By the surface area of A ⊆ X we mean the following limit +µ+(A) = lim +ε→0+ +µ(Aε) − µ(A) +ε +Lower bounds on µ+(A) might allow us to get results on the growth of +µ(Aε), but all we know is the measure µ(A) of our subset A. So we want to +know the least possible value of µ+(A) when µ(A) is fixed, or at least bound +µ+(A) below. +Definition 4. By the isoperimetric profile we mean a function that maps t +to the infimum of possible values that µ+(A) could take when µ(A) = t. +Iµ(t) = +inf +µ(A)=t µ+(A) +We are no longer interested in the growth of µ(Aε) after we reach the measure +of one half, also our initial µ(A) is greater than 0. This means that we are +only interested in the values of Iµ(t) when 0 < t < 1 +2. That is why throughout +this paper by default the domain of the isoperimetric profile is the interval +(0; 1 +2). +A region, which has the minimal surface area amongst all the regions +of the same measure, is called an isoperimetric region, and its boundary is +called an isoperimetric hypersurface. +8 + +3 +Euclidean balls. +Introduction. The euclidean ball is a perfect candidate for applying the +symmetrization techniques. An argument([16, Theorems 1 and 5]) involving +them completely classifies the optimal isoperimetric regions of the euclidean +ball. Lower bounds on the isoperimetric profile thus could be extracted by +considering these optimal regions. +By Bn we denote the unit n-ball. Its volume is +√πn +Γ +� n +2 + 1 +� +So the unit-volume n-ball will be of radius +ωn = Γ +�n +2 + 1 +� 1 +n +√π +∼ +� n +2πe +By µ denote the Lebesgue measure on ωnBn. +Combination of theorems 1 and 5 from [16] provides a classification of +optimal isoperimetric regions in ωnBn. +Theorem 3.1 ([16, Theorems 1 and 5]). Isoperimetric hypersurfaces in a +ball are either hyperplanes passing through the origin or spherical caps which +are orthogonal to the surface of ωnBn. +We would like to find lower bounds on the isoperimetric profile Iµ of +ωnBn, by Theorem 3.1 it would suffice to consider intersections with balls +orthogonal to ωnBn. +By Ψ(x) we denote Φ(√ex). Note that function Ψ(x) has a finite Lipschitz +constant C > 0, since its derivative is a bounded function. On the interval +(−∞; 0) both Ψ(x) and Ψ′(x) are increasing functions. +Theorem 3.2. For every ε0 ∈ (0; 1 +2) and τ > 0 there is a number N such +that the isoperimetric inequality +Iµ(Ψ(t) + τ) ≥ Ψ′(t) +would hold for all n > N and Ψ(t) ∈ (ε0; 1 +2 − τ). +9 + +Proof. First, note that Iµ is a non-decreasing function on the interval (0; 1 +2). +Indeed, if one would take a ball orthogonal to ωnBn whose intersection with +ωnBn is of volume ε ∈ (0; 1 +2) and replace it with a ball that has the same +radius but whose center is further away from the center of ωnBn, one would +get a region of ωnBn of smaller volume and smaller surface area. +Here we are going to prove that for any D, δ1, δ2 > 0 there is a number N +such that for every n > N and 0 < d ≤ D there is an optimal isoperimetric +region in ωnBn whose volume V and surface area S satisfy +Ψ(−d) + δ1 ≥ V +S ≥ Ψ′(−d − δ2) +Note that our theorem follows from this last claim. Indeed, in the state- +ment of the theorem we require Ψ(t) to be in range (ε0; 1 +2 − τ), which means +that t ≥ Ψ−1(ε0). Set D = −Ψ−1(ε0) and pick δ1, δ2 > 0 so that +Cδ2 + δ1 ≤ τ +t < Ψ−1 +�1 +2 − τ +� +≤ −δ2 +By our claim there will be a number N such that for every n > N and +d ∈ (0; D] there would be an optimal isoperimetric region in ωnBn of volume +V not greater than Ψ(−d)+δ1 and surface area S at least Ψ′(−d−δ2). Since +this region is optimal, +Iµ(V ) = S +And our bounds imply +Iµ(Ψ(−d) + δ1) ≥ Iµ(V ) = S ≥ Ψ′(−d − δ2) +(1) +For an arbitrary t satisfying Ψ(t) ∈ (ε0; 1 +2 −τ) we can set d = −t−δ2 ∈ (0; D], +but then +Ψ(−d) + δ1 = Ψ(t + δ2) + δ1 ≤ Ψ(t) + Cδ2 + δ1 ≤ Ψ(t) + τ +We combine this with (1) and get the desired isoperimetric inequality +Iµ(Ψ(t) + τ) ≥ Iµ(Ψ(−d) + δ1) ≥ Ψ′(−d − δ2) = Ψ′(t) +Now we need to prove our claim. Fix D > 0. We would be considering +the intersection of ωnBn and a ball Bn +r (A) orthogonal to it such that the +10 + +distance from the origin, which we will denote here as O, to Bn +r (A) is some +number d +2 in range (0; D +2 ]. +Two balls are orthogonal if their centers together with an arbitrary point +on the intersection of the corresponding spheres form a right triangle. +O +A +X +Y +N +H +M +Figure 2: The right triangle described above +In the figure above ball ωnBn corresponds to the circle with center at O +and radius OX, ball Bn +r (A) – to the circle with center at A and radius AX. +Since we have a right triangle, +OA = +√ +OX2 + AX2 = +� +ω2n + r2 +We require that ON = d +2 +ON = OA − AN = +� +ω2n + r2 − r = d +2 +� +ω2n + r2 = r + d +2 +ω2 +n = rd + d2 +4 +11 + +r = ω2 +n +d − d +4 +Consider the altitude XH of the right triangle OAX and note that +OH = OX2 +OA = +ω2 +n +� +ω2 +n + r2 = +1 +� +1 +ω2n + +� +r +ω2n +�2 = +1 +� +1 +ω2n + +� +1 +d − +d +4ω2n +�2 += +1 +� +1 +d2 + +1 +2ω2n + +d2 +16ω4n += +1 +�� +1 +d + +d +4ω2n +�2 = +1 +1 +d + +d +4ω2n += +d +1 + +d2 +4ω2n +≤ d +(2) +Let P be a hyperplane at a distance x from the origin. By Sn(x) denote +the volume of the hyperplane section of ωnBn by P. Hyperplane P cuts ωnBn +into two parts, at least one which is of volume not greater than 1 +2, denote that +volume by Vn(x). Both Vn(x) and Sn(x) are decreasing functions defined on +[0; ∞). +It follows from the proof of theorem 1 from [20] that the sequence of func- +tions Vn(x) uniformly converges to Ψ(−x) and that the sequence of functions +Sn(x) uniformly converges to Ψ′(−x)(see Appendix C). +On the interval [0; D] positive continuous function Ψ′(−x) − Ψ′(−x − δ2) +reaches its minimum value ε1 > 0. By uniform convergence for all sufficiently +large n we shall have +Sn(x) ≥ Ψ′(−x) − ε1, +which gives us +Sn(OH) ≥ Sn(d) ≥ Ψ′(−d) − ε1 ≥ Ψ′(−d − δ2) +for all d ∈ (0; D]. +We also know that Sn(OH) ≤ Sn(0) and Sn(0) converge towards Ψ′(0). +Thus Sn(OH) is always bounded above by some constant S0. +Hyperplane passing through the point H orthogonal to OA divides the +intersection of two balls ωnBn and Bn +r (A) into two spherical domes: Ω1 +belonging to ωnBn and Ω2 belonging to Bn +r (A). The volume of ωnBn∩Bn +r (A) +is equal to the sum of volumes of Ω1 and Ω2. +We would like to bound above the volume of Ω2. Its base, the hyperplane +section passing through H, has area not greater than S0. Fix some number +1 > ε2 > 0. On the segment HN pick a point M such that HM : HN = ε2. +12 + +By S1 denote the area of the hyperplane section of Bn +r (A) passing through +M orthogonal to OA. Volume of Ω2 could be bounded above as +S1NM + S0MH = S1NH(1 − ε2) + S0NHε2 +(3) +Radius of the (n − 1)-ball corresponding to S1 equals to +√ +AX2 − AM2 = +� +AX2 − (AH + HM)2 += +� +(AX2 − AH2) − 2AH · HM − HM2 ≤ +� +XH2 − 2ε2AH · HN +Radius of the (n−1)-ball corresponding to S0 is XH, so the ratio of the two +radii is +� +1 − 2ε2 +AH +XH2HN +Since XH is the altitude in the right triangle, XH2 is equal to OH · HA, +and the ratio could be rewritten as +� +1 − 2ε2 +HN +OH +Note that HN = OH − ON and that by formula (2) +� +1 − 2ε2 +HN +OH = +� +� +� +� +� +�1 − 2ε2 + + +1 − ON +2ON +1+ d2 +4ω2n + + + = +� +� +� +�1 − 2ε2 +� +1 − +1 + +d2 +4ω2n +2 +� +For all sufficiently large n +d2 +4ω2 +n +≤ 1 +2 ⇒ +� +� +� +�1 − 2ε2 +� +1 − +1 + +d2 +4ω2n +2 +� +≤ +� +1 − ε2 +2 +We conclude +S1 ≤ S0 +� +1 − ε2 +2 +� n−1 +2 +Clearly, NH ≤ OH ≤ d ≤ D, and by (3) the volume of Ω2 is not greater +than +DS0 +�� +1 − ε2 +2 +� n−1 +2 (1 − ε2) + ε2 +� +13 + +Note that we could pick ε2 ∈ (0; 1) so that for all sufficiently large n +DS0 +�� +1 − ε2 +2 +� n−1 +2 (1 − ε2) + ε2 +� +≤ δ1 +2 +(4) +The volume of Ω1 is Vn(OH). And by (2) +d − OH = d − +d +1 + +d2 +4ω2n += d · +d2 +4ω2n +1 + +d2 +4ω2n +≤ D3 +4ω2n +We noted that the sequence of functions Vn(x) uniformly converges to Ψ(−x). +Thus for all sufficiently large n +Ψ(−OH) + δ1 +4 ≥ Vn(OH) +Note that for all x ≥ 0 +0 < Ψ′(−x) ≤ Ψ′(0), +which means +Ψ(−d) + Ψ′(0) D3 +4ω2 +n ++ δ1 +4 ≥ Ψ(−OH) + δ1 +4 ≥ Vn(OH) +(5) +And for large enough n we have +Ψ′(0) D3 +4ω2n +≤ δ1 +4 +(6) +We conclude that the volume V of ωnBn ∩ Bn +r (A) is equal to the sum of +volumes of Ω1 and Ω2, and thus by inequalities (4), (5), (6) +Ψ(−d) + δ1 ≥ V +By Theorem 3.1 the intersection between ωnBn and Bn +r is an optimal +isoperimetric region in ωnBn. Its surface area S is equal to the surface area +of the spherical cap corresponding to Ω2, which can be bounded below by the +area of the base of Ω2, i. e. Sn(OH). We have thus shown that the volume V +and the free surface area S of ωnBn ∩ Bn +r (A) satisfy +Ψ(−d) + δ1 ≥ V +S ≥ Sn(OH) ≥ Ψ′(−d − δ2) +for all sufficiently large n and that the number d here can be chosen here as +an arbitrary number from (0; D], which proves our claim. +14 + +By dn(ε) denote the supremum of all possible distances between two sub- +sets of volume ε ∈ (0; 1 +2) inside ωnBn. +Using our isoperimetric inequality we derive +Theorem 3.3. For every ε ∈ (0; 1 +2) +lim sup +n→∞ dn(ε) ≤ −2 1 +√eΦ−1(ε), +where the function −2 1 +√eΦ−1(ε) is asymptotically equivalent to +−2 1 +√πe +√ +− ln ε +as ε → 0. +Proof. Pick any ε0 ∈ (0; ε) and τ ∈ (0; ε). By Theorem 3.2 there is a number +N such that inequality +Iµ(Ψ(t) + τ) ≥ Ψ′(t) +(7) +holds for all n > N and t such that Ψ(t) ∈ (ε0; 1 +2 − τ). +Assume that n > N. We consider two bodies A and B of volume ε inside +the unit-volume euclidean ball ωnBn. +We are interested in the least values δA, δB such that the δA-enlargement +of body A in ωnBn will be of volume 1 +2 and δB-enlargement of body B will +be of volume 1 +2 too. For these enlargements we shall have +dist(AδA, BδB) = 0, +from which +dist(A, B) ≤ δA + δB +follows. +Isoperimetric inequality (7) provides an estimate on the growth of δ- +enlargements of our bodies: +∂+µ(Aδ) ≥ Ψ′(Ψ−1(µ(Aδ) − τ)) +(8) +when µ(Aδ) ≤ 1 +2. +By δ0 denote Ψ−1(ε − τ). Now consider the function +y(δ) = Ψ(δ0 + δ) + τ +15 + +By δM denote the moment when y reaches 1 +2, i. e. y(δM) = 1 +2. Assume that +δM < δA. Functions µ(Aδ) and y(δ) coincide at δ = 0. Furthermore, because +of inequality (8), we should have +µ(Aδ) ≥ y(δ) and ∂+µ(Aδ) ≥ Ψ′(Ψ−1(µ(Aδ) − τ)) +≥ Ψ′(Ψ−1(y(δ) − τ)) = Ψ′(Ψ−1(Ψ(δ0 + δ))) = ∂+y(δ) +for all δ ∈ [0; δM]. But then we have a contradiction +1 +2 = µ(AδA) > µ(AδM) ≥ y(δM) = 1 +2 +Thus δA and similarly δB are bounded above by δM, which implies +dist(A, B) ≤ δA + δB ≤ 2δM +The value δM satisfies +1 +2 = y(δM) = Ψ(δ0 + δM) + τ +1 +2 − τ = Ψ(δ0 + δM) +δM = Ψ−1 +�1 +2 − τ +� +− Ψ−1 (ε − τ) +As τ tends to 0 +Ψ−1 +�1 +2 − τ +� +− Ψ−1 (ε − τ) → −Ψ−1(ε) = − 1 +√eΦ−1(ε) +And since we can choose τ to be an arbitrary number in (0; ε) +lim sup +n→∞ dn(ε) ≤ −2 1 +√eΦ−1(ε) +Conclusions. +Symmetrization techniques lead to the solution of the +isoperimetric problem in other different cases: the classical isoperimetric +problem in Rn; isoperimetric inequality on the sphere([8, Appendix], [10, +Theorem 2.2.1]), from which the gaussian isoperimetric inequality could be +derived([10, Theorem 2.2.3], [16, Theorem 20]). +16 + +4 +Unit cubes. +Introduction. Unlike the euclidean ball the cube does not have «many» +symmetries. To derive lower bounds on the isoperimetric profile we are going +to perform a «transfer» to a different space. Descriptions of this idea could +be found in [16, Theorem 7], [13, Proposition 2.8]. In this section we will +follow the approach presented in [16]. +Consider an n-dimensional unit cube (0; 1)n. We can think of it as of a +space with Lebesgue measure µ and Euclidean metric. Now we would like +to be able to show estimates on the isoperimetric profile of µ. However, it +is quite unclear how to deal with the corresponding space. +For example, +the cube only has a finite number of symmetries, so symmetrization methods +would not get us far. That is why it makes sense to consider a way to transfer +to a different, «better» space – an idea that plays a key role in our approach. +Proposition 4.1 ([16, Proposition 1]). Assume that for a pair of spaces M +and M′ with measures υ and υ′, respectively, we have a map φ : M → M′ +which transforms measure υ into υ′, i. e. µ′(A) = µ(φ−1(A)), and that is also +c-Lipschitz for some c > 0, i. e. a pair of points in M at a distance d has +images at distance at most c · d. The following inequality holds +Iυ ≤ c · Iυ′ +Proof. Consider a closed R′ ⊆ M′ and its preimage R = φ−1(R′). Since +φ transforms υ into υ′, we shall have υ′(R′) = υ(R). The fact that φ is +c-Lipschitz gives us φ(Rε) ⊆ R′ +cε, from which it follows that +υ′(R′ +cε) = υ(φ−1(R′ +cε)) ≥ υ(φ−1(φ(Rε))) ≥ υ(Rε) +By combining this inequality with υ′(R′) = υ(R) we get +υ′(R′ +cε) − υ′(R′) +cε +≥ υ(Rε) − υ(R) +cε +And by taking limit ε → 0 we reach conclusion +(υ′)+(R′) ≥ 1 +cυ+(R) +So for every closed R′ ⊆ M′ we can find R ⊆ M that has the same measure +and whose surface area is at most c times the surface area of R′. Thus we +shall have +Iυ(t) ≤ cIυ′(t) +17 + +We are going to apply the above lemma to get lower bounds on Iµ. The +role of M′ will play our cube with the Lebesgue measure µ on it. The role +of M will play the space Rn with Gaussian measure γn defined by its density +at a point x = (x1, . . . , xn) as +dγn +dx = e−π(x2 +1+...+x2 +n) +In the one-dimensional case the map Φ defined by +Φ(a) = +� a +−∞ +e−πx2dx +transforms (−∞; +∞) into (0; 1). +Also Φ turns Gaussian measure γ1 on +(−∞; +∞) into the Lebesgue measure on (0; 1). Indeed, the Gaussian mea- +sure of the segment [a; b] is equal to the integral +� b +a +e−πx2dx +of its density, which in turn is equal to Φ(b) − Φ(a), but the image of [a; b] +under our map Φ is [Φ(a); Φ(b)], whose Lebesgue measure is equal to Φ(b) − +Φ(a). +Now the role of φ in the above lemma will be played by +φ(x1, . . . , xn) = (Φ(x1), . . . , Φ(xn)), +i. e. we are applying Φ coordinatewise. It indeed transforms (−∞; +∞)n = +Rn into (0; 1)n. For every box [a1; b1] × . . . × [an; bn] we could note that +γn([a1; b1] × . . . × [an; bn]) = +� b1 +a1 +. . . +� bn +an +e−πx2 +1 · . . . · e−πx2 +ndxn . . . dx1 += +�� b1 +a1 +e−πx2 +1dx1 +� +. . . +�� bn +an +e−πx2 +ndxn +� += (Φ(b1) − Φ(a1)) . . . (Φ(bn) − Φ(an)) = µ(φ([a1; b1] × . . . × [an; bn])), +so φ turns measure γn into µ. And finally, map φ is 1-Lipschitz since |Φ′(x)| = +|e−πx2| ≤ 1 and φ applies Φ coordinatewise. +As we see the requirements of Proposition 4.1 are met. So the isoperi- +metric profile Iµ of our unit cube could be bounded below by Iγn. But what +do we know about Iγn? Well, there are tight inequalities on the isoperimet- +ric profile of Gaussian measures, but here we would only need the following +theorem. +18 + +Theorem 4.1 ([10, Lemma 2.2.2]). Let γn be the standard Gaussian measure +defined by its density at a point x = (x1, . . . , xn) ∈ Rn as +dγn +dx = +1 +√ +2π +ne− 1 +2(x2 +1+...+x2 +n) +Amongst all subsets A ⊂ Rn with fixed measure γn(A) ∈ (0; 1 +2) the minimum +surface area is attained at half-spaces. +In general, a Gaussian measure γn +µ,σ2 is a measure defined by its density +at a point x ∈ Rn as +dγn +µ,σ2 +dx += +1 +√ +2πσ2ne− +1 +2σ2 ∥x−µ∥2 +But Gaussian measures are equivalent to each other under translation and +scaling. For example, if we shrink the standard Gaussian measure γn by a +factor of +√ +2π, the density ρ′ of the resulting measure is related to the density +ρ of γn as +ρ′(x) = +√ +2π +nρ( +√ +2πx) = +√ +2π +n +1 +√ +2π +ne− 1 +2 ∥ +√ +2πx∥2 = dγn +dx +And we could also note that for A ⊂ Rn +γn(A) = γn +� +1 +√ +2π +A +� +(γn)+(A) = +1 +√ +2π +γ+ +n +� +1 +√ +2π +A +� +So shrinking everything by a factor of +√ +2π in Theorem 4.1 would not change +the fact that half-spaces are optimal solutions to the isoperimetric problem. +And that is why to figure out lower bounds on Iγn we would only need to +consider half-spaces. +The density of γn at a point x only depends on ∥x∥, so our measure is +rotation-invariant, which means that we could only consider half-spaces Ha +defined by xn ≤ a for some a. First, we could note that +γn(Ha) = γn((−∞; +∞) × . . . × (−∞; +∞) +� +�� +� +n−1 times +×(−∞; a]) += γ1((−∞; +∞)) × . . . × γ1((−∞; +∞)) +� +�� +� +n−1 times +×γ1((−∞; a]) += γ1((−∞; a]) = Φ(a) +19 + +And, because of this last observation, +γ+ +n (Ha) = lim +ε→0 +γn((Ha)ε) − γn(Ha) +ε += lim +ε→0 +γn(Ha+ε) − γn(Ha) +ε += lim +ε→0 +γ1((−∞; a + ε]) − γ1((−∞; a]) +ε += lim +ε→0 +γ1((−∞; a]ε) − γ1((−∞; a)) +ε += γ+ +1 ((−∞; a]) +Equalities +γn(Ha) = γ1((−∞; a]) +γ+ +n (Ha) = γ+ +1 ((−∞; a]) +imply +Iγn = Iγ1 +And now we only have to estimate Iγ1. +To do that we need to consider +intervals (−∞; a] for a < 0. The measure of such an interval is Φ(a) and the +surface area is Φ′(a) = e−πa2, from which we get +Iγ1(Φ(a)) = e−πa2 +By combining our observations we conclude +Theorem 4.2 ([16, Theorem 7]). For the Lebesgue measure µ on the unit +cube (0; 1)n isoperimetric inequality +Iµ(t) ≥ e−πΦ−1(t)2 +holds for all t ∈ (0; 1 +2). +Proof. Note that for all a < 0 +Iµ(Φ(a)) ≥ Iγn(Φ(a)) = Iγ1(Φ(a)) = e−πa2 +Using this isoperimetric inequality we derive +Theorem 4.3. Inside a unit cube (0; 1)n two bodies A and B of volume +ε ∈ (0; 1 +2) are at a distance at most +−2Φ−1(ε) +20 + +Function −2Φ−1(ε) is asymptotically equivalent to +2 +√π +√ +− ln ε +as ε → 0. +Proof. We are interested in the least values δA, δB such that the δA-enlargement +of body A in the unit cube (0; 1)n will be of volume 1 +2 and the δB-enlargement +of body B will be of volume 1 +2 too. For these enlargements we shall have +dist(AδA, BδB) = 0, +from which +dist(A, B) ≤ δA + δB +follows. +Isoperimetric inequality from Theorem 4.2 provides an estimate on the +growth of δ-enlargements of our bodies: +∂+µ(Aδ) ≥ e−πΦ−1(µ(Aδ))2 +(9) +when µ(Aδ) < 1 +2. +By δM denote −Φ−1(ε). Now consider the function +y(δ) = Φ(−δM + δ) +Assume that δM < δA. Functions µ(Aδ) and y(δ) coincide at δ = 0. Further- +more, because of inequality (9), we should have +µ(Aδ) ≥ y(δ) and ∂+µ(Aδ) ≥ e−πΦ−1(µ(Aδ))2 ≥ e−πΦ−1(y(δ))2 += e−π(−δM +δ)2 = ∂+y(δ) +for all δ ≤ δM. But then we have a contradiction +1 +2 = µ(AδA) > µ(AδM) ≥ y(δM) = Φ(0) = 1 +2 +Thus δA and similarly δB are bounded above δM, which implies +dist(A, B) ≤ δA + δB ≤ 2δM = −2Φ−1(ε) +21 + +Conclusions. The transfer from one space to another might have lead +to the loss of accuracy to some extent, so there is not much of what we could +say about how precise our estimates are. The problem of finding optimal +hypersurfaces in the n-cube also seems to be quite complicated. More de- +tails about the isoperimetric inequalities in a cube and their applications the +reader may find in section 1.5 of [16]. +In a similar way we could derive lower bounds on the isoperimetric profile +of the euclidean ball, since the transition to the space Rn with gaussian +measure is possible(see [13, Proposition 2.9]). +22 + +5 +Simplexes and ℓp-balls. +Introduction. In this section we are going to describe the approach used in +[19] to prove an isoperimetric inequality for ℓp balls(p ∈ [1; 2]) by presenting +a very similar proof of an isoperimetric inequality for simplexes. Much like +in the case of the unit cube we will be performing a transfer to a different +space(see Lemmas 5.1, 5.2). +But to address the problems with Lipschitz +continuity(see Lemma 5.3) a number of new ideas and methods needs to be +introduced. +5.1 +Simplexes. +By R+ we denote the interval (0; +∞). Consider a regular simplex ∆n defined +as +∆n = {(x1, . . . , xn) ∈ Rn ++ | x1 + . . . + xn = 1} +By µ we will denote the normalized Lebesgue measure on ∆n. Note that +µ(∆n) = 1. Simple calculations show that the area of ∆n is equal to n√n +n! . So +if we set +ωn = +� n! +n√n +� +1 +n−1 +∼ n +e , +the area of ωn∆n will be equal to 1. By λ denote the Lebesgue measure on +ωn∆n. One could note +µ+ +� 1 +ωn +A +� += lim +ε→0 +µ +�� +1 +ωnA +� +ε +� +− µ +� +1 +ωnA +� +ε += lim +ε→0 +λ(Aωnε) − λ(A) +ε += ωnλ+(A) +(10) +for A ⊆ ∆n. +In this section we will be using a slightly different notion of an isoperi- +metric profile. +Definition 5. By the isoperimetric function we mean a function that maps +t ∈ (0; 1 +2) to the infimum of possible values that µ+(A) could take when t ≤ +µ(A) < 1 +2 +Iµ(t) = +inf +t≤µ(A)< 1 +2 +µ+(A) +23 + +Our observation (10) implies +Iµ = ωnIλ, +(11) +i. e. the isoperimetric functions are proportional. +To solve our problem we would need estimates on Iλ, but for the sake of +simplicity we would be working with Iµ instead. Yet again it is quite unclear +how to deal with ∆n as a space, so we would like to be able to transfer to a +«better» space. +The map T : Rn ++ → ∆n defined as +T(x1, . . . , xn) = +� +x1 +x1 + . . . + xn +, . . . , +xn +x1 + . . . + xn +� +transforms the measure νn on Rn ++ defined by its density at a point (x1, . . . , xn) ∈ +Rn ++ as +dνn +dx = e−x1−...−xn +into the normalized Lebesgue measure µ on ∆n as a corollary of the following +lemma. +Lemma 5.1 ([18, Lemma 2.1]). Let X1, . . . , Xn be independent random +variables each with density function +1 +2e−|t| and put S = � +i |Xi|. +Then +( X1 +S , . . . , Xn +S ) induces the normalized Lebesgue measure on the surface of ℓn +1 +ball. Moreover, ( X1 +S , . . . , Xn +S ) is independent of S. +Indeed, because of the symmetry amongst the orthants of Rn,2 we can restrict +our attention to the positive orthant Rn ++ in Lemma 5.1 and reach the desired +conclusion. +But what do we know about the isoperimetric profile of Iνn? The fol- +lowing lemma completely determines this isoperimetric profile in the one- +dimensional case. +Lemma 5.2 ([21, Remark 1]). By ν denote ν1, then +Iν(t) = min(t, 1 − t), +where the domain of Iν is the whole interval (0; 1). +2the orthants of Rn are multidimensional analogues of the quadrants of R2 +24 + +For a measure υ we could consider its isoperimetric constant – the largest +value Is(υ) for which the following holds for all subsets A with υ(A) ∈ (0; 1) +υ+(A) ≥ Is(υ) min(υ(A), 1 − υ(A)) +And by Lemma 5.2 we have Is(ν) = 1. Now we could note that3 νn = νn +since the density of νn at a point (x1, . . . , xn) ∈ Rn ++ could be written as a +product +e−x1 . . . e−xn +And thus the following theorem +Theorem 5.1 ([2, Theorem 1.1]). For triple (X, d, ψ) – space, metric, mea- +sure, +Is(ψn) ≥ +1 +2 +√ +6 +Is(ψ) +gives us +Iνn(t) ≥ +1 +2 +√ +6 min(t, 1 − t) +or4 +Iνn(t) ≥ +1 +2 +√ +6t +(12) +So we have a map T : Rn ++ → ∆n that transforms νn into µ and a lower +bound on Iνn(t). But to use Proposition 4.1 we would also need our mapping +T to be Lipschitz continuous. +In the neighborhood of a point x ∈ Rn ++ the behavior of our map T could +be described by a linear operator defined by the matrix, whose entries are +∂Tj(x) +∂xi +, +where Tj(x) is the j-th coordinate of T(x). We are interested in the norm of +this linear operator. The next lemma gives an upper bound +Lemma 5.3 (corresponds to Lemma 1 from [19]). +���� +∂Tj(x) +∂xi +���� +2 +≤ +1 +∥x∥1 +(1 + √n∥T(x)∥2) +3by this we mean the product measure ν × . . . × ν +� +�� +� +n times +4recall that, generally, when we talk about isoperimetric profiles we are only interested +in the values of t ∈ (0; 1 +2), i. e. the domain of Iνn is (0; 1 +2) +25 + +Proof. First, we will calculate the entries of our matrix, i. e. the partial deriva- +tives +∂ +∂xi +Tj(x) = ∂ +∂xi +xj +x1 + . . . + xn += +(x1 + . . . + xn) ∂ +∂xixj − xj +∂ +∂xi(x1 + . . . + xn) +(x1 + . . . + xn)2 += +1 +x1 + . . . + xn +� +δij − +xj +x1 + . . . + xn +� +If ∆y is the image of ∆x, then +∆yj = +� +i +∂Tj +∂xi +∆xi = +1 +∥x∥1 +� +∆xj − +xj +∥x∥1 +� +i +∆xi +� +(13) +The length of the vector, whose coordinates are +xj +∥x∥1 +� +i ∆xi, could be esti- +mated as +∥T(x)∥2 +����� +� +i +∆xi +����� ≤ √n∥T(x)∥2∥∆x∥2 +since coordinates +xj +∥x∥1 define T(x) and, clearly, +� +i +|∆xi| ≤ √n +�� +i +(∆xi)2 +� 1 +2 +And now by triangle inequality from (13) we get +∥∆y∥2 ≤ +1 +∥x∥1 +� +∥∆x∥2 + √n∥T(x)∥2∥∆x∥2 +� += +1 +∥x∥1 +� +1 + √n∥T(x)∥2 +� +∥∆x∥2, +from which the statement of the lemma follows. +Indeed, we cannot apply Proposition 4.1 in our case since there are places +where the image of the mapping T varies wildly. For example, the simplex +in Rn ++ defined by x1 + . . . + xn = δ, where δ is a very small number. +26 + +However, the upper bound from Lemma 5.3 +���� +∂Tj(x) +∂xi +���� +2 +≤ +1 +∥x∥1 +(1 + √n∥T(x)∥2) +tells us that the only parts of Rn ++, where the variance of T could be high, are +the regions where ∥x∥1 is too small, which corresponds to a little corner of +the orthant Rn ++, or where ∥T(x)∥2 is too large, and, since the distance from +the origin to the center of ∆n – +1 +√n is much smaller than the distance from +the origin to the vertices of ∆n – 1, the region where ∥T(x)∥2 is too large +corresponds to the little corners of ∆n near the vertices.5 +Our last observation suggests that the parts of our space, where T does +not behave the way we want it to, i. e. high variance, might be negligible. +And now we are going to present an argument that will allow us to «get rid» +of these regions, perform the transfer to the space Rn ++ with measure νn and +then apply the lower bound for Iνn. +But, first of all, since we want to estimate Iµ(t) on the whole half-interval +(0; 1 +2], including the small values of t, and since we are intending to remove +some negligible parts of our space in the main argument, we need to employ +a different approach for those values of t that might, perhaps, be lesser than +the measure of the regions that we are getting rid of. In other words, an +estimate on the surface area of the «small» subsets of ∆n. +This could be done with the help of the following theorem. +Theorem 5.2 ([1, Theorem 1.1]). Let ψ be a log-concave probability measure +on Rn. For all measurable sets A ⊂ Rn, for every point x0 ∈ Rn and every +number r > 0, +ψ+(A) ≥ 1 +2r +� +ψ(A) ln +1 +ψ(A) + (1 − ψ(A)) ln +1 +1 − ψ(A) ++ ln ψ({|x − x0| ≤ r}) +� +Convexity of ∆n ensures that the normalized Lebesgue measure µ on it is +log-concave. +In the application of the above theorem to measure µ on ∆n it is possible +to set x0 = (0, . . . , 0), even though x0 /∈ ∆n, since when r > +1 +√n for x1 = +5here we are talking about the image of T +27 + +( 1 +n, . . . , 1 +n) and r′ = +� +r2 − 1 +n we shall have +ln µ({|x| ≤ r}) = ln µ({|x − x1| ≤ r′}) +1 +2r′ ≥ 1 +2r +One question that arises after examining the lower bound is: how do we +estimate ψ({|x − x0| ≤ r}) – the measure of the ball with center at x0 and +radius r? In terms of Lemma 5.1 we have the following result +Theorem 5.3 ([18, Theorem 2.2]). There are absolute positive constants T, c +such that for all t > +T +√n, putting X = (X1, . . . , Xn) and S = X1 + . . . + Xn, +Pr +�∥X∥2 +S +> t +� +≤ e−ctn +Again, because of the symmetry, we can restrict our attention to the positive +orthant Rn ++. +By combining Theorem 5.3 with Lemma 5.1 we get +µ({|x| ≤ r}) ≥ 1 − e−cnr +for r > +T +√n. Now note +r > T +√n ⇒ cnr ≥ c√nT ≥ cT ⇒ e−cnr ≤ e−cT, +which implies that for r > +T +√n +ln µ({|x| ≤ r}) ≥ ln(1 − e−cnr) ≥ −Ce−cnr +for some constant C > 0. +If we assume that µ(A) < c′ for some constant 0 < c′ < 1, then we will +have +(1 − µ(A)) ln +� +1 +1 − µ(A) +� +≥ C1µ(A) +for some constant C1 > 0. +We would like the sum of the two last terms of +µ(A) ln +1 +µ(A) + (1 − µ(A)) ln +1 +1 − µ(A) + ln µ({|x − x0| ≤ r}) +28 + +to be non-negative, for this the following will be sufficient +C1µ(A) ≥ Ce−cnr, +which could be rewritten as +ln C1µ(A) +C +≥ −cnr ⇔ r ≥ 1 +cn ln +C +C1µ(A) = 1 +cn +� +ln +1 +µ(A) + ln C +C1 +� +If we replace ≥ above with equality and apply Theorem 5.2 with x0 = 0, we +will get +µ+(A) ≥ 1 +2cnµ(A) +ln +1 +µ(A) +ln +1 +µ(A) + ln C +C1 +The condition r > +T +√n means that we need +ln +1 +µ(A) + ln C +C1 +> cT√n +to apply Theorem 5.2 here. And so these last inequalities imply +Proposition 5.1. There are universal constants cs, C > 0 such that 6 +µ+(A) ≥ csnµ(A) +for all subsets A ⊂ ∆n with µ(A) < e−C√n. +Now that we have dealt with the case of «small» sets, we can proceed to +the main argument. +Definition 6. The gradient modulus ∥∇f∥2 of a locally Lipschitz function f +is +∥∇f(x)∥2 = lim sup +∥x−y∥2→0+ +∥f(x) − f(y)∥2 +∥x − y∥2 +The next lemma would allow us to switch between the two equivalents of +the isoperimetric problem. +6here for the sake of simplicity we forget about the auxiliary constants c′, C, C1 intro- +duced during the proof of this proposition +29 + +Lemma 5.4 ([19, Proposition A]). Let µ be a probability measure, 0 < a < 1 +2 +and b > 0. The following are equivalent +(a) Iµ(a) ≥ b +(b) for any locally Lipschitz function φ : supp µ → [0; 1] such that µ{φ = +0} ≥ 1 +2 and µ{φ = 1} ≥ a, +� +∥∇φ∥2dµ ≥ b +Remark 1. In this theorem and its applications in this article we can replace +«locally Lipschitz» in (b) with just «Lipschitz», since it would not affect the +implication (a) ⇒ (b), and in the proof([19]) of the implication (b) ⇒ (a) +only Lipschitz functions φ: supp µ → [0; 1] were considered. +The idea of «getting rid» of unwanted parts of our space would be realized +through the so-called cut-off functions. A cut-off function maps our space +to [0; 1], where 0 corresponds to the regions we are getting rid of, 1 – to the +regions we want to keep, and we would also need our function to take values +in between 0 and 1 to ensure continuity. +The following lemma shows how a cut-off function h can be applied. +Lemma 5.5 ([19, Lemma 2]). If k, h: Rn → [0; 1] are two locally Lipschitz +functions, then +∥∇k∥2 ≥ ∥∇(kh)∥2 − ∥∇h∥2 +By Lemma 5.4 we can speak about the isoperimetric problem in terms of the +integral of gradient modulus ∥∇φ∥2. Lemma 5.5 would allow us to pass from +φ to function φ · h, which vanishes on the unwanted region of our space, at +the cost of an error term ∥∇h∥2 +� +∥∇φ∥2dµ ≥ +� +∥∇(φ · h)∥2dµ − +� +∥∇h∥2dµ +Recall that our upper bound on gradient modulus of mapping T : Rn ++ → +∆n is +∥∇T∥2 = +���� +∂Tj(x) +∂xi +���� +2 +≤ +1 +∥x∥1 +(1 + √n∥T(x)∥2) +So we are going to need two cut-off functions: one for parts of ∆n that are +too far from the origin will be of the form +h1 : Rn → [0; 1], +h1(x) = max(0, min(1, 2 − c1 +√n∥x∥2)), +30 + +and will take care of large values of ∥T(x)∥2; another for the region of Rn +with low ∥x∥1 will be of the form +h2 : Rn → [0; 1], +h2(x) = max(0, min(1, c2n−1∥x∥1 − 1)). +Constants c1 and c2 will be chosen later. Note that the two cut-off functions +are meant for different domains: h1 for ∆n and h2 for Rn ++, but since both Rn ++ +and ∆n lie inside Rn we choose Rn as their domain of definition. +To employ Lemma 5.5 in our argument we are going to need some es- +timates related to the arising error terms. +In the following lemma some +properties of our cut-off functions and their gradients will be established. +Lemma 5.6 (corresponds to Lemma 3 from [19]). The cut-off function h1 +has the following properties +h1(x) = 1 ⇔ ∥x∥2 ≤ +1 +c1 +√n +(14) +h1(x) = 0 ⇔ ∥x∥2 ≥ +2 +c1 +√n +(15) +∥∇h1∥2 ≤ c1 +√n +(16) +The cut-off function h2 has the following properties +h2(x) = 1 ⇔ ∥x∥1 ≥ 2 +c2 +n +(17) +h2(x) = 0 ⇔ ∥x∥1 ≤ n +c2 +(18) +∥∇h2∥2 ≤ c2 +√n +(19) +Proof. Properties (14), (15), (17), (18) immediately follow from the definition +of our cut-off functions. +By the triangle inequality we note that the gradient modulus of ∥x∥2 +considered as a function from Rn ++ to R+ is not greater than 1, thus +∥∇h1∥2 ≤ c1 +√n +Inequalities +∥x + ∆x∥1 ≤ ∥x∥1 + ∥∆x∥1, +31 + +∥∆x∥1 ≤ √n∥∆x∥2 +imply that the gradient modulus of ∥x∥1 is not greater than √n, from which +we derive +∥∇h2∥2 ≤ c2 +√n +Lemma 5.7 (corresponds to Lemma 4 from [19]). For α ≥ 0 we have +νn{∥x∥1 ≤ αn} ≤ +1 +√ +2πn(αe)n +And for every α > T by Lemma 5.3 we have +µ +� +∥x∥2 ≥ α +√n +� +≤ e−αc√n +Proof. Since the density of νn everywhere in Rn ++ is not greater than 1, we +can bound νn{∥x∥1 ≤ αn} above by the volume of the region of Rn ++ defined +by ∥x∥1 ≤ αn, which is equal to +1 +n!(αn)n +By Stirling’s approximation +n! ≥ +√ +2πn +�n +e +�n +e +1 +12n+1 +And so we arrive at +νn{∥x∥1 ≤ αn} ≤ 1 +n!(αn)n ≤ +1 +√ +2πn +� e +n +�n +(αn)ne− +1 +12n+1 ≤ +1 +√ +2πn(αe)n +Note that +α +√n > +T +√n, so by Theorem 5.3 +µ +� +∥x∥2 ≥ α +√n +� +≤ e−c α +√nn = e−αc√n +Now we are ready to present the main argument. +32 + +Proposition 5.2 (corresponds to Proposition 2 from [19]). There is a uni- +versal constant cb > 0 such that for all e−C√n ≤ t < 1 +2 +Iµ(t) ≥ cbnt. +Proof. Pick e−C√n ≤ a < 1 +2. According to Lemma 5.4 and Remark 1 the +problem of finding lower bounds on Iµ(a) is equivalent to the estimation of +� +∆n +∥∇f∥2dµ +for a Lipschitz function f : ∆n → [0; 1] such that +µ{f = 0} ≥ 1 +2 and µ{f = 1} ≥ a +(20) +To «get rid» of the parts of ∆n that are too far from the origin we can +use our cut-off function h1 and by Lemma 5.5 we will get +� +∆n +∥∇f∥2dµ ≥ +� +∆n +∥∇(fh1)∥2dµ − +� +∆n +∥∇h1∥2dµ +(21) +Here by Lemma 5.6 we can estimate the error term as +� +∆n +∥∇h1∥2dµ ≤ c1 +√n µ +� +∥x∥2 ≥ +1 +c1 +√n +� +(22) +Mapping T : Rn ++ → ∆n transforms measure νn into µ, which allows us to +replace integrals over ∆n with integrals over Rn ++ as follows +� +∆n +w dµ = +� +Rn ++ +(w ◦ T)dνn +Denote (fh1) ◦ T by g. As we already noted +� +∆n +∥∇(fh1)∥2dµ = +� +Rn ++ +∥∇(fh1) ◦ T∥2dνn +(23) +An observation similar to the chain rule of differentiation could be made +∥∇(a ◦ b)∥2 ≤ ∥(∇a) ◦ b∥2 · ∥∇b∥2, +33 + +which in our case would mean that +� +Rn ++ +∥∇(fh1) ◦ T∥2dνn ≥ +� +Rn ++ +∥∇g∥2 +∥∇T∥2 +dνn, +(24) +and since ∥∇T∥2 ̸= 0 by Lemma 5.3 could be bounded above by +1 +∥x∥1(1 + +√n∥T(x)∥2) we have +� +Rn ++ +∥∇g∥2 +∥∇T∥2 +dνn ≥ +� +Rn ++ +∥∇g∥2∥x∥1 +1 + √n∥T(x)∥2 +dνn +(25) +But h1 is zero when ∥x∥2 ≥ +2 +c1 +√n by Lemma 5.6. So the gradient modulus +∥∇g∥2 is equal to zero when ∥T(x)∥2 > +2 +c1 +√n. And because of this, +� +Rn ++ +∥∇g∥2∥x∥1 +1 + √n∥T(x)∥2 +dνn ≥ +� +Rn ++ +∥∇g∥2∥x∥1 +1 + √n +2 +c1 +√n +dνn += +1 +1 + 2 +c1 +� +Rn ++ +∥∇g∥2∥x∥1dνn +(26) +Now to «get rid» of the region of Rn ++ where ∥x∥1 is too small we will +apply our cut-off function h2 and by Lemma 5.5 get +� +Rn ++ +∥∇g∥2∥x∥1dνn ≥ +� +Rn ++ +∥∇(gh2)∥2∥x∥1dνn − +� +Rn ++ +∥∇h2∥2∥x∥1dνn +(27) +By Lemma 5.6 we have the following upper bound on the error term +� +Rn ++ +∥∇h2∥2∥x∥1dνn ≤ c2 +√n +2n +c2 +νn +� +∥x∥1 ≤ 2n +c2 +� += 2√nνn +� +∥x∥1 ≤ 2n +c2 +� +(28) +Cut-off function h2 is zero when ∥x∥1 ≤ n +c2, thus ∥∇(gh2)∥2 is equal to zero +when ∥x∥1 < n +c2, from which it follows that +� +Rn ++ +∥∇(gh2)∥2∥x∥1dνn ≥ n +c2 +� +Rn ++ +∥∇(gh2)∥2dνn +(29) +34 + +Now consider function gh2: Rn ++ → [0; 1] +gh2 = ((f · h1) ◦ T) · h2 +Note that if (f ◦ T)(x) = 0 for x ∈ Rn ++, then (gh2)(x) = 0 too. By our +assumption µ{f = 0} ≥ 1 +2, which implies νn{gh2 = 0} ≥ 1 +2. Function gh2 +equals to 1 at a point x ∈ Rn ++ if and only if +(f ◦ T)(x) = 1, and (h1 ◦ T)(x) = 1, and h2(x) = 1. +To estimate νn{gh2 = 1} we will subtract νn{(h1 ◦ T) < 1} = µ{h1 < 1} and +νn{h2 < 1} from νn{(f ◦T) = 1} = µ{f = 1}, which by our assumption (20) +is greater than a, and get +νn{gh2 = 1} ≥ a − µ{h1 < 1} − νn{h2 < 1} +By Lemma 5.6 +µ{h1 < 1} = µ +� +∥x∥2 > +1 +c1 +√n +� +νn{h2 < 1} = νn +� +∥x∥1 < 2n +c2 +� +Isoperimetric inequality (12) on νn combined with Lemma 5.4 would give us +� +Rn ++ +∥∇(gh2)∥2dνn ≥ +1 +2 +√ +6 +� +a +− µ +� +∥x∥2 > +1 +c1 +√n +� +− νn +� +∥x∥1 < 2n +c2 +� � +(30) +Putting inequalities (21), (22), (23), (24), (25), (26), (27), (28), (29) +together, we arrive at +� +∆n +∥∇f∥2dµ ≥ 1 +c2 +1 +1 + 2 +c1 +n +� +Rn ++ +∥∇(gh2)∥2dνn +− c1 +√nµ +� +∥x∥2 ≥ +1 +c1 +√n +� +− +2 +1 + 2 +c1 +√nνn +� +∥x∥1 ≤ 2n +c2 +� +35 + +We combine this with inequality (30) and get +� +∆n +∥∇f∥2dµ ≥ +1 +2 +√ +6 +1 +c2 +1 +1 + 2 +c1 +na +− +� +1 +2 +√ +6 +1 +c2 +1 +1 + 2 +c1 +n + c1 +√n +� +µ +� +∥x∥2 ≥ +1 +c1 +√n +� +− +� +1 +2 +√ +6 +1 +c2 +1 +1 + 2 +c1 +n + +2 +1 + 2 +c1 +√n +� +νn +� +∥x∥1 ≤ 2n +c2 +� +, +which could be rewritten as +� +∆n +∥∇f∥2dµ ≥ +1 +2 +√ +6 +1 +c2 +1 +1 + 2 +c1 +n +� +a +− +� +1 + 2 +√ +6c1c2 +� +1 + 2 +c1 +� 1 +√n +� +µ +� +∥x∥2 ≥ +1 +c1 +√n +� +− +� +1 + 4 +√ +6c2 +1 +√n +� +νn +� +∥x∥1 ≤ 2n +c2 +� � +If +1 +c1 > T ⇔ c1 < 1 +T , then by Lemma 5.7 we should have +� +∆n +∥∇f∥2dµ ≥ +1 +2 +√ +6 +1 +c2 +1 +1 + 2 +c1 +n +� +a +− +� +1 + 2 +√ +6c2 (c1 + 2) 1 +√n +� +e− c +c1 +√n +− +� +1 + 4 +√ +6c2 +1 +√n +� +1 +√ +2πn +�2e +c2 +�n � +Now we can choose appropriate values for constants c1 and c2. We choose c2 +to be large enough so that +� +1 + 4 +√ +6c2 +1 +√n +� +1 +√ +2πn +�2e +c2 +�n +≤ 1 +3e−C√n +holds for all natural n. This is possible since one can note that +�2e +c2 +�n += e +− ln +� +2e +c2 +� +n +36 + +After that we choose c1 to be small enough so that +� +1 + 2 +√ +6c2 (c1 + 2) 1 +√n +� +e− c +c1 +√n ≤ 1 +3e−C√n +holds for all natural n. +Our a is at least e−C√n, which means +� +∆n +∥∇f∥2dµ ≥ +1 +2 +√ +6 +1 +c2 +1 +1 + 2 +c1 +n +� +a − 2 +3e−C√n +� +≥ +1 +2 +√ +6 +1 +c2 +1 +1 + 2 +c1 +n +�1 +3a +� +And since f here can be an arbitrary Lipschitz function f : ∆n → [0; 1] with +µ{f = 0} ≥ 1 +2 and µ{f = 1} ≥ a +we by Lemma 5.4 conclude +Iµ(a) ≥ +1 +6 +√ +6 +1 +c2 +1 +1 + 2 +c1 +na +Propositions 5.1 and 5.2 imply +Theorem 5.4. For the Lebesgue measure λ on the unit-volume simplex ωn∆n +the following isoperimetric inequality +Iλ(t) ≥ cλt +holds for all t ∈ (0; 1 +2), where cλ > 0 is a universal constant independent of +the dimension n. +Proof. By Proposition 5.1 +µ+(A) ≥ csnµ(A) +for all A ⊂ ∆n with µ(A) ∈ (0; e−C√n), and by Proposition 5.2 +Iµ(t) ≥ cbnt +for all t ∈ [e−C√n; 1 +2), which means that +Iµ(t) ≥ min(cs, cb)nt +37 + +for all t ∈ (0; 1 +2). +Equation (11) relates Iµ and Iλ to each other as +Iλ = 1 +ωn +Iµ +Thus for all t ∈ (0; 1 +2) we must have +Iλ(t) ≥ min(cs, cb) n +ωn +t. +Here we could note that +n +ωn is positive for all n and that by Stirling’s +approximation +lim +n→∞ +n +ωn += e, +which must imply that +inf +n +n +ωn +> 0 +So we can take +min(cs, cb) inf +n +n +ωn +as our constant cλ. +From this isoperimetric inequality we conclude +Theorem 5.5. Inside a unit-volume simplex ωn∆n two bodies A and B of +volume ε ∈ (0; 1 +2) are at a distance at most +−c ln ε +for some universal constant c > 0 independent of the dimension n and volume +ε. +Proof. We are interested in the least values δA, δB such that the δA-enlargement +of body A in ωn∆n will be of volume 1 +2 and the δB-enlargement of body B +will be of volume 1 +2 too. For these enlargements we shall have +dist(AδA, BδB) = 0, +from which +dist(A, B) ≤ δA + δB +38 + +follows. +Isoperimetric inequality from Theorem 5.4 provides an estimate on the +growth of δ-enlargements of our bodies +δ+λ(Aδ) ≥ cλλ(Aδ) +which holds as long as λ(Aδ) < 1 +2. +And so to bound δA above we would like consider a function y(δ) that +behaves in accordance with our lower bound +y(0) = λ(A) +(31) +y′ = cλy +(32) +If by δM we will denote the moment when y reaches 1 +2, i. e. y(δM) = 1 +2, then +δA ≤ δM. Indeed, otherwise δM < δA, but functions λ(Aδ) and y(δ) coincide +at δ = 0 and for all δ ∈ [0; δM] we should have +λ(Aδ) ≥ y(δ) and δ+λ(Aδ) ≥ cλλ(Aδ) ≥ cλy(δ) = δ+y(δ) +And so we reach contradiction +1 +2 = λ(AδA) > λ(AδM) ≥ y(δM) = 1 +2 +A solution to differential equation (32) should be of the form +Cecλδ +and since at δ = 0 by our initial condition (31) we should have y(0) = λ(A) +we reach conclusion +y(δ) = λ(A)ecλδ +So δM will be a solution to equation +λ(A)ecλδM = 1 +2, +which after taking logarithm on both sides turns into +ln λ(A) + cλδM = − ln 2 +δM = − 1 +cλ +(ln λ(A) + ln 2) +By the same reasoning δB ≤ δM, and thus +dist(A, B) ≤ δA + δB ≤ − 2 +cλ +(ln λ(A) + ln 2) ≤ − 2 +cλ +ln λ(A) +39 + +5.2 +ℓp-balls. +By the ℓn +p unit ball we mean +ℓn +p = {(x1, . . . , xn) ∈ Rn | |x1|p + . . . + |xn|p ≤ 1} +Let µ be a normalized Lebesgue measure on it. Note that µ(ℓn +p) = 1. The +volume of ℓn +p is equal to +2nΓ +� +1 + 1 +p +�n +Γ +� +1 + n +p +� +by theorem 1 from [23]. So in order to get a unit-volume ℓn +p ball we would +need to stretch the ℓn +p unit ball by a factor of +ωn = +Γ +� +1 + n +p +� 1 +n +2Γ +� +1 + 1 +p +� ∼ +n +1 +p +2Γ +� +1 + 1 +p +� +(pe) +1 +p +By λ denote the Lebesgue measure on ωnℓn +p. Yet again by (11) we should +have proportionality of the isoperimetric functions +Iµ = ωnIλ +(33) +The following theorem was proven by Sasha Sodin in [19]. +Theorem 5.6 ([19, Theorem 1]). There exists a universal constant c > 0 +such that for 1 ≤ p ≤ 2, 0 < a < 1 +2 +Iµ(a) ≥ cn +1 +pa log1− 1 +p a +It follows that +Theorem 5.7. For every p ∈ [1; 2] there exists a positive constant cp > 0 +such that +Iλ(a) > cpa log1− 1 +p a +Proof. By (33) we already now that +Iλ(a) ≥ c 1 +ωn +n +1 +pa log1− 1 +p a +40 + +The number +c +ωnn +1 +p is positive for all n and by Stirling’s approximation +lim +n→∞ +c +ωn +n +1 +p = 2cΓ +� +1 + 1 +p +� +(pe) +1 +p > 0 +Thus +cp = inf +n c 1 +ωn +n +1 +p > 0 +and +Iλ(a) ≥ cpa log1− 1 +p a +From this isoperimetric inequality we derive +Theorem 5.8. Inside a unit-volume ℓp +n ball ωnℓn +p two bodies A and B of +volume ε ∈ (0; 1 +2) are at a distance at most +Cp log +1 +p 1 +ε +for some constant Cp > 0 independent of dimension n and volume ε. +Proof. We are interested in the least values δA, δB such that the δA-enlargement +of body A in ωnℓn +p will be of volume 1 +2 and the δB-enlargement of body B +will be of volume 1 +2 too. For these enlargements we shall have +dist(AδA, BδB) = 0, +from which +dist(A, B) ≤ δA + δB +follows. +The isoperimetric inequality from Theorem 5.7 allows to estimate the +growth of λ(Aδ) as +δ+λ(Aδ) ≥ cpλ(Aδ) log1− 1 +p +1 +λ(Aδ) +while λ(Aδ) < 1 +2. +So we would like to consider a function y(δ) that behaves in accordance +with our lower bound +y(0) = ε +(34) +41 + +y′ = cpy log1− 1 +p 1 +y +(35) +If by δM we will denote the moment when y reaches one half, i. e. y(δM) = 1 +2, +then δA ≤ δM. Indeed, otherwise δM < δA, but functions λ(Aδ) and y(δ) +coincide at δ = 0 and for all δ ∈ [0; δM] we should have +λ(Aδ) ≥ y(δ) and δ+λ(Aδ) ≥ cpλ(Aδ) log1− 1 +p +1 +λ(Aδ) +≥ cpy(δ) log1− 1 +p +1 +y(δ) = δ+y(δ), +since x(− log x)1− 1 +p is increasing on (0; 1 +2](see Appendix B). And so we reach +a contradiction +1 +2 = µ(AδA) > µ(AδM) ≥ y(δM) = 1 +2 +Differential equation (35) is separable +dy = cpy(− log y)1− 1 +pdδ +−(− log y) +1 +p −1 +� +−1 +ydy +� += cpdδ +− +� +(− log y) +1 +p −1d(− log y) = +� +cpdδ +−p(− log y) +1 +p = cpδ + C0 +Our initial condition (34) gives us +−p(− log ε) +1 +p = C0 +And for δ = δM we should have +−p(log 2) +1 +p = cpδM − p(− log ε) +1 +p +δM = 1 +cp +� +p(− log ε) +1 +p − p(log 2) +1 +p +� +≤ p +cp +(− log ε) +1 +p +By the same reasoning δB ≤ δM, and we conclude +dist(A, B) ≤ δA + δB ≤ 2δM ≤ 2p +cp +(− log ε) +1 +p +42 + +6 +Lower bounds. +Introduction. Here we are going to be concerned with the lower bounds +on the largest distance between two subsets of volume 0 < ε < 1 +2. We will +derive the lower bounds simply by considering certain hyperplane cuts of +our convex bodies. For families of convex bodies such as the euclidean balls, +cubes, hyperoctahedrons, simplexes and ℓp balls specific lower bounds will be +shown in Theorems 6.1, 6.3, 6.4, 6.5. It turns out that for euclidean balls our +lower bounds coincide with the upper bounds(see Theorem 6.2). In Theorem +6.6 a general lower bound will be established, showing that in a way the +family of euclidean balls is optimal in regard to our problem. +It was already shown that for unit-volume cube, ball, simplex and ℓp +balls with p ∈ [1; 2] the largest distance is bounded above by some constant +dependent on ε but not on the dimension n. That is why it makes sense to +consider the lower bounds on the supremum of all possible distances between +two subsets of volume ε that take place as n tends to infinity. +For a family of convex bodies Kn by dn(ε) here we denote the supremum +of all possible distances between two subsets of volume ε ∈ (0; 1 +2) in Kn. +Theorem 6.1. When Kn are the unit-volume euclidean balls we have +lim inf +n→∞ dn(ε) ≥ −2 1 +√eΦ−1(ε) +The function −2 1 +√eΦ−1(ε) is asymptotically equivalent to +−2 1 +√πe +√ +− ln ε +as ε → 0. +Proof. In the unit-volume n-ball ωnBn ⊂ Rn, where the radii is +ωn = Γ +�n +2 + 1 +� 1 +n +√π +∼ +� n +2πe, +consider the diagonal from (−ωn, 0, . . . , 0) to (ωn, 0, . . . , 0), i. e. a diagonal +corresponding to the X1-axis. We will be interested in the hyperplanes or- +thogonal to this diagonal, i. e. hyperplanes defined by X1 = t. +43 + +Pick a number a such that +1 +√eΦ−1(ε) < −a < 0 +If we consider the uniform probability distribution on ωnBn, then we could +think of X1 as of a random variable. We would like to consider the part of +our ball that corresponds to X1 ≤ −a. The volume would be equal to +Pr(X1 ≤ −a) = Pr(√nω−1 +n X1 ≤ −√nω−1 +n a) +By theorem 1 of [20] as n tends to infinity the distribution of n +1 +2ω−1 +n X1 +converges in total variation to the standard normal distribution on R, whose +probability density function is +1 +√ +2πe− 1 +2x2 +Furthermore, note that +lim +n→∞ −√nω−1 +n a = lim +n→∞ −√n +√ +2πe +√n a = − +√ +2πea > +√ +2πΦ−1(ε) +So for all sufficiently large n we shall have +√ +2πΦ−1(ε) + δ < −√nω−1 +n a +⇒ Pr(√nω−1 +n X1 ≤ −√nω−1 +n a) ≥ Pr(√nω−1 +n X1 ≤ +√ +2πΦ−1(ε) + δ) +for some δ > 0. Because distribution of n +1 +2ω−1 +n X1 converges in total variation +to the standard normal distribution, for all sufficiently large n we have +Pr(X1 ≤ −a) ≥ Pr(√nω−1 +n X1 ≤ +√ +2πΦ−1(ε) + δ) +≥ +1 +√ +2π +� √ +2πΦ−1(ε) +−∞ +e− 1 +2x2dx = +� √ +2πΦ−1(ε) +−∞ +e +−π +� +1 +√ +2π x +�2 +d +� +1 +√ +2π +x +� += +� Φ−1(ε) +−∞ +e−πx2dx = ε +By symmetry we have a similar result for the part of our ball defined +by X1 ≥ a. That means that for sufficiently large n we are going to have +44 + +two subsets of volume at least ε at a distance 2a. But a was chosen as an +arbitrary number lesser than +− 1 +√eΦ−1(ε), +from which the statement of the theorem follows. +By combining this with Theorem 3.3 we get +Theorem 6.2. When Kn are the unit-volume euclidean balls +lim +n→∞ dn(ε) = −2 1 +√eΦ−1(ε) +Theorem 6.3. When Kn are the unit cubes we have +lim inf +n→∞ dn(ε) ≥ −2 +�π +6 Φ−1(ε) +The function −2� π +6Φ−1(ε) is asymptotically equivalent to +2 +√ +6 +√ +− ln ε +as ε → 0. +Proof. By the main diagonal of a cube (0; 1)n we mean a segment from the +origin to (1, . . . , 1). We would be considering hyperplanes orthogonal to the +main diagonal. +For each point on the main diagonal we could consider the area of the +corresponding orthogonal hyperplane section of (0; 1)n. This gives rise to a +probability distribution on the segment from (0, . . . , 0) to (1, . . . , 1), whose +length is √n. We will take the midpoint of this segment as the origin, i. e. +we have a probability distribution on [− +√n +2 ; + +√n +2 ]. +The variance of the uniform distribution on the segment [0; 1] is equal to +σ2 = +� 1 +0 +� +x − 1 +2 +�2 +dx = 2 +� +1 +2 +0 +x2dx = 21 +3 +1 +23 = 1 +12 +And our distribution on [− +√n +2 ; + +√n +2 ] could be produced by n random variables +X1, . . . , Xn uniformly distributed on [0; 1] as +√n +�X1 + . . . + Xn +n +− 1 +2 +� +45 + +So by central limit theorem as n goes to infinity our distribution converges +to a normal distribution N (0, σ2), whose probability density function would +be +1 +σ +√ +2πe− 1 +2 +x2 +σ2 = +� +6 +πe−6x2 +Pick a number a such that +�π +6 Φ−1(ε) < −a < 0 +And consider the part of the unit cube (0; 1)n whose orthogonal projection on +the main diagonal lies inside [0; 1 +2 +√n − a], i. e. a certain hyperplane section. +As n goes to infinity the volume of this region would converge to the value +of the cumulative distribution function of N (0, σ2) at −a, which is equal to +� −a +−∞ +� +6 +πe−6x2dx = +� −a +−∞ +e +−π +�√ 6 +π x +�2 +d +�� +6 +πx +� += +� −√ 6 +π a +−∞ +e−πx2dx = Φ +� +− +� +6 +πa +� +> Φ +�� +6 +π +�π +6 Φ−1(ε) +� += ε +Thus for large enough n a part of volume at least ε is going to be cut off. By +symmetry the same is true for an orthogonal hyperplane section of our cube +corresponding to [ 1 +2 +√n + a; √n] on the main diagonal. +So for n large enough we get two subsets of volume at least ε at a distance +2a. But a was chosen as an arbitrary number smaller than +− +�π +6 Φ−1(ε), +from which the statement of the theorem follows. +Theorem 6.4. When Kn are the unit-volume simplexes we have +lim inf +n→∞ dn(ε) ≥ − +√ +2 +e ln(2ε). +Proof. The volume of the unit simplex ∆n defined by +∆n = {(x1, . . . , xn) ∈ Rn ++ | x1 + . . . + xn = 1} +46 + +is equal to +n√n +n! +If we set +ωn = +� n! +n√n +� +1 +n−1 +∼ n +e , +then ωn∆n will be a regular unit-volume simplex, whose side length is +√ +2ωn +Now consider any number ε′ such that +ε < ε′ < 1 +2 +Let P and Q be two vertices of ωn∆n. +Hyperplane passing through the +midpoint of the side PQ and orthogonal to it divides our simplex into two +parts of equal volume. We set +α = (2ε′) +1 +n−1. +Consider the image of the part containing P after a homotethy with cen- +ter at P and coefficient α, the resulting subset of our simplex will be of +volume 1 +2αn−1 = ε′ > ε. Analogously, we construct a subset of volume ε′ +corresponding to the vertex Q. +Note that we have two homotethies with coefficient α applied to the halves +of PQ. The distance between our subsets will be equal to +√ +2ωn (1 − α) = +√ +2ωn +� +1 − (2ε′) +1 +n−1 +� +Now we take limit +lim +n→∞ +√ +2ωn +� +1 − (2ε′) +1 +n−1 +� += lim +n→∞ +√ +2 +e n +� +1 − (2ε′) +1 +n−1 +� += lim +n→∞ − +√ +2 +e +n +n − 1 +(2ε′)0 − (2ε′) +1 +n−1 +0 − +1 +n−1 += − +√ +2 +e +d +dt(2ε′)t��� +t=0 = − +√ +2 +e ln(2ε′) +In other words, for each n we have two subsets of ωn∆n of volume at least +ε, and the distance between them tends to +− +√ +2 +e ln(2ε′) +47 + +as n goes to infinity. But ε′ was chosen as an arbitrary number from (ε; 1 +2), +from which the statement of the theorem follows. +Theorem 6.5. When Kn are the unit-volume ℓp balls for p ∈ [1; 2] we have +lim inf +n→∞ dn(ε) ≥ −2Ψ−1 +p (ε), +where function −2Ψ−1 +p (ε)(see Appendix C) is asymptotically equivalent to +1 +e +1 +pΓ +� +1 + 1 +p +�(− ln ε) +1 +p +as ε → 0. +Proof. In the unit-volume ℓp ball ωnℓn +p, where +ωn = +Γ +� +1 + n +p +� 1 +n +2Γ +� +1 + 1 +p +� ∼ +n +1 +p +2Γ +� +1 + 1 +p +� +(pe) +1 +p +, +consider the segment from (−ωn, 0, . . . , 0) to (ωn, 0, . . . , 0). We will be con- +sidering the hyperplanes orthogonal to this segment, i. e. hyperplanes defined +by X1 = t. By Vn(t) denote the function that measures the volume of the +part given by X1 ≥ t. +We use notation from Appendix C, where it was established that func- +tions Vn(x) uniformly converge to Ψp(−x) on (−∞; +∞). Pick an arbitrary +number a such that +Ψ−1 +p (ε) < −a < 0 +Then for all sufficiently large n we shall have +Vn(a) > ε +Because of symmetry, for all sufficiently large n we would have two bodies in +ωnℓn +p of volume at least ε at a distance at least 2a. And, since the choice of +a above was arbitrary, we would have +lim inf +n→∞ dn(ε) ≥ −2Ψ−1 +p (ε) +48 + +Assume that we have a family Kn of bounded unit-volume centrally sym- +metric bodies that are not necessarily convex, where the origin will be the +center of symmetry for each Kn. By µn denote the uniform probability mea- +sure on Kn. +Fix ε ∈ (0; 1 +2). Since Kn is bounded, it has a finite diameter, thus for +each Kn we could consider dn(ε) – the supremum of all possible distances +between two subsets of Kn, whose volume is ε. +Turns out that a simple averaging argument gives us. +Theorem 6.6. For every ε ∈ (0; 1 +2) +lim inf +n→∞ dn(ε) ≥ −2 1 +√eΦ−1(ε), +where function −2 1 +√eΦ−1(ε) is asymptotically equivalent to +−2 1 +√πe +√ +− ln ε +as ε → 0. +Proof. Consider a number d such that +1 +√eΦ−1(ε) < −d < 0 +We would like to show that for all sufficiently large n there would be a +direction defined by a unit vector u on the sphere Sn−1 such that the part of +our body Kn where ⟨x, u⟩ ≤ −d would be of volume at least ε. By central +symmetry the part where ⟨x, u⟩ ≥ d would be of the same volume. But then +we will be having two subsets of Kn of volume at least ε at a distance at +least 2d, where d was chosen as an arbitrary number lesser than +− 1 +√eΦ−1(ε), +from which the statement of the theorem would follow. +We are looking for a direction u ∈ Sn−1 with +µn(⟨x, u⟩ ≤ −d) ≥ ε +49 + +If the average value of µn(⟨x, u⟩ ≤ −d) is greater than ε, then such a direction +will exist. +By ϑn denote the uniform probability measure on Sn−1. We would like +to have +� +Sn−1 µn(⟨x, u⟩ ≤ −d)dϑn ≥ ε +(36) +Volume of the hyperplane cut defined by ⟨x, u⟩ ≤ −d is the integral of the +indicator function [⟨x, u⟩ ≤ −d] on Kn, which allows to rewrite (36) as +� +Sn−1 +� +Kn +[⟨x, u⟩ ≤ −d]dµndϑn ≥ ε +We may switch the order of integration +� +Kn +�� +Sn−1[⟨x, u⟩ ≤ −d]dϑn +� +dµn ≥ ε +(37) +We would like to consider the integrand of (37) for x ̸= 0. First, we rewrite +[⟨x, u⟩ ≤ −d] as +�√n +� x +∥x∥2 +, u +� +≤ − +√nd +∥x∥2 +� +Now we could think of u as of a random vector on the unit sphere Sn−1. +Then +� x +∥x∥2 +, u +� +corresponds to the projection of u on the diameter from − +x +∥x∥2 to + +x +∥x∥2. And +thus by theorem 1 from [20] as n goes to infinity the distribution of random +variable +√n +� x +∥x∥2 +, u +� +converges in total variation to the standard normal distribution. +Note that the value of +� +Sn−1 +�√n +� x +∥x∥2 +, u +� +≤ − +√nd +∥x∥2 +� +dϑn +only depends on − +√nd +∥x∥2 . So we define the function +Ψn(x) = +� +Sn−1 +�√n +� x +∥x∥2 +, u +� +≤ x +� +dϑn +50 + +As was remarked above, for every x +lim +n→∞ Ψn(x) = +1 +√ +2π +� x +−∞ +e− 1 +2x2dx +Since functions Ψn are monotonic, we should also have +lim +n→∞ Ψn(xn) = +1 +√ +2π +� x +−∞ +e− 1 +2 x2dx +for any sequence xn that tends to x as n → ∞. +We rewrite (37) as +� +Kn\{0} +Ψn +� +− +√nd +∥x∥2 +� +dµn ≥ ε +(38) +The function Ψn is non-decreasing, thus +∥x∥2 ≥ r > 0 ⇒ Ψn +� +− +√nd +∥x∥2 +� +≥ Ψn +� +− +√nd +r +� +So for (38) condition +� +Kn\Bn +r +Ψn +� +− +√nd +r +� +dµn ≥ ε +(39) +would be sufficient, we may go further and require +(1 − V (Bn +r ))Ψn +� +− +√nd +r +� +≥ ε, +(40) +where V (Bn +r ) is the volume of the n-ball with radius r. +The volume of the unit n-ball Bn +1 is +√πn +Γ +�n +2 + 1 +�. +The unit volume corresponds to the radius +ωn = Γ +�n +2 + 1 +� 1 +n +√π +∼ +� n +2πe +51 + +Pick a number α ∈ (0; 1) such that +α 1 +√eΦ−1(ε) < −d +(41) +and set +r = αωn +Inequality (40) turns into +(1 − αn)Ψn +� +− +√nd +αωn +� +≥ ε +(42) +Note that +lim +n→∞ +√nd +αωn += lim +n→∞ +� +2πe +n +√nd +α += +√ +2πed +α +Now take limit of the left side of (42) and apply condition (41) +lim +n→∞(1 − αn)Ψn +� +− +√nd +αωn +� += +1 +√ +2π +� − +√ +2πed +α +−∞ +e− 1 +2 x2dx += +� − +√ +2πed +α +−∞ +e +−π +� +1 +√ +2π x +�2 +d +� +1 +√ +2πx +� += +� − +√ed +α +−∞ +e−πx2dx = Φ +� +− +√ed +α +� +> Φ(Φ−1(ε)) = ε +Thus for all sufficiently large n +(1 − αn)Ψn +� +− +√nd +αωn +� +> ε +and, consequently, +� +Sn−1 µn(⟨x, u⟩ ≤ −d)dϑn ≥ ε, +(43) +which means that for all sufficiently large n the desired direction u ∈ Sn−1 +with property +µn(⟨x, u⟩ ≤ −d) ≥ ε +exists. +52 + +Theorem 6.1 is an immediate corollary of the Theorem 6.6. +Conclusions. +Roughly speaking, the results established in [12], [11] +tell us that generally hyperplane sections of convex bodies across arbitrary +directions lead to a gaussian distribution, suggesting that the asymptotic +behavior of dn(ε) different from Φ−1(ε) is probably caused by a few degenerate +directions. For example, Theorems 6.4 and 5.5 tell us that the asymptotic +behavior of dn(ε) for simplexes corresponds to a function − ln ε, but one +could also note that the simplex ∆n is unusually «stretched» in n directions +corresponding to its corners, and that almost all of its volume by Theorem +5.3 is concentrated in the euclidean ball of a diameter much smaller than the +diameter of ∆n. +53 + +7 +Discrete isoperimetric problem. +Introduction. In this section instead of considering the euclidean distance +between two subsets of volume ε we will be considering the Manhattan dis- +tance. We can no longer say that for fixed ε this distance is bounded, but +we will derive a result(Theorem 7.3) concerning its asymptotic behaviour in +the case of the unit cube [0; 1]n. The key idea in the proof of it would be +to replace the unit cube [0; 1]n with a lattice, for which the solution of our +problem is already known(see Theorem 7.2). +We consider the Manhattan distance d in the unit cube [0; 1]n. We may +ask a similar question: what is the largest Manhattan distance between two +bodies of volume ε > 0 in the unit cube [0; 1]n? +Turns out this problem could be dealt with by discretization. Consider a +lattice L = { 0 +m, 1 +m, . . . , m +m}n inside [0; 1]n, two points x and y in L are adjacent +whenever d(x, y) = +1 +m. By t-boundary A(t) of a subset A ⊆ L we mean the +set of all points of L that are at a distance at most +t +m from A. The latter +concept is analagous to δ-enlargements, and one could think of |A(1) \ A| as +an analogue of the surface area. This gives rise to the discrete isoperimetirc +problem: how large the t-boundary of a set A ⊆ L with fixed size can be? +This question was answered in [4] +Theorem 7.1 ([4, Corollary 9]). Let A ⊂ [k]n. For any t = 0, 1, . . . , the +t-boundary of A is at least as large as the t-boundary of the first |A| elements +of [k]n in the simplicial order. +Here [k]n is the lattice {0, . . . , k − 1}n. Simplicial order on [k]n is defined +by setting x < y if either � xi < � yi, or � xi = � yi and for some j we +have xj > yj and xi = yi for all i < j. This discrete isoperimetric inequality +leads to +Theorem 7.2 ([4, Corollary 10]). There are sets A, B ⊂ [k]n with |A| = +r, |B| = s, and d(A, B) ≥ d iff the distance between the first r and the last s +elements of the simplicial order on [k]n is at least d. +From this discrete version of the problem considered in the first pararaph +we can derive +Theorem 7.3. If by dn(ε) we denote the largest Manhattan distance between +two bodies of volume ε ∈ (0; 1 +2) in the unit cube [0; 1]n, then +lim +n→∞ +dn(ε) +√n = −2 +�π +6Φ−1(ε) +54 + +The function −2� π +6Φ−1(ε) is asymptotically equivalent to +2 +√ +6 +√ +− ln ε +as ε → 0. +Proof. Pick a number a such that +−a < +�π +6 Φ−1(ε) < 0 +If by Vn and Wn we denote the subsets of [0; 1]n defined by inequalities +� xi ≤ 1 +2 +√n − a and � xi ≥ 1 +2 +√n + a, respectively, then by applying the +central limit theorem just as we did in Theorem 6.3 we shall get +lim +n→∞ µ(Vn) = lim +n→∞ µ(Wn) = +� −a +−∞ +� +6 +πe−6x2dx = Φ +� +− +� +6 +πa +� +< Φ +�� +6 +π +�π +6 Φ−1(ε) +� += ε +So there is a number N such that for all n > N the volumes of Vn and Wn are +going to be lesser than ε, and it could be noted that the Manhattan distance +between them is equal to 2a√n. From now on we assume that n > N. +Now consider two bodies A and B inside [0; 1]n of volume ε and a lat- +tice Lm = { 0 +m, . . . , m +m}n. The discretization is justified by the fact that the +distance does not decrease when we restrict our attention to the lattice Lm +d(A, B) ≤ d(A ∩ Lm, B ∩ Lm) +Since µ(Vn) < µ(A) and µ(Wn) < µ(B), for all m large enough we shall have +|Vn ∩ Lm| ≤ |A ∩ Lm| +|Wm ∩ Lm| ≤ |B ∩ Lm| +And by Theorem 7.2 this implies +d(A ∩ Lm, B ∩ Lm) ≤ d(Vn ∩ Lm, Wn ∩ Lm), +because Vn ∩ Lm and Wn ∩ Lm are the first |Vn ∩ Lm| and the last |Wn ∩ Lm| +elements of the simplicial order on Lm, respectively. +55 + +We conclude that for all m large enough +d(A, B) ≤ d(Vn ∩ Lm, Wn ∩ Lm) +Also +lim +m→∞ d(Vn ∩ Lm, Wn ∩ Lm) = d(Vn, Wn) = 2a√n +Thus for all n > N +d(A, B) ≤ 2a√n, +but a was chosen here as an arbitrary number greater than −� π +6Φ−1(ε), +from which +lim sup +n→∞ +dn(ε) +√n ≤ −2 +�π +6 Φ−1(ε) +(44) +follows. +If we assume that +�π +6 Φ−1(ε) < −a < 0, +then as it was already shown in the proof of Theorem 6.3 for all n large +enough +µ(Vn) = µ(Wn) > ε +But the Manhattan distance between Vn and Wn is 2a√n. +From this observation we can conclude +lim inf +n→∞ +dn(ε) +√n ≥ −2 +�π +6 Φ−1(ε), +which together with inequality (44) gives us the statement of the theorem. +Conclusions. There are also other variations of the discrete isoperimet- +ric problem. For example, Theorem 7.2 has an analogue for the non-negative +orthant of the integer lattice Zn ++(see [22] and [4, Theorem 4]). +56 + +8 +Conclusions. +Upper bounds on the distance between subsets in unit-volume ℓp balls with +p ∈ [1; 2] ∪ {+∞} have been established. A case of p ∈ (2; +∞) remains. +Since asymptotically our estimates for unit-volume euclidean balls(p = 2) +and for unit cubes(p = +∞) are the same, we expect to have a similar +asymptotic behavior for all p ∈ [2; +∞]. As was remarked before both cases +of p = 2 and p = +∞ could be approached by providing a Lipschitz map that +transforms Gaussian measure into the uniform measure on the corresponding +ℓp balls. Perhaps, the same approach might work out for p ∈ (2; +∞). On +the page 4 of [3] it was remarked that the uniform measure on ℓn +p balls +with p ∈ [2; +∞] «can be obtained from the canonical Gaussian measure as +Lipschitz transform». Although, we are not sure about the exact meaning +and implications of this statement. +In Theorem 6.6 we established a sort of a general lower bound regarding +our problem. But is it possible to find a general upper bound on the distance +between two subsets of volume ε ∈ (0; 1 +2) in a unit-volume convex body? +Clearly, we are going to have to consider only some certain «good» convex +bodies. For example, in a convex body stretched far in a particular direc- +tion two subsets of fixed volume could be at an arbitrarily large distance. +One such notion of a «good» convex body is related to the isotropic position. +But this means that we are looking for general estimates on the isoperimetric +problem in isotropic convex bodies, this appears to be a complicated open +question(Kannan-Lov´asz-Simonovits conjecture, see [14]). Furthermore, even +the relation between the volume and the isotropic constant of a convex body +seems to be at the core of another open problem(isotropic constant conjec- +ture, see [5], [7]). In paper [6] a very good lower bound related to the KLS +conjecture was proven. There is a lot of material on isotropic convex bodies, +for example, [9]. +Log-concave probability measures generalize uniform measures on convex +bodies. So one might try to consider the problem in a more general setting. +Gaussian measures are log-concave, which means that in a more general +setting our estimates may be sharp for some specific distributions in the one- +dimensional case. In paper [15] some general estimates on the specific version +of the isoperimetric problem were proven(see, for example, theorem 1.3). +57 + +References +[1] +S. G. Bobkov. “Isoperimetric and Analytic Inequalities for Log-Concave +Probability Measures”. 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In: Mathematics +Magazine 78.5 (2005), pp. 390–395. issn: 0025570X, 19300980. url: +http://www.jstor.org/stable/30044198 (visited on 11/24/2022). +60 + +A +Asymptotic behavior of Φ−1. +We want to show that +lim +ε→0 +1 +√π +√ +− ln ε +−Φ−1(ε) = 1 +(45) +We begin with the following observation +d +dx +� +e−πx2 +−2πx +� += e−πx2 + e−πx2 +2πx2 +By integrating both parts from −∞ to a < 0 we get +e−πa2 +−2πa = +� +e−πx2 +−2πx +������ +a +−∞ += +� a +−∞ +e−πx2 � +1 + +1 +2πx2 +� +dx +(46) +From this we can derive the following upper bound on Φ +Φ(a) ≤ +� a +−∞ +e−πx2 � +1 + +1 +2πx2 +� +dx = e−πa2 +−2πa +Now we apply −Φ−1 to both sides +−a ≥ −Φ−1 +� +e−πa2 +−2πa +� +Keep in mind that a < 0. By putting e−πa2 +−2πa instead of ε into the expression +from (45) we arrive at +1 +√π +� +πa2 + ln(−2πa) +−Φ−1( e−πa2 +−2πa ) +≥ +1 +√π +� +πa2 + ln(−2πa) +−a +But as a goes to −∞ +e−πa2 +−2πa → 0 and +1 +√π +� +πa2 + ln(−2πa) +−a +→ 1 +So +lim inf +ε→0 +1 +√π +√ +− ln ε +−Φ−1(ε) ≥ 1 +(47) +61 + +if defined. +From (46) we can also derive a lower bound on Φ +Φ(a) ≥ +� +1 + +1 +2πa2 +�−1 � a +−∞ +e−πx2 � +1 + +1 +2πx2 +� +dx = e−πa2 +−2πa +� +1 + +1 +2πa2 +�−1 +Again we apply −Φ−1 to both sides and get +−a ≤ −Φ−1 +� +e−πa2 +−2πa +� +1 + +1 +2πa2 +�−1� +By putting e−πa2 +−2πa +� +1 + +1 +2πa2 +�−1 instead of ε into the expression from (45) we +arrive at +1 +√π +� +πa2 + ln(−2πa) + ln +� +1 + +1 +2πa2 +� +−Φ−1 +� +e−πa2 +−2πa +� +1 + +1 +2πa2 +�−1� +≤ +1 +√π +� +πa2 + ln(−2πa) + ln +� +1 + +1 +2πa2 +� +−a +But as a goes to −∞ +e−πa2 +−2πa +� +1 + +1 +2πa2 +�−1 +→ 0, +1 +√π +� +πa2 + ln(−2πa) + ln +� +1 + +1 +2πa2 +� +−a +→ 1 +Thus +lim sup +ε→0 +1 +√π +√ +− ln ε +−Φ−1(ε) ≤ 1 +(48) +Together (47) and (48) give +lim +ε→0 +1 +√π +√ +− ln ε +−Φ−1(ε) = 1 +62 + +B +Function x(− log x)1−1 +p is increasing on (0; 1 +2]. +To show that function x(− log x)1− 1 +p is increasing on (0; 1 +2] for p ∈ [1; 2] we +simply take the derivative +d +dx +� +x(− log x)1− 1 +p +� += (− log x)1− 1 +p + x d +dx(− log x)1− 1 +p += (− log x)1− 1 +p + xd(− log x)1− 1 +p +d(− log x) +d(− log x) +dx += (− log x)1− 1 +p + +� +1 − 1 +p +� +x(− log x)− 1 +p +� +−1 +x +� += (− log x)1− 1 +p − +� +1 − 1 +p +� +(− log x)− 1 +p += (− log x)− 1 +p +� +(− log x) − +� +1 − 1 +p +�� +Clearly, (− log x)− 1 +p > 0. On the half-interval (0; 1 +2] we should have +(− log x) − +� +1 − 1 +p +� +≥ +� +− log 1 +2 +� +− 1 +2 = log 2 − 1 +2 > 0, +which means that for all x ∈ (0; 1 +2] +d +dx +� +x(− log x)1− 1 +p +� +> 0 +C +Functions Vn(x) and Sn(x) in limit. +For p ∈ [1; ∞) consider the unit ℓn +p ball defined by +|x1|p + . . . + |xn|p = 1 +It has volume +� +2Γ +� +1 + 1 +p +��n +Γ +� +1 + n +p +� +63 + +Let +ωn = +Γ +� +1 + n +p +� 1 +n +2Γ +� +1 + 1 +p +� ∼ +n +1 +p +2Γ +� +1 + 1 +p +� +(pe) +1 +p +(49) +For a non-negative number x by Sn(x) denote the volume of the section +x1 = x of the unit-volume body ωnℓn +p. For x > ωn we assume Sn(x) = 0. For +x ≤ ωn the ℓn−1 +p +ball corresponding to this section is defined by equation +|x2|p + . . . + |xn|p = ωp +n − xp +It has volume +Sn(x) = (ωp +n − xp) +n−1 +p +� +2Γ +� +1 + 1 +p +��n−1 +Γ +� +1 + n−1 +p +� += +� +1 − xp +ωp +n +� n−1 +p +1 +ωn +ωn +n +� +2Γ +� +1 + 1 +p +��n−1 +Γ +� +1 + n−1 +p +� += +� +1 − xp +ωp +n +� n−1 +p +1 +ωn +Γ +� +1 + n +p +� +� +2Γ +� +1 + 1 +p +��n +� +2Γ +� +1 + 1 +p +��n−1 +Γ +� +1 + n−1 +p +� += +� +1 − xp +ωp +n +� n−1 +p +Γ +� +1 + n +p +� +2ωnΓ +� +1 + 1 +p +� +Γ +� +1 + n−1 +p +� +Since +lim +x→∞ +Γ(x + α) +Γ(x)xα += 1, +we shall have +lim +n→∞ +Γ +� +1 + n +p +� +2ωnΓ +� +1 + 1 +p +� +Γ +� +1 + n−1 +p +� = lim +n→∞ +� +ωn2Γ +� +1 + 1 +p +��−1 Γ +� +1 + n +p +� +Γ +� +1 + n−1 +p +� += lim +n→∞ + + +n +1 +p +2Γ +� +1 + 1 +p +� +(pe) +1 +p +2Γ +� +1 + 1 +p +� + +−1 �n +p +� 1 +p += e +1 +p +64 + +Also +lim +n→∞ +n +pωp +n = lim +n→∞ n + +p +n +2pΓ +� +1 + 1 +p +�p +pe + + +−1 += 2pΓ +� +1 + 1 +p +�p +e +Thus on the segment [0; D] the sequence of functions +� +1 − xp +ωp +n +� n−1 +p += +� +1 − xp +ωp +n +� � +1 − +xp +n +pωp +n +n +p +� n +p +uniformly converges to +e−xp2pΓ(1+ 1 +p) +pe = e +− +� +2Γ(1+ 1 +p)e +1 +p x +�p +as n tends to ∞. +Thus on the segment [0; D] functions Sn(x) uniformly converge to +ψp(x) = e +1 +p e−xp2pΓ(1+ 1 +p) +pe = e +1 +p e +− +� +2Γ(1+ 1 +p)e +1 +p x +�p +For an arbitrary ε > 0 pick a number D such that +ψp(D) < ε, +then for all sufficiently large n +Sn(D) < 2ε +And, since both functions Sn(x) and ψp(x) are non-increasing, we will have +|ψp(x) − Sn(x)| < 3ε +for every x ≥ D for all sufficiently large n. This implies that functions Sn(x) +uniformly converge to ψp(x) on the whole [0; +∞). +For a non-negative x by Vn(x) denote the volume of the part of ωnℓn +p +defined by x1 ≥ x. For x > ωn we assume that Vn(x) = 0. +Note that +Vn(x) = 1 +2 − +� d +0 +Sn(t)dt +65 + +And, since Sn(t) uniformly converge to ψp(x) on [0; D], we shall have that +Vn(x) uniformly converge to +Λp(x) = 1 +2 − +� d +0 +ψp(t)dt +on [0; D]. +Also +� ∞ +0 +ψp(x)dx = +� ∞ +0 +e +1 +pe +− +� +2Γ(1+ 1 +p)e +1 +p x +�p +dx += +1 +2Γ +� +1 + 1 +p +� +� ∞ +0 +e +− +� +2Γ(1+ 1 +p)e +1 +p x +�p  + +1 +2Γ +� +1 + 1 +p +� +e +1 +p +dx + + += +1 +2Γ +� +1 + 1 +p +� +� ∞ +0 +e−xpdx = +1 +2Γ +� +1 + 1 +p +� +� ∞ +0 +e−tdt +1 +p += +1 +2pΓ +� +1 + 1 +p +� +� ∞ +0 +t +1 +p −1e−tdt = +Γ +� +1 +p +� +2pΓ +� +1 + 1 +p +� = 1 +2 +For an arbitrary ε there is a number D such that +Λp(D) < ε +Since Vn(D) converge to Λp(D), we shall have +Vn(D) < 2ε +for all sufficiently large n. And, since both functions Vn(x) and Λp(x) are +non-increasing, +|Λp(x) − Vn(x)| < 3ε +for every x ≥ D for all sufficiently large n. This means that functions Vn(x) +uniformly converge to +Λp(x) = +� ∞ +x +φp(t)dt +on the whole [0; +∞). +66 + +For Section 3 note that +ψ2(x) = √ee−πex2 = Ψ′(−x) +� ∞ +x +√ee−πet2dt = +� ∞ +x +e−π(√et)2d(√et) = +� ∞ +√ex +e−πx2dx = Ψ(−x) +For convenience we introduce new functions defined on (−∞; +∞) +φp(x) = e−cp|x|p +cp = 2pΓ +� +1 + 1 +p +�p +Φp(a) = +� a +−∞ +e−cp|x|pdx +Ψp(a) = +� a +−∞ +φp(x)dx +The latter function is the reflection of Λp. +We would like to show that +lim +ε→0 +c +− 1 +p +p (− ln ε) +1 +p +−Φ−1 +p (ε) += 1 +(50) +Note that +d +dx +e−cpxp +pcpxp−1 = −pcpxp−1e−cpxp +pcpxp−1 +− e−cpxpp(p − 1)cpxp−2 +p2c2px2(p−1) += −e−cpxp � +1 + p − 1 +p +1 +cp +1 +xp +� +Let +A = p − 1 +p +1 +cp +We conclude that for a negative x we shall have +� e−cp|x|p +pcp|x|p−1 +�′ += e−cp|x|p � +1 + A +|x|p +� +Integrating both sides from −∞ to a < 0 gives us +e−cp|x|p +pcp|x|p−1 = +� a +−∞ +e−cp|x|p � +1 + A +|x|p +� +dx +(51) +67 + +From this we can derive an upper bound on Φp +Φp(a) = +� a +−∞ +e−cp|x|pdx ≤ +e−cp|x|p +pcp|x|p−1 +Applying −Φ−1 +p +to both sides should give us +−a ≥ −Φ−1 +p +� e−cp|x|p +pcp|x|p−1 +� +If we put ε = +e−cp|x|p +pcp|x|p−1 into the expression from (50), we get +c +− 1 +p +p (cp|x|p + ln(pcp|x|p−1)) +1 +p +−Φ−1 +p +� +e−cp|x|p +pcp|x|p−1 +� +≥ c +− 1 +p +p (cp|x|p + ln(pcp|x|p−1)) +1 +p +−a +But as a goes to −∞ +e−cp|x|p +pcp|x|p−1 → 0 and c +− 1 +p +p (cp|x|p + ln(pcp|x|p−1)) +1 +p +−a +→ 1, +from which we conclude +lim inf +ε→0 +c +− 1 +p +p (− ln ε) +1 +p +−Φ−1 +p (ε) +≥ 1 +(52) +Equality (51) also leads to a lower bound on Φp +Φp(a) ≥ +� +1 + A +|a|p +�−1 e−cp|x|p +pcp|x|p−1 +By applying −Φ−1 +p +to both sides again we get +−a ≤ −Φ−1 +p +�� +1 + A +|a|p +�−1 e−cp|x|p +pcp|x|p−1 +� +68 + +By putting ε = +� +1 + +A +|a|p +�−1 e−cp|x|p +pcp|x|p−1 into the expression from (50) we arrive +at +c +− 1 +p +p +� +cp|x|p + ln(pcp|x|p−1) + ln +� +1 + +A +|a|p +�� 1 +p +−Φ−1 +p +�� +1 + +A +|a|p +�−1 e−cp|x|p +pcp|x|p−1 +� +≤ +c +− 1 +p +p +� +cp|x|p + ln(pcp|x|p−1) + ln +� +1 + +A +|a|p +�� 1 +p +−a +As a goes to −∞ +� +1 + A +|a|p +�−1 e−cp|x|p +pcp|x|p−1 → 0 +c +− 1 +p +p +� +cp|x|p + ln(pcp|x|p−1) + ln +� +1 + +A +|a|p +�� 1 +p +−a +→ 1 +And we conclude +lim sup +ε→0 +c +− 1 +p +p (− ln ε) +1 +p +−Φ−1 +p (ε) +≤ 1 +Together with (52) this leads to +lim +ε→0 +c +− 1 +p +p (− ln ε) +1 +p +−Φ−1 +p (ε) += 1 +If by Ψp we denote the function +Ψp(a) = +� a +∞ +e +1 +pe +− +� +2Γ(1+ 1 +p)e +1 +p x +�p +dx = Φp(e +1 +p a), +then we shall have +−Ψ−1 +p (ε) ∼ +1 +2e +1 +p Γ +� +1 + 1 +p +�(− ln ε) +1 +p +69 + +D +Average distance. +The radius of the unit-volume euclidean n-ball is +ωn = Γ +�n +2 + 1 +� 1 +n +√π +∼ +� n +2πe +(53) +For an arbitrary α ∈ (0; 1) define +rn = α +� n +2πe +(54) +By (53) we have +lim +n→∞ +rn +ωn += α < 1 +If by Vn we denote the volume of the n-ball of radius rn, then +lim +n→∞ Vn = 0 +(55) +Now consider an arbitrary unit-volume convex body K. For each point +p ∈ K the set of points of K that are at a distance at most rn from p is of +volume not greater than Vn, so the average distance d(K) between points in +K should be at least +d(K) ≥ (1 − Vn)rn +And by (54), (55) +(1 − Vn)rn ∼ α +� n +2πe +Since the choice of α ∈ (0; 1) above was arbitrary, for a family of unit- +volume convex bodies Kn we should have +lim inf +n→∞ +d(Kn) +� n +2πe +≥ 1 +70 + diff --git a/P9FRT4oBgHgl3EQfJTdB/content/tmp_files/load_file.txt b/P9FRT4oBgHgl3EQfJTdB/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..4227d0d6ad3742530f7aec58ffe1ee03fcf803c7 --- /dev/null +++ b/P9FRT4oBgHgl3EQfJTdB/content/tmp_files/load_file.txt @@ -0,0 +1,1894 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf,len=1893 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='13495v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='MG] 31 Jan 2023 Dimension-free estimates on distances between subsets of volume ε inside a unit-volume body Abdulamin Ismailov1 Alexei Kanel-Belov Fyodor Ivlev Abstract Average distance between two points in a unit-volume body K ⊂ Rn tends to infinity as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' However, for two small subsets of volume ε > 0 the situation is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' For unit-volume cubes and euclidean balls the largest distance is of order √ − ln ε, for simplexes and hyperoctahedrons – of order − ln ε, for ℓp balls with p ∈ [1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 2] – of order (− ln ε) 1 p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' These estimates are not dependent on the di- mensionality n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The goal of the paper is to study this phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Isoperimetric inequalities will play a key role in our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Contents 1 Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 3 2 Preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 7 3 Euclidean balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 9 4 Unit cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 17 5 Simplexes and ℓp-balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 23 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 Simplexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 40 6 Lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 43 7 Discrete isoperimetric problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 54 8 Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 57 A Asymptotic behavior of Φ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 61 1E-mail: nameundefinednamovich@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='com 1 B Function x(− log x)1− 1 p is increasing on (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 63 C Functions Vn(x) and Sn(x) in limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 63 D Average distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 70 2 1 Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' In high dimensions we observe a variety of different phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' For ex- ample, Vladimir Igorevich Arnold liked to ask his students the following question: «What percent of the overall mass is occupied by the pulp of the 100-dimensional watermelon of diameter 1 meter, if the crust is of width 1 centimeter?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='» The answer is approximately 1−e−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' This question in a simple way demonstrates the concentration of measure phenomenon: how most of the mass of a body could lie inside a thin shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Here is another example, the volume of a euclidean ball of radius 2023 tends to 0 as the dimensionality goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' More generally, we have the isodiametric inequality, which suggests that in high dimensions the diameter of a unit-volume body shall become arbitrarily large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The goal of this paper is to achieve a better understanding how things work in high-dimensional spaces by studying the following phenomenon: two points in a unit-volume convex body could be at an arbitrarily large distance from each other;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' consider, for example, the unit cubes (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1)n – as n tends to +∞ the diameter equal to √n also tends to infinity, similarly, the average distance would be of order at least √n(see Appendix D);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' even the distance between a point and a subset of some fixed volume ε < 1 could be arbitrarily big, but it turns out that the distance between two subsets of some fixed volume ε > 0 in the unit cube is bounded above by some constant dependent on ε but not on the dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' What about convex bodies other than the unit cubes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Consider a family of unit-volume bounded convex bodies Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' For each Kn it makes sense to consider the supremum of all possible distances between two subsets of some fixed volume ε ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Denote this value by dn(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' By Φ we mean the function Φ(a) = � a −∞ e−πx2dx Function Φ−1(ε) is asymptotically equivalent to − 1 √π √ − ln ε as ε tends to 0(see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 3 Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' When Kn are the unit-volume euclidean balls lim n→∞ dn(ε) = −2 1 √eΦ−1(ε) Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='3 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' When Kn are the unit cubes we have −2 �π 6 Φ−1(ε) ≤ lim inf n→∞ dn(ε) ≤ lim sup n→∞ dn(ε) ≤ −2Φ−1(ε) Theorems 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='5 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' When Kn are the unit-volume simplexes we have − √ 2 e ln(2ε) ≤ lim inf n→∞ dn(ε) ≤ lim sup n→∞ dn(ε) ≤ −c ln ε for some universal constant c > 0 independent of n and ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Theorems 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='8 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Fix some p ∈ [1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' When Kn are the unit-volume ℓp balls −2Ψ−1 p (ε) ≤ lim inf n→∞ dn(ε) ≤ lim sup n→∞ dn(ε) ≤ Cp(− ln ε) 1 p, where Cp is some universal constant determined by p, and function −2Ψ−1 p (ε)(see Appendix C) is asymptotically equivalent to 1 e 1 pΓ � 1 + 1 p �(− ln ε) 1 p as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' A version of our problem, in which the euclidean distance is replaced by the Manhattan distance, can be approached by discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' If by dn(ε) we denote the largest Manhattan distance between two bodies of volume ε ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2) in the unit cube [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1]n, then lim n→∞ dn(ε) √n = −2 �π 6Φ−1(ε) We also establish a sort of a general lower bound, showing that in a way euclidean balls are optimal in regard to our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 4 Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' When Kn are unit-volume centrally symmetric bounded con- vex bodies −2 1 √eΦ−1(ε) ≤ lim inf n→∞ dn(ε) Lower bounds on our problem could be derived simply by considering some hyperplane cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' But how can we bound above the distance between two subsets A and B in a unit-volume convex body?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Well, first, we observe that if both A and B are of volume at least 1 2, then the distance between them is zero(see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' That is why we assume that both A and B are of some volume ε ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Next we introduce the concept of a δ-enlargement of a body defined as the set of all points at a distance at most δ from our body, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Aδ = {x ∈ X | ∃y ∈ A: d(x, y) ≤ δ} What happens if we replace A with its δ-enlargement for a small enough value of δ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Roughly speaking, a layer of width δ will be added to our body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The volume of this layer Aδ \\ A could be approximated as δ · S(A), where S(A) is the surface area of the body A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' So the volume of A increases by approximately δ·S(A), but the distance between bodies A and B will decrease exactly by δ after we enlarge A(or might become zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' To estimate the distance between A and B we will be slowly enlarging them simultaneously until both bodies would be of volume 1 2 at least, at which point the distance between them is already zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The double of the amount of time it took both bodies to reach volume 1 2 would be an upper bound on the distance between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' This was just a rough description of how we approach the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' To make this idea work we are going to need more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' We have not said anything about how our bodies may look like, at this point they could be arbitrary subsets of volume ε, which may present a problem, since we plan to rely on concepts such as surface area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' In part, these issues might be mitigated by the following observation: after enlarging both A and B by a little δ distances and volumes would not change much, but smoothness properties might improve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Anyway, throughout this whole text we assume that the bodies we are dealing with are as smooth as needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Now consider the process of a slow enlargement of a body A at its very beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Instead of talking about the approximate volume of the layer Aδ \\ A it would be better to take the right derivative at the point δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 5 What we will get is called the Minkowski–Steiner formula for the free surface area µ+(A) = lim δ→0+ µ(Aδ) − µ(A) δ Thus it is vital to our approach to be able to estimate this surface area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' But we are only aware of the initial volumes of A and B, which leads us to the isoperimetric problem: given the information about the initial volume of a body, find a lower bound on its surface area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Euclidean balls have really good symmetry properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Symmetrization techniques could be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Isoperimetric regions inside the euclidean balls have been completely classified([16, Theorems 1 and 5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' This allows us to get tight enough estimates that lead to the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Unit cubes, however, are not as good as euclidean balls in that regard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' That is why instead of dealing with the interior of the unit cube we perform a transfer(Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1) to a different space, where the situation with the isoperimetric problem is better, and by doing so derive the lower bounds on the initial space([16, Theorem 7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' At last, to derive lower bounds in the case of simplex new ideas and methods would need to be introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Here we repeat the approach from an article by Sasha Sodin [19], where an isoperimetric inequality for ℓp balls with p ∈ [1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 2] was proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' In particular, in case p = 1 we get hyperoctahedrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='8 is an immediate consequence of the isoperimetric inequality established in article [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Even though our method does provide asymptotically correct estimates, we should not expect it to lead to exact constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' We bound the growth of µ(Aδ) below by considering the isoperimetric problem for volume µ(Aδ), but that might lead to suboptimal estimates, since as δ varies Aδ does not have to look like an optimal isoperimetric region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' (a) region A (b) region Aδ (c) optimal region of volume µ(Aδ) 6 2 Preliminaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Assume that we are working in the space X with metric d and probability measure µ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' µ(X) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' In this section we are going to introduce some basic concepts related to our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The distance between a pair of non-empty subsets A, B ⊆ X is the infimum of distances between points from A and B dist(A, B) = inf x∈A,y∈B d(x, y) Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' A point x ∈ X belongs to the δ-enlargement of a subset A ⊆ X if it is at distance at most δ from some point of A Aδ = {x ∈ X | ∃y ∈ A: d(x, y) ≤ δ} We want to know how far apart from each other two subsets A, B ⊂ X of measure µ(A) = µ(B) = ε > 0 could be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' To that end, note that, if their δ-enlargements intersect, then the distance is bounded above by 2δ Aδ ∩ Bδ ̸= ∅ ⇒ dist(A, B) ≤ 2δ Our problem is concerned with the case of X being an open convex bounded subset of Rn of unit volume, d being the euclidean metric, and µ being the Lebesgue measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' In that case the following lemma holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' If A and B are two subsets of X with µ(A) + µ(B) ≥ 1, then they are at a distance 0 from each other dist(A, B) = 0 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Assume the contrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Let the distance between A and B be a positive number r = dist(A, B) > 0 This would mean that our subsets do not intersect, and thus µ(A) + µ(B) = µ(A ∩ B) + µ(A ∪ B) = µ(A ∪ B) = 1 Pick a pair of points a ∈ A and b ∈ B at a distance less than 2r d(a, b) < 2r 7 By c denote the midpoint of the segment between a and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The distance from c to both subsets A and B is strictly less than r, so the point c does not belong to any of our subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Now consider a δ-neighborhood of c that lies inside X with δ < r − d(a,b) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Clearly, it could not intersect neither A nor B, and at the same time it has a non-zero measure, so µ(X \\ (A ∪ B)) > 0, which leads to contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' To ensure the nonemptiness of the intersection of Aδ and Bδ the following condition would suffice µ(Aδ) ≥ 1 2 and µ(Bδ) ≥ 1 2 Thus we are interested in the growth of µ(Aε) considered as a function of ε, since that might lead to an upper bound on δ and consequently on dist(A, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The derivative of µ(Aε) at ε = 0 gives us Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' By the surface area of A ⊆ X we mean the following limit µ+(A) = lim ε→0+ µ(Aε) − µ(A) ε Lower bounds on µ+(A) might allow us to get results on the growth of µ(Aε), but all we know is the measure µ(A) of our subset A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' So we want to know the least possible value of µ+(A) when µ(A) is fixed, or at least bound µ+(A) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' By the isoperimetric profile we mean a function that maps t to the infimum of possible values that µ+(A) could take when µ(A) = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Iµ(t) = inf µ(A)=t µ+(A) We are no longer interested in the growth of µ(Aε) after we reach the measure of one half, also our initial µ(A) is greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' This means that we are only interested in the values of Iµ(t) when 0 < t < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' That is why throughout this paper by default the domain of the isoperimetric profile is the interval (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' A region, which has the minimal surface area amongst all the regions of the same measure, is called an isoperimetric region, and its boundary is called an isoperimetric hypersurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 8 3 Euclidean balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The euclidean ball is a perfect candidate for applying the symmetrization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' An argument([16, Theorems 1 and 5]) involving them completely classifies the optimal isoperimetric regions of the euclidean ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Lower bounds on the isoperimetric profile thus could be extracted by considering these optimal regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' By Bn we denote the unit n-ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Its volume is √πn Γ � n 2 + 1 � So the unit-volume n-ball will be of radius ωn = Γ �n 2 + 1 � 1 n √π ∼ � n 2πe By µ denote the Lebesgue measure on ωnBn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Combination of theorems 1 and 5 from [16] provides a classification of optimal isoperimetric regions in ωnBn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ([16, Theorems 1 and 5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Isoperimetric hypersurfaces in a ball are either hyperplanes passing through the origin or spherical caps which are orthogonal to the surface of ωnBn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' We would like to find lower bounds on the isoperimetric profile Iµ of ωnBn, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 it would suffice to consider intersections with balls orthogonal to ωnBn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' By Ψ(x) we denote Φ(√ex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Note that function Ψ(x) has a finite Lipschitz constant C > 0, since its derivative is a bounded function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' On the interval (−∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 0) both Ψ(x) and Ψ′(x) are increasing functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' For every ε0 ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2) and τ > 0 there is a number N such that the isoperimetric inequality Iµ(Ψ(t) + τ) ≥ Ψ′(t) would hold for all n > N and Ψ(t) ∈ (ε0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2 − τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' First, note that Iµ is a non-decreasing function on the interval (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Indeed, if one would take a ball orthogonal to ωnBn whose intersection with ωnBn is of volume ε ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2) and replace it with a ball that has the same radius but whose center is further away from the center of ωnBn, one would get a region of ωnBn of smaller volume and smaller surface area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Here we are going to prove that for any D, δ1, δ2 > 0 there is a number N such that for every n > N and 0 < d ≤ D there is an optimal isoperimetric region in ωnBn whose volume V and surface area S satisfy Ψ(−d) + δ1 ≥ V S ≥ Ψ′(−d − δ2) Note that our theorem follows from this last claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Indeed, in the state- ment of the theorem we require Ψ(t) to be in range (ε0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2 − τ), which means that t ≥ Ψ−1(ε0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Set D = −Ψ−1(ε0) and pick δ1, δ2 > 0 so that Cδ2 + δ1 ≤ τ t < Ψ−1 �1 2 − τ � ≤ −δ2 By our claim there will be a number N such that for every n > N and d ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' D] there would be an optimal isoperimetric region in ωnBn of volume V not greater than Ψ(−d)+δ1 and surface area S at least Ψ′(−d−δ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Since this region is optimal, Iµ(V ) = S And our bounds imply Iµ(Ψ(−d) + δ1) ≥ Iµ(V ) = S ≥ Ψ′(−d − δ2) (1) For an arbitrary t satisfying Ψ(t) ∈ (ε0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2 −τ) we can set d = −t−δ2 ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' D], but then Ψ(−d) + δ1 = Ψ(t + δ2) + δ1 ≤ Ψ(t) + Cδ2 + δ1 ≤ Ψ(t) + τ We combine this with (1) and get the desired isoperimetric inequality Iµ(Ψ(t) + τ) ≥ Iµ(Ψ(−d) + δ1) ≥ Ψ′(−d − δ2) = Ψ′(t) Now we need to prove our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Fix D > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' We would be considering the intersection of ωnBn and a ball Bn r (A) orthogonal to it such that the 10 distance from the origin, which we will denote here as O, to Bn r (A) is some number d 2 in range (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' D 2 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Two balls are orthogonal if their centers together with an arbitrary point on the intersection of the corresponding spheres form a right triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' O A X Y N H M Figure 2: The right triangle described above In the figure above ball ωnBn corresponds to the circle with center at O and radius OX, ball Bn r (A) – to the circle with center at A and radius AX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Since we have a right triangle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='OA = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='OX2 + AX2 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='ω2n + r2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='We require that ON = d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='ON = OA − AN = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='ω2n + r2 − r = d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='ω2n + r2 = r + d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='ω2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='n = rd + d2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='r = ω2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='d − d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='Consider the altitude XH of the right triangle OAX and note that ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='OH = OX2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='OA = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='ω2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='ω2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='n + r2 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='ω2n + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='ω2n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='�2 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='ω2n + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='d − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='4ω2n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='d2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2ω2n + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='d2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='16ω4n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='d + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='4ω2n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='�2 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='d + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='4ω2n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='d2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='4ω2n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='≤ d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='(2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='Let P be a hyperplane at a distance x from the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' By Sn(x) denote the volume of the hyperplane section of ωnBn by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Hyperplane P cuts ωnBn into two parts, at least one which is of volume not greater than 1 2, denote that volume by Vn(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Both Vn(x) and Sn(x) are decreasing functions defined on [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' It follows from the proof of theorem 1 from [20] that the sequence of func- tions Vn(x) uniformly converges to Ψ(−x) and that the sequence of functions Sn(x) uniformly converges to Ψ′(−x)(see Appendix C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' On the interval [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' D] positive continuous function Ψ′(−x) − Ψ′(−x − δ2) reaches its minimum value ε1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' By uniform convergence for all sufficiently large n we shall have Sn(x) ≥ Ψ′(−x) − ε1, which gives us Sn(OH) ≥ Sn(d) ≥ Ψ′(−d) − ε1 ≥ Ψ′(−d − δ2) for all d ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' D].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' We also know that Sn(OH) ≤ Sn(0) and Sn(0) converge towards Ψ′(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Thus Sn(OH) is always bounded above by some constant S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Hyperplane passing through the point H orthogonal to OA divides the intersection of two balls ωnBn and Bn r (A) into two spherical domes: Ω1 belonging to ωnBn and Ω2 belonging to Bn r (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The volume of ωnBn∩Bn r (A) is equal to the sum of volumes of Ω1 and Ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' We would like to bound above the volume of Ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Its base, the hyperplane section passing through H, has area not greater than S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Fix some number 1 > ε2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' On the segment HN pick a point M such that HM : HN = ε2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 12 By S1 denote the area of the hyperplane section of Bn r (A) passing through M orthogonal to OA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Volume of Ω2 could be bounded above as S1NM + S0MH = S1NH(1 − ε2) + S0NHε2 (3) Radius of the (n − 1)-ball corresponding to S1 equals to √ AX2 − AM2 = � AX2 − (AH + HM)2 = � (AX2 − AH2) − 2AH · HM − HM2 ≤ � XH2 − 2ε2AH · HN Radius of the (n−1)-ball corresponding to S0 is XH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' so the ratio of the two radii is � 1 − 2ε2 AH XH2HN Since XH is the altitude in the right triangle,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' XH2 is equal to OH · HA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' and the ratio could be rewritten as � 1 − 2ε2 HN OH Note that HN = OH − ON and that by formula (2) � 1 − 2ε2 HN OH = � � � � � �1 − 2ε2 \uf8eb \uf8ec \uf8ed1 − ON 2ON 1+ d2 4ω2n \uf8f6 \uf8f7 \uf8f8 = � � � �1 − 2ε2 � 1 − 1 + d2 4ω2n 2 � For all sufficiently large n d2 4ω2 n ≤ 1 2 ⇒ � � � �1 − 2ε2 � 1 − 1 + d2 4ω2n 2 � ≤ � 1 − ε2 2 We conclude S1 ≤ S0 � 1 − ε2 2 � n−1 2 Clearly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' NH ≤ OH ≤ d ≤ D,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' and by (3) the volume of Ω2 is not greater than DS0 �� 1 − ε2 2 � n−1 2 (1 − ε2) + ε2 � 13 Note that we could pick ε2 ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1) so that for all sufficiently large n DS0 �� 1 − ε2 2 � n−1 2 (1 − ε2) + ε2 � ≤ δ1 2 (4) The volume of Ω1 is Vn(OH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' And by (2) d − OH = d − d 1 + d2 4ω2n = d · d2 4ω2n 1 + d2 4ω2n ≤ D3 4ω2n We noted that the sequence of functions Vn(x) uniformly converges to Ψ(−x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Thus for all sufficiently large n Ψ(−OH) + δ1 4 ≥ Vn(OH) Note that for all x ≥ 0 0 < Ψ′(−x) ≤ Ψ′(0), which means Ψ(−d) + Ψ′(0) D3 4ω2 n + δ1 4 ≥ Ψ(−OH) + δ1 4 ≥ Vn(OH) (5) And for large enough n we have Ψ′(0) D3 4ω2n ≤ δ1 4 (6) We conclude that the volume V of ωnBn ∩ Bn r (A) is equal to the sum of volumes of Ω1 and Ω2, and thus by inequalities (4), (5), (6) Ψ(−d) + δ1 ≥ V By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 the intersection between ωnBn and Bn r is an optimal isoperimetric region in ωnBn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Its surface area S is equal to the surface area of the spherical cap corresponding to Ω2, which can be bounded below by the area of the base of Ω2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Sn(OH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' We have thus shown that the volume V and the free surface area S of ωnBn ∩ Bn r (A) satisfy Ψ(−d) + δ1 ≥ V S ≥ Sn(OH) ≥ Ψ′(−d − δ2) for all sufficiently large n and that the number d here can be chosen here as an arbitrary number from (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' D], which proves our claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 14 By dn(ε) denote the supremum of all possible distances between two sub- sets of volume ε ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2) inside ωnBn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Using our isoperimetric inequality we derive Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' For every ε ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2) lim sup n→∞ dn(ε) ≤ −2 1 √eΦ−1(ε), where the function −2 1 √eΦ−1(ε) is asymptotically equivalent to −2 1 √πe √ − ln ε as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Pick any ε0 ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' ε) and τ ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2 there is a number N such that inequality Iµ(Ψ(t) + τ) ≥ Ψ′(t) (7) holds for all n > N and t such that Ψ(t) ∈ (ε0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2 − τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Assume that n > N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' We consider two bodies A and B of volume ε inside the unit-volume euclidean ball ωnBn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' We are interested in the least values δA, δB such that the δA-enlargement of body A in ωnBn will be of volume 1 2 and δB-enlargement of body B will be of volume 1 2 too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' For these enlargements we shall have dist(AδA, BδB) = 0, from which dist(A, B) ≤ δA + δB follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Isoperimetric inequality (7) provides an estimate on the growth of δ- enlargements of our bodies: ∂+µ(Aδ) ≥ Ψ′(Ψ−1(µ(Aδ) − τ)) (8) when µ(Aδ) ≤ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' By δ0 denote Ψ−1(ε − τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Now consider the function y(δ) = Ψ(δ0 + δ) + τ 15 By δM denote the moment when y reaches 1 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' y(δM) = 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Assume that δM < δA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Functions µ(Aδ) and y(δ) coincide at δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Furthermore, because of inequality (8), we should have µ(Aδ) ≥ y(δ) and ∂+µ(Aδ) ≥ Ψ′(Ψ−1(µ(Aδ) − τ)) ≥ Ψ′(Ψ−1(y(δ) − τ)) = Ψ′(Ψ−1(Ψ(δ0 + δ))) = ∂+y(δ) for all δ ∈ [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' δM].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' But then we have a contradiction 1 2 = µ(AδA) > µ(AδM) ≥ y(δM) = 1 2 Thus δA and similarly δB are bounded above by δM, which implies dist(A, B) ≤ δA + δB ≤ 2δM The value δM satisfies 1 2 = y(δM) = Ψ(δ0 + δM) + τ 1 2 − τ = Ψ(δ0 + δM) δM = Ψ−1 �1 2 − τ � − Ψ−1 (ε − τ) As τ tends to 0 Ψ−1 �1 2 − τ � − Ψ−1 (ε − τ) → −Ψ−1(ε) = − 1 √eΦ−1(ε) And since we can choose τ to be an arbitrary number in (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' ε) lim sup n→∞ dn(ε) ≤ −2 1 √eΦ−1(ε) Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Symmetrization techniques lead to the solution of the isoperimetric problem in other different cases: the classical isoperimetric problem in Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' isoperimetric inequality on the sphere([8, Appendix], [10, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1]), from which the gaussian isoperimetric inequality could be derived([10, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='3], [16, Theorem 20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 16 4 Unit cubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Unlike the euclidean ball the cube does not have «many» symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' To derive lower bounds on the isoperimetric profile we are going to perform a «transfer» to a different space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Descriptions of this idea could be found in [16, Theorem 7], [13, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' In this section we will follow the approach presented in [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Consider an n-dimensional unit cube (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' We can think of it as of a space with Lebesgue measure µ and Euclidean metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Now we would like to be able to show estimates on the isoperimetric profile of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' However, it is quite unclear how to deal with the corresponding space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' For example, the cube only has a finite number of symmetries, so symmetrization methods would not get us far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' That is why it makes sense to consider a way to transfer to a different, «better» space – an idea that plays a key role in our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ([16, Proposition 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Assume that for a pair of spaces M and M′ with measures υ and υ′, respectively, we have a map φ : M → M′ which transforms measure υ into υ′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' µ′(A) = µ(φ−1(A)), and that is also c-Lipschitz for some c > 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' a pair of points in M at a distance d has images at distance at most c · d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The following inequality holds Iυ ≤ c · Iυ′ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Consider a closed R′ ⊆ M′ and its preimage R = φ−1(R′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Since φ transforms υ into υ′, we shall have υ′(R′) = υ(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The fact that φ is c-Lipschitz gives us φ(Rε) ⊆ R′ cε, from which it follows that υ′(R′ cε) = υ(φ−1(R′ cε)) ≥ υ(φ−1(φ(Rε))) ≥ υ(Rε) By combining this inequality with υ′(R′) = υ(R) we get υ′(R′ cε) − υ′(R′) cε ≥ υ(Rε) − υ(R) cε And by taking limit ε → 0 we reach conclusion (υ′)+(R′) ≥ 1 cυ+(R) So for every closed R′ ⊆ M′ we can find R ⊆ M that has the same measure and whose surface area is at most c times the surface area of R′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Thus we shall have Iυ(t) ≤ cIυ′(t) 17 We are going to apply the above lemma to get lower bounds on Iµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The role of M′ will play our cube with the Lebesgue measure µ on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The role of M will play the space Rn with Gaussian measure γn defined by its density at a point x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' , xn) as dγn dx = e−π(x2 1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='+x2 n) In the one-dimensional case the map Φ defined by Φ(a) = � a −∞ e−πx2dx transforms (−∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' +∞) into (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Also Φ turns Gaussian measure γ1 on (−∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' +∞) into the Lebesgue measure on (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Indeed, the Gaussian mea- sure of the segment [a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' b] is equal to the integral � b a e−πx2dx of its density, which in turn is equal to Φ(b) − Φ(a), but the image of [a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' b] under our map Φ is [Φ(a);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Φ(b)], whose Lebesgue measure is equal to Φ(b) − Φ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Now the role of φ in the above lemma will be played by φ(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' , xn) = (Φ(x1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' , Φ(xn)), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' we are applying Φ coordinatewise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' It indeed transforms (−∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' +∞)n = Rn into (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' For every box [a1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' b1] × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' × [an;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' bn] we could note that γn([a1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' b1] × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' × [an;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' bn]) = � b1 a1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' � bn an e−πx2 1 · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' · e−πx2 ndxn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' dx1 = �� b1 a1 e−πx2 1dx1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' �� bn an e−πx2 ndxn � = (Φ(b1) − Φ(a1)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' (Φ(bn) − Φ(an)) = µ(φ([a1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' b1] × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' × [an;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' bn])), so φ turns measure γn into µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' And finally, map φ is 1-Lipschitz since |Φ′(x)| = |e−πx2| ≤ 1 and φ applies Φ coordinatewise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' As we see the requirements of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 are met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' So the isoperi- metric profile Iµ of our unit cube could be bounded below by Iγn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' But what do we know about Iγn?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Well, there are tight inequalities on the isoperimet- ric profile of Gaussian measures, but here we would only need the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 18 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ([10, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Let γn be the standard Gaussian measure defined by its density at a point x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' , xn) ∈ Rn as dγn dx = 1 √ 2π ne− 1 2(x2 1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='+x2 n) Amongst all subsets A ⊂ Rn with fixed measure γn(A) ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2) the minimum surface area is attained at half-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' In general, a Gaussian measure γn µ,σ2 is a measure defined by its density at a point x ∈ Rn as dγn µ,σ2 dx = 1 √ 2πσ2ne− 1 2σ2 ∥x−µ∥2 But Gaussian measures are equivalent to each other under translation and scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' For example, if we shrink the standard Gaussian measure γn by a factor of √ 2π, the density ρ′ of the resulting measure is related to the density ρ of γn as ρ′(x) = √ 2π nρ( √ 2πx) = √ 2π n 1 √ 2π ne− 1 2 ∥ √ 2πx∥2 = dγn dx And we could also note that for A ⊂ Rn γn(A) = γn � 1 √ 2π A � (γn)+(A) = 1 √ 2π γ+ n � 1 √ 2π A � So shrinking everything by a factor of √ 2π in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 would not change the fact that half-spaces are optimal solutions to the isoperimetric problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' And that is why to figure out lower bounds on Iγn we would only need to consider half-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The density of γn at a point x only depends on ∥x∥, so our measure is rotation-invariant, which means that we could only consider half-spaces Ha defined by xn ≤ a for some a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' First, we could note that γn(Ha) = γn((−∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' +∞) × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' × (−∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' +∞) � �� � n−1 times ×(−∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' a]) = γ1((−∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' +∞)) × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' × γ1((−∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' +∞)) � �� � n−1 times ×γ1((−∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' a]) = γ1((−∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' a]) = Φ(a) 19 And, because of this last observation, γ+ n (Ha) = lim ε→0 γn((Ha)ε) − γn(Ha) ε = lim ε→0 γn(Ha+ε) − γn(Ha) ε = lim ε→0 γ1((−∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' a + ε]) − γ1((−∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' a]) ε = lim ε→0 γ1((−∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' a]ε) − γ1((−∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' a)) ε = γ+ 1 ((−∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' a]) Equalities γn(Ha) = γ1((−∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' a]) γ+ n (Ha) = γ+ 1 ((−∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' a]) imply Iγn = Iγ1 And now we only have to estimate Iγ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' To do that we need to consider intervals (−∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' a] for a < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The measure of such an interval is Φ(a) and the surface area is Φ′(a) = e−πa2, from which we get Iγ1(Φ(a)) = e−πa2 By combining our observations we conclude Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2 ([16, Theorem 7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' For the Lebesgue measure µ on the unit cube (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1)n isoperimetric inequality Iµ(t) ≥ e−πΦ−1(t)2 holds for all t ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Note that for all a < 0 Iµ(Φ(a)) ≥ Iγn(Φ(a)) = Iγ1(Φ(a)) = e−πa2 Using this isoperimetric inequality we derive Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Inside a unit cube (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1)n two bodies A and B of volume ε ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2) are at a distance at most −2Φ−1(ε) 20 Function −2Φ−1(ε) is asymptotically equivalent to 2 √π √ − ln ε as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' We are interested in the least values δA, δB such that the δA-enlargement of body A in the unit cube (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1)n will be of volume 1 2 and the δB-enlargement of body B will be of volume 1 2 too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' For these enlargements we shall have dist(AδA, BδB) = 0, from which dist(A, B) ≤ δA + δB follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Isoperimetric inequality from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2 provides an estimate on the growth of δ-enlargements of our bodies: ∂+µ(Aδ) ≥ e−πΦ−1(µ(Aδ))2 (9) when µ(Aδ) < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' By δM denote −Φ−1(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Now consider the function y(δ) = Φ(−δM + δ) Assume that δM < δA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Functions µ(Aδ) and y(δ) coincide at δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Further- more, because of inequality (9), we should have µ(Aδ) ≥ y(δ) and ∂+µ(Aδ) ≥ e−πΦ−1(µ(Aδ))2 ≥ e−πΦ−1(y(δ))2 = e−π(−δM +δ)2 = ∂+y(δ) for all δ ≤ δM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' But then we have a contradiction 1 2 = µ(AδA) > µ(AδM) ≥ y(δM) = Φ(0) = 1 2 Thus δA and similarly δB are bounded above δM, which implies dist(A, B) ≤ δA + δB ≤ 2δM = −2Φ−1(ε) 21 Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The transfer from one space to another might have lead to the loss of accuracy to some extent, so there is not much of what we could say about how precise our estimates are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The problem of finding optimal hypersurfaces in the n-cube also seems to be quite complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' More de- tails about the isoperimetric inequalities in a cube and their applications the reader may find in section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='5 of [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' In a similar way we could derive lower bounds on the isoperimetric profile of the euclidean ball, since the transition to the space Rn with gaussian measure is possible(see [13, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 22 5 Simplexes and ℓp-balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' In this section we are going to describe the approach used in [19] to prove an isoperimetric inequality for ℓp balls(p ∈ [1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 2]) by presenting a very similar proof of an isoperimetric inequality for simplexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Much like in the case of the unit cube we will be performing a transfer to a different space(see Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' But to address the problems with Lipschitz continuity(see Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='3) a number of new ideas and methods needs to be introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 Simplexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' By R+ we denote the interval (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Consider a regular simplex ∆n defined as ∆n = {(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' , xn) ∈ Rn + | x1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' + xn = 1} By µ we will denote the normalized Lebesgue measure on ∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Note that µ(∆n) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Simple calculations show that the area of ∆n is equal to n√n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' So if we set ωn = � n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' n√n � 1 n−1 ∼ n e , the area of ωn∆n will be equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' By λ denote the Lebesgue measure on ωn∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' One could note µ+ � 1 ωn A � = lim ε→0 µ �� 1 ωnA � ε � − µ � 1 ωnA � ε = lim ε→0 λ(Aωnε) − λ(A) ε = ωnλ+(A) (10) for A ⊆ ∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' In this section we will be using a slightly different notion of an isoperi- metric profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' By the isoperimetric function we mean a function that maps t ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2) to the infimum of possible values that µ+(A) could take when t ≤ µ(A) < 1 2 Iµ(t) = inf t≤µ(A)< 1 2 µ+(A) 23 Our observation (10) implies Iµ = ωnIλ, (11) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' the isoperimetric functions are proportional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' To solve our problem we would need estimates on Iλ, but for the sake of simplicity we would be working with Iµ instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Yet again it is quite unclear how to deal with ∆n as a space, so we would like to be able to transfer to a «better» space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The map T : Rn + → ∆n defined as T(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' , xn) = � x1 x1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' + xn , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' , xn x1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' + xn � transforms the measure νn on Rn + defined by its density at a point (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' , xn) ∈ Rn + as dνn dx = e−x1−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−xn into the normalized Lebesgue measure µ on ∆n as a corollary of the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ([18, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Let X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' , Xn be independent random variables each with density function 1 2e−|t| and put S = � i |Xi|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Then ( X1 S , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' , Xn S ) induces the normalized Lebesgue measure on the surface of ℓn 1 ball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Moreover, ( X1 S , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' , Xn S ) is independent of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Indeed, because of the symmetry amongst the orthants of Rn,2 we can restrict our attention to the positive orthant Rn + in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 and reach the desired conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' But what do we know about the isoperimetric profile of Iνn?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The fol- lowing lemma completely determines this isoperimetric profile in the one- dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2 ([21, Remark 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' By ν denote ν1, then Iν(t) = min(t, 1 − t), where the domain of Iν is the whole interval (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 2the orthants of Rn are multidimensional analogues of the quadrants of R2 24 For a measure υ we could consider its isoperimetric constant – the largest value Is(υ) for which the following holds for all subsets A with υ(A) ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1) υ+(A) ≥ Is(υ) min(υ(A), 1 − υ(A)) And by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2 we have Is(ν) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Now we could note that3 νn = νn since the density of νn at a point (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' , xn) ∈ Rn + could be written as a product e−x1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' e−xn And thus the following theorem Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ([2, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' For triple (X, d, ψ) – space, metric, mea- sure, Is(ψn) ≥ 1 2 √ 6 Is(ψ) gives us Iνn(t) ≥ 1 2 √ 6 min(t, 1 − t) or4 Iνn(t) ≥ 1 2 √ 6t (12) So we have a map T : Rn + → ∆n that transforms νn into µ and a lower bound on Iνn(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' But to use Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 we would also need our mapping T to be Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' In the neighborhood of a point x ∈ Rn + the behavior of our map T could be described by a linear operator defined by the matrix, whose entries are ∂Tj(x) ∂xi , where Tj(x) is the j-th coordinate of T(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' We are interested in the norm of this linear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The next lemma gives an upper bound Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='3 (corresponds to Lemma 1 from [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' ���� ∂Tj(x) ∂xi ���� 2 ≤ 1 ∥x∥1 (1 + √n∥T(x)∥2) 3by this we mean the product measure ν × .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' × ν � �� � n times 4recall that, generally, when we talk about isoperimetric profiles we are only interested in the values of t ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' the domain of Iνn is (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2) 25 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' First, we will calculate the entries of our matrix, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' the partial deriva- tives ∂ ∂xi Tj(x) = ∂ ∂xi xj x1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' + xn = (x1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' + xn) ∂ ∂xixj − xj ∂ ∂xi(x1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' + xn) (x1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' + xn)2 = 1 x1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' + xn � δij − xj x1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' + xn � If ∆y is the image of ∆x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' then ∆yj = � i ∂Tj ∂xi ∆xi = 1 ∥x∥1 � ∆xj − xj ∥x∥1 � i ∆xi � (13) The length of the vector,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' whose coordinates are xj ∥x∥1 � i ∆xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' could be esti- mated as ∥T(x)∥2 ����� � i ∆xi ����� ≤ √n∥T(x)∥2∥∆x∥2 since coordinates xj ∥x∥1 define T(x) and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' clearly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' � i |∆xi| ≤ √n �� i (∆xi)2 � 1 2 And now by triangle inequality from (13) we get ∥∆y∥2 ≤ 1 ∥x∥1 � ∥∆x∥2 + √n∥T(x)∥2∥∆x∥2 � = 1 ∥x∥1 � 1 + √n∥T(x)∥2 � ∥∆x∥2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' from which the statement of the lemma follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Indeed, we cannot apply Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 in our case since there are places where the image of the mapping T varies wildly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' For example, the simplex in Rn + defined by x1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' + xn = δ, where δ is a very small number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 26 However, the upper bound from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='3 ���� ∂Tj(x) ∂xi ���� 2 ≤ 1 ∥x∥1 (1 + √n∥T(x)∥2) tells us that the only parts of Rn +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' where the variance of T could be high,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' are the regions where ∥x∥1 is too small,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' which corresponds to a little corner of the orthant Rn +,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' or where ∥T(x)∥2 is too large,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' since the distance from the origin to the center of ∆n – 1 √n is much smaller than the distance from the origin to the vertices of ∆n – 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' the region where ∥T(x)∥2 is too large corresponds to the little corners of ∆n near the vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='5 Our last observation suggests that the parts of our space, where T does not behave the way we want it to, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' high variance, might be negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' And now we are going to present an argument that will allow us to «get rid» of these regions, perform the transfer to the space Rn + with measure νn and then apply the lower bound for Iνn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' But, first of all, since we want to estimate Iµ(t) on the whole half-interval (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2], including the small values of t, and since we are intending to remove some negligible parts of our space in the main argument, we need to employ a different approach for those values of t that might, perhaps, be lesser than the measure of the regions that we are getting rid of.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' In other words, an estimate on the surface area of the «small» subsets of ∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' This could be done with the help of the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2 ([1, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Let ψ be a log-concave probability measure on Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' For all measurable sets A ⊂ Rn, for every point x0 ∈ Rn and every number r > 0, ψ+(A) ≥ 1 2r � ψ(A) ln 1 ψ(A) + (1 − ψ(A)) ln 1 1 − ψ(A) + ln ψ({|x − x0| ≤ r}) � Convexity of ∆n ensures that the normalized Lebesgue measure µ on it is log-concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' In the application of the above theorem to measure µ on ∆n it is possible to set x0 = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' , 0), even though x0 /∈ ∆n, since when r > 1 √n for x1 = 5here we are talking about the image of T 27 ( 1 n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' , 1 n) and r′ = � r2 − 1 n we shall have ln µ({|x| ≤ r}) = ln µ({|x − x1| ≤ r′}) 1 2r′ ≥ 1 2r One question that arises after examining the lower bound is: how do we estimate ψ({|x − x0| ≤ r}) – the measure of the ball with center at x0 and radius r?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' In terms of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 we have the following result Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='3 ([18, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' There are absolute positive constants T, c such that for all t > T √n, putting X = (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' , Xn) and S = X1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' + Xn, Pr �∥X∥2 S > t � ≤ e−ctn Again, because of the symmetry, we can restrict our attention to the positive orthant Rn +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' By combining Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='3 with Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 we get µ({|x| ≤ r}) ≥ 1 − e−cnr for r > T √n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Now note r > T √n ⇒ cnr ≥ c√nT ≥ cT ⇒ e−cnr ≤ e−cT, which implies that for r > T √n ln µ({|x| ≤ r}) ≥ ln(1 − e−cnr) ≥ −Ce−cnr for some constant C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' If we assume that µ(A) < c′ for some constant 0 < c′ < 1, then we will have (1 − µ(A)) ln � 1 1 − µ(A) � ≥ C1µ(A) for some constant C1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' We would like the sum of the two last terms of µ(A) ln 1 µ(A) + (1 − µ(A)) ln 1 1 − µ(A) + ln µ({|x − x0| ≤ r}) 28 to be non-negative, for this the following will be sufficient C1µ(A) ≥ Ce−cnr, which could be rewritten as ln C1µ(A) C ≥ −cnr ⇔ r ≥ 1 cn ln C C1µ(A) = 1 cn � ln 1 µ(A) + ln C C1 � If we replace ≥ above with equality and apply Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2 with x0 = 0, we will get µ+(A) ≥ 1 2cnµ(A) ln 1 µ(A) ln 1 µ(A) + ln C C1 The condition r > T √n means that we need ln 1 µ(A) + ln C C1 > cT√n to apply Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2 here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' And so these last inequalities imply Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' There are universal constants cs, C > 0 such that 6 µ+(A) ≥ csnµ(A) for all subsets A ⊂ ∆n with µ(A) < e−C√n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Now that we have dealt with the case of «small» sets, we can proceed to the main argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The gradient modulus ∥∇f∥2 of a locally Lipschitz function f is ∥∇f(x)∥2 = lim sup ∥x−y∥2→0+ ∥f(x) − f(y)∥2 ∥x − y∥2 The next lemma would allow us to switch between the two equivalents of the isoperimetric problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 6here for the sake of simplicity we forget about the auxiliary constants c′, C, C1 intro- duced during the proof of this proposition 29 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='4 ([19, Proposition A]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Let µ be a probability measure, 0 < a < 1 2 and b > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The following are equivalent (a) Iµ(a) ≥ b (b) for any locally Lipschitz function φ : supp µ → [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1] such that µ{φ = 0} ≥ 1 2 and µ{φ = 1} ≥ a, � ∥∇φ∥2dµ ≥ b Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' In this theorem and its applications in this article we can replace «locally Lipschitz» in (b) with just «Lipschitz», since it would not affect the implication (a) ⇒ (b), and in the proof([19]) of the implication (b) ⇒ (a) only Lipschitz functions φ: supp µ → [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1] were considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The idea of «getting rid» of unwanted parts of our space would be realized through the so-called cut-off functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' A cut-off function maps our space to [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1], where 0 corresponds to the regions we are getting rid of, 1 – to the regions we want to keep, and we would also need our function to take values in between 0 and 1 to ensure continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The following lemma shows how a cut-off function h can be applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='5 ([19, Lemma 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' If k, h: Rn → [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1] are two locally Lipschitz functions, then ∥∇k∥2 ≥ ∥∇(kh)∥2 − ∥∇h∥2 By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='4 we can speak about the isoperimetric problem in terms of the integral of gradient modulus ∥∇φ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='5 would allow us to pass from φ to function φ · h, which vanishes on the unwanted region of our space, at the cost of an error term ∥∇h∥2 � ∥∇φ∥2dµ ≥ � ∥∇(φ · h)∥2dµ − � ∥∇h∥2dµ Recall that our upper bound on gradient modulus of mapping T : Rn + → ∆n is ∥∇T∥2 = ���� ∂Tj(x) ∂xi ���� 2 ≤ 1 ∥x∥1 (1 + √n∥T(x)∥2) So we are going to need two cut-off functions: one for parts of ∆n that are too far from the origin will be of the form h1 : Rn → [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1], h1(x) = max(0, min(1, 2 − c1 √n∥x∥2)), 30 and will take care of large values of ∥T(x)∥2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' another for the region of Rn with low ∥x∥1 will be of the form h2 : Rn → [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1], h2(x) = max(0, min(1, c2n−1∥x∥1 − 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Constants c1 and c2 will be chosen later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Note that the two cut-off functions are meant for different domains: h1 for ∆n and h2 for Rn +, but since both Rn + and ∆n lie inside Rn we choose Rn as their domain of definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' To employ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='5 in our argument we are going to need some es- timates related to the arising error terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' In the following lemma some properties of our cut-off functions and their gradients will be established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='6 (corresponds to Lemma 3 from [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The cut-off function h1 has the following properties h1(x) = 1 ⇔ ∥x∥2 ≤ 1 c1 √n (14) h1(x) = 0 ⇔ ∥x∥2 ≥ 2 c1 √n (15) ∥∇h1∥2 ≤ c1 √n (16) The cut-off function h2 has the following properties h2(x) = 1 ⇔ ∥x∥1 ≥ 2 c2 n (17) h2(x) = 0 ⇔ ∥x∥1 ≤ n c2 (18) ∥∇h2∥2 ≤ c2 √n (19) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Properties (14), (15), (17), (18) immediately follow from the definition of our cut-off functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' By the triangle inequality we note that the gradient modulus of ∥x∥2 considered as a function from Rn + to R+ is not greater than 1, thus ∥∇h1∥2 ≤ c1 √n Inequalities ∥x + ∆x∥1 ≤ ∥x∥1 + ∥∆x∥1, 31 ∥∆x∥1 ≤ √n∥∆x∥2 imply that the gradient modulus of ∥x∥1 is not greater than √n, from which we derive ∥∇h2∥2 ≤ c2 √n Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='7 (corresponds to Lemma 4 from [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' For α ≥ 0 we have νn{∥x∥1 ≤ αn} ≤ 1 √ 2πn(αe)n And for every α > T by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='3 we have µ � ∥x∥2 ≥ α √n � ≤ e−αc√n Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Since the density of νn everywhere in Rn + is not greater than 1, we can bound νn{∥x∥1 ≤ αn} above by the volume of the region of Rn + defined by ∥x∥1 ≤ αn, which is equal to 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' (αn)n By Stirling’s approximation n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' ≥ √ 2πn �n e �n e 1 12n+1 And so we arrive at νn{∥x∥1 ≤ αn} ≤ 1 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' (αn)n ≤ 1 √ 2πn � e n �n (αn)ne− 1 12n+1 ≤ 1 √ 2πn(αe)n Note that α √n > T √n, so by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='3 µ � ∥x∥2 ≥ α √n � ≤ e−c α √nn = e−αc√n Now we are ready to present the main argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 32 Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2 (corresponds to Proposition 2 from [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' There is a uni- versal constant cb > 0 such that for all e−C√n ≤ t < 1 2 Iµ(t) ≥ cbnt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Pick e−C√n ≤ a < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' According to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='4 and Remark 1 the problem of finding lower bounds on Iµ(a) is equivalent to the estimation of � ∆n ∥∇f∥2dµ for a Lipschitz function f : ∆n → [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1] such that µ{f = 0} ≥ 1 2 and µ{f = 1} ≥ a (20) To «get rid» of the parts of ∆n that are too far from the origin we can use our cut-off function h1 and by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='5 we will get � ∆n ∥∇f∥2dµ ≥ � ∆n ∥∇(fh1)∥2dµ − � ∆n ∥∇h1∥2dµ (21) Here by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='6 we can estimate the error term as � ∆n ∥∇h1∥2dµ ≤ c1 √n µ � ∥x∥2 ≥ 1 c1 √n � (22) Mapping T : Rn + → ∆n transforms measure νn into µ, which allows us to replace integrals over ∆n with integrals over Rn + as follows � ∆n w dµ = � Rn + (w ◦ T)dνn Denote (fh1) ◦ T by g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' As we already noted � ∆n ∥∇(fh1)∥2dµ = � Rn + ∥∇(fh1) ◦ T∥2dνn (23) An observation similar to the chain rule of differentiation could be made ∥∇(a ◦ b)∥2 ≤ ∥(∇a) ◦ b∥2 · ∥∇b∥2, 33 which in our case would mean that � Rn + ∥∇(fh1) ◦ T∥2dνn ≥ � Rn + ∥∇g∥2 ∥∇T∥2 dνn, (24) and since ∥∇T∥2 ̸= 0 by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='3 could be bounded above by 1 ∥x∥1(1 + √n∥T(x)∥2) we have � Rn + ∥∇g∥2 ∥∇T∥2 dνn ≥ � Rn + ∥∇g∥2∥x∥1 1 + √n∥T(x)∥2 dνn (25) But h1 is zero when ∥x∥2 ≥ 2 c1 √n by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' So the gradient modulus ∥∇g∥2 is equal to zero when ∥T(x)∥2 > 2 c1 √n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' And because of this, � Rn + ∥∇g∥2∥x∥1 1 + √n∥T(x)∥2 dνn ≥ � Rn + ∥∇g∥2∥x∥1 1 + √n 2 c1 √n dνn = 1 1 + 2 c1 � Rn + ∥∇g∥2∥x∥1dνn (26) Now to «get rid» of the region of Rn + where ∥x∥1 is too small we will apply our cut-off function h2 and by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='5 get � Rn + ∥∇g∥2∥x∥1dνn ≥ � Rn + ∥∇(gh2)∥2∥x∥1dνn − � Rn + ∥∇h2∥2∥x∥1dνn (27) By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='6 we have the following upper bound on the error term � Rn + ∥∇h2∥2∥x∥1dνn ≤ c2 √n 2n c2 νn � ∥x∥1 ≤ 2n c2 � = 2√nνn � ∥x∥1 ≤ 2n c2 � (28) Cut-off function h2 is zero when ∥x∥1 ≤ n c2, thus ∥∇(gh2)∥2 is equal to zero when ∥x∥1 < n c2, from which it follows that � Rn + ∥∇(gh2)∥2∥x∥1dνn ≥ n c2 � Rn + ∥∇(gh2)∥2dνn (29) 34 Now consider function gh2: Rn + → [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1] gh2 = ((f · h1) ◦ T) · h2 Note that if (f ◦ T)(x) = 0 for x ∈ Rn +, then (gh2)(x) = 0 too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' By our assumption µ{f = 0} ≥ 1 2, which implies νn{gh2 = 0} ≥ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Function gh2 equals to 1 at a point x ∈ Rn + if and only if (f ◦ T)(x) = 1, and (h1 ◦ T)(x) = 1, and h2(x) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' To estimate νn{gh2 = 1} we will subtract νn{(h1 ◦ T) < 1} = µ{h1 < 1} and νn{h2 < 1} from νn{(f ◦T) = 1} = µ{f = 1}, which by our assumption (20) is greater than a, and get νn{gh2 = 1} ≥ a − µ{h1 < 1} − νn{h2 < 1} By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='6 µ{h1 < 1} = µ � ∥x∥2 > 1 c1 √n � νn{h2 < 1} = νn � ∥x∥1 < 2n c2 � Isoperimetric inequality (12) on νn combined with Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='4 would give us � Rn + ∥∇(gh2)∥2dνn ≥ 1 2 √ 6 � a − µ � ∥x∥2 > 1 c1 √n � − νn � ∥x∥1 < 2n c2 � � (30) Putting inequalities (21),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' (22),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' (23),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' (24),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' (25),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' (26),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' (27),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' (28),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' (29) together,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' we arrive at � ∆n ∥∇f∥2dµ ≥ 1 c2 1 1 + 2 c1 n � Rn + ∥∇(gh2)∥2dνn − c1 √nµ � ∥x∥2 ≥ 1 c1 √n � − 2 1 + 2 c1 √nνn � ∥x∥1 ≤ 2n c2 � 35 We combine this with inequality (30) and get � ∆n ∥∇f∥2dµ ≥ 1 2 √ 6 1 c2 1 1 + 2 c1 na − � 1 2 √ 6 1 c2 1 1 + 2 c1 n + c1 √n � µ � ∥x∥2 ≥ 1 c1 √n � − � 1 2 √ 6 1 c2 1 1 + 2 c1 n + 2 1 + 2 c1 √n � νn � ∥x∥1 ≤ 2n c2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' which could be rewritten as � ∆n ∥∇f∥2dµ ≥ 1 2 √ 6 1 c2 1 1 + 2 c1 n � a − � 1 + 2 √ 6c1c2 � 1 + 2 c1 � 1 √n � µ � ∥x∥2 ≥ 1 c1 √n � − � 1 + 4 √ 6c2 1 √n � νn � ∥x∥1 ≤ 2n c2 � � If 1 c1 > T ⇔ c1 < 1 T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' then by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='7 we should have � ∆n ∥∇f∥2dµ ≥ 1 2 √ 6 1 c2 1 1 + 2 c1 n � a − � 1 + 2 √ 6c2 (c1 + 2) 1 √n � e− c c1 √n − � 1 + 4 √ 6c2 1 √n � 1 √ 2πn �2e c2 �n � Now we can choose appropriate values for constants c1 and c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' We choose c2 to be large enough so that � 1 + 4 √ 6c2 1 √n � 1 √ 2πn �2e c2 �n ≤ 1 3e−C√n holds for all natural n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' This is possible since one can note that �2e c2 �n = e − ln � 2e c2 � n 36 After that we choose c1 to be small enough so that � 1 + 2 √ 6c2 (c1 + 2) 1 √n � e− c c1 √n ≤ 1 3e−C√n holds for all natural n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Our a is at least e−C√n, which means � ∆n ∥∇f∥2dµ ≥ 1 2 √ 6 1 c2 1 1 + 2 c1 n � a − 2 3e−C√n � ≥ 1 2 √ 6 1 c2 1 1 + 2 c1 n �1 3a � And since f here can be an arbitrary Lipschitz function f : ∆n → [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1] with µ{f = 0} ≥ 1 2 and µ{f = 1} ≥ a we by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='4 conclude Iµ(a) ≥ 1 6 √ 6 1 c2 1 1 + 2 c1 na Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2 imply Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' For the Lebesgue measure λ on the unit-volume simplex ωn∆n the following isoperimetric inequality Iλ(t) ≥ cλt holds for all t ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2), where cλ > 0 is a universal constant independent of the dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' By Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 µ+(A) ≥ csnµ(A) for all A ⊂ ∆n with µ(A) ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' e−C√n), and by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2 Iµ(t) ≥ cbnt for all t ∈ [e−C√n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2), which means that Iµ(t) ≥ min(cs, cb)nt 37 for all t ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Equation (11) relates Iµ and Iλ to each other as Iλ = 1 ωn Iµ Thus for all t ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2) we must have Iλ(t) ≥ min(cs, cb) n ωn t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Here we could note that n ωn is positive for all n and that by Stirling’s approximation lim n→∞ n ωn = e, which must imply that inf n n ωn > 0 So we can take min(cs, cb) inf n n ωn as our constant cλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' From this isoperimetric inequality we conclude Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Inside a unit-volume simplex ωn∆n two bodies A and B of volume ε ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2) are at a distance at most −c ln ε for some universal constant c > 0 independent of the dimension n and volume ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' We are interested in the least values δA, δB such that the δA-enlargement of body A in ωn∆n will be of volume 1 2 and the δB-enlargement of body B will be of volume 1 2 too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' For these enlargements we shall have dist(AδA, BδB) = 0, from which dist(A, B) ≤ δA + δB 38 follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Isoperimetric inequality from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='4 provides an estimate on the growth of δ-enlargements of our bodies δ+λ(Aδ) ≥ cλλ(Aδ) which holds as long as λ(Aδ) < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' And so to bound δA above we would like consider a function y(δ) that behaves in accordance with our lower bound y(0) = λ(A) (31) y′ = cλy (32) If by δM we will denote the moment when y reaches 1 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' y(δM) = 1 2, then δA ≤ δM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Indeed, otherwise δM < δA, but functions λ(Aδ) and y(δ) coincide at δ = 0 and for all δ ∈ [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' δM] we should have λ(Aδ) ≥ y(δ) and δ+λ(Aδ) ≥ cλλ(Aδ) ≥ cλy(δ) = δ+y(δ) And so we reach contradiction 1 2 = λ(AδA) > λ(AδM) ≥ y(δM) = 1 2 A solution to differential equation (32) should be of the form Cecλδ and since at δ = 0 by our initial condition (31) we should have y(0) = λ(A) we reach conclusion y(δ) = λ(A)ecλδ So δM will be a solution to equation λ(A)ecλδM = 1 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' which after taking logarithm on both sides turns into ln λ(A) + cλδM = − ln 2 δM = − 1 cλ (ln λ(A) + ln 2) By the same reasoning δB ≤ δM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' and thus dist(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' B) ≤ δA + δB ≤ − 2 cλ (ln λ(A) + ln 2) ≤ − 2 cλ ln λ(A) 39 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2 ℓp-balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' By the ℓn p unit ball we mean ℓn p = {(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' , xn) ∈ Rn | |x1|p + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' + |xn|p ≤ 1} Let µ be a normalized Lebesgue measure on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Note that µ(ℓn p) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The volume of ℓn p is equal to 2nΓ � 1 + 1 p �n Γ � 1 + n p � by theorem 1 from [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' So in order to get a unit-volume ℓn p ball we would need to stretch the ℓn p unit ball by a factor of ωn = Γ � 1 + n p � 1 n 2Γ � 1 + 1 p � ∼ n 1 p 2Γ � 1 + 1 p � (pe) 1 p By λ denote the Lebesgue measure on ωnℓn p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Yet again by (11) we should have proportionality of the isoperimetric functions Iµ = ωnIλ (33) The following theorem was proven by Sasha Sodin in [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='6 ([19, Theorem 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' There exists a universal constant c > 0 such that for 1 ≤ p ≤ 2, 0 < a < 1 2 Iµ(a) ≥ cn 1 pa log1− 1 p a It follows that Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' For every p ∈ [1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 2] there exists a positive constant cp > 0 such that Iλ(a) > cpa log1− 1 p a Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' By (33) we already now that Iλ(a) ≥ c 1 ωn n 1 pa log1− 1 p a 40 The number c ωnn 1 p is positive for all n and by Stirling’s approximation lim n→∞ c ωn n 1 p = 2cΓ � 1 + 1 p � (pe) 1 p > 0 Thus cp = inf n c 1 ωn n 1 p > 0 and Iλ(a) ≥ cpa log1− 1 p a From this isoperimetric inequality we derive Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Inside a unit-volume ℓp n ball ωnℓn p two bodies A and B of volume ε ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2) are at a distance at most Cp log 1 p 1 ε for some constant Cp > 0 independent of dimension n and volume ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' We are interested in the least values δA, δB such that the δA-enlargement of body A in ωnℓn p will be of volume 1 2 and the δB-enlargement of body B will be of volume 1 2 too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' For these enlargements we shall have dist(AδA, BδB) = 0, from which dist(A, B) ≤ δA + δB follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The isoperimetric inequality from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='7 allows to estimate the growth of λ(Aδ) as δ+λ(Aδ) ≥ cpλ(Aδ) log1− 1 p 1 λ(Aδ) while λ(Aδ) < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' So we would like to consider a function y(δ) that behaves in accordance with our lower bound y(0) = ε (34) 41 y′ = cpy log1− 1 p 1 y (35) If by δM we will denote the moment when y reaches one half, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' y(δM) = 1 2, then δA ≤ δM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Indeed, otherwise δM < δA, but functions λ(Aδ) and y(δ) coincide at δ = 0 and for all δ ∈ [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' δM] we should have λ(Aδ) ≥ y(δ) and δ+λ(Aδ) ≥ cpλ(Aδ) log1− 1 p 1 λ(Aδ) ≥ cpy(δ) log1− 1 p 1 y(δ) = δ+y(δ), since x(− log x)1− 1 p is increasing on (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2](see Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' And so we reach ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='a contradiction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2 = µ(AδA) > µ(AδM) ≥ y(δM) = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='Differential equation (35) is separable ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='dy = cpy(− log y)1− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='pdδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−(− log y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p −1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='ydy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='= cpdδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='(− log y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p −1d(− log y) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='cpdδ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−p(− log y) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p = cpδ + C0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='Our initial condition (34) gives us ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−p(− log ε) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p = C0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='And for δ = δM we should have ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−p(log 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p = cpδM − p(− log ε) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='δM = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='cp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p(− log ε) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p − p(log 2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='≤ p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='cp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='(− log ε) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='By the same reasoning δB ≤ δM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' and we conclude dist(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' B) ≤ δA + δB ≤ 2δM ≤ 2p cp (− log ε) 1 p 42 6 Lower bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Here we are going to be concerned with the lower bounds on the largest distance between two subsets of volume 0 < ε < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' We will derive the lower bounds simply by considering certain hyperplane cuts of our convex bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' For families of convex bodies such as the euclidean balls, cubes, hyperoctahedrons, simplexes and ℓp balls specific lower bounds will be shown in Theorems 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='3, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='4, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' It turns out that for euclidean balls our lower bounds coincide with the upper bounds(see Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' In Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='6 a general lower bound will be established, showing that in a way the family of euclidean balls is optimal in regard to our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' It was already shown that for unit-volume cube, ball, simplex and ℓp balls with p ∈ [1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 2] the largest distance is bounded above by some constant dependent on ε but not on the dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' That is why it makes sense to consider the lower bounds on the supremum of all possible distances between two subsets of volume ε that take place as n tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' For a family of convex bodies Kn by dn(ε) here we denote the supremum of all possible distances between two subsets of volume ε ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2) in Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' When Kn are the unit-volume euclidean balls we have lim inf n→∞ dn(ε) ≥ −2 1 √eΦ−1(ε) The function −2 1 √eΦ−1(ε) is asymptotically equivalent to −2 1 √πe √ − ln ε as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' In the unit-volume n-ball ωnBn ⊂ Rn, where the radii is ωn = Γ �n 2 + 1 � 1 n √π ∼ � n 2πe, consider the diagonal from (−ωn, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' , 0) to (ωn, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' , 0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' a diagonal corresponding to the X1-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' We will be interested in the hyperplanes or- thogonal to this diagonal, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' hyperplanes defined by X1 = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 43 Pick a number a such that 1 √eΦ−1(ε) < −a < 0 If we consider the uniform probability distribution on ωnBn, then we could think of X1 as of a random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' We would like to consider the part of our ball that corresponds to X1 ≤ −a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The volume would be equal to Pr(X1 ≤ −a) = Pr(√nω−1 n X1 ≤ −√nω−1 n a) By theorem 1 of [20] as n tends to infinity the distribution of n 1 2ω−1 n X1 converges in total variation to the standard normal distribution on R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' whose probability density function is 1 √ 2πe− 1 2x2 Furthermore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' note that lim n→∞ −√nω−1 n a = lim n→∞ −√n √ 2πe √n a = − √ 2πea > √ 2πΦ−1(ε) So for all sufficiently large n we shall have √ 2πΦ−1(ε) + δ < −√nω−1 n a ⇒ Pr(√nω−1 n X1 ≤ −√nω−1 n a) ≥ Pr(√nω−1 n X1 ≤ √ 2πΦ−1(ε) + δ) for some δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Because distribution of n 1 2ω−1 n X1 converges in total variation to the standard normal distribution, for all sufficiently large n we have Pr(X1 ≤ −a) ≥ Pr(√nω−1 n X1 ≤ √ 2πΦ−1(ε) + δ) ≥ 1 √ 2π � √ 2πΦ−1(ε) −∞ e− 1 2x2dx = � √ 2πΦ−1(ε) −∞ e −π � 1 √ 2π x �2 d � 1 √ 2π x � = � Φ−1(ε) −∞ e−πx2dx = ε By symmetry we have a similar result for the part of our ball defined by X1 ≥ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' That means that for sufficiently large n we are going to have 44 two subsets of volume at least ε at a distance 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' But a was chosen as an arbitrary number lesser than − 1 √eΦ−1(ε), from which the statement of the theorem follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' By combining this with Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='3 we get Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' When Kn are the unit-volume euclidean balls lim n→∞ dn(ε) = −2 1 √eΦ−1(ε) Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' When Kn are the unit cubes we have lim inf n→∞ dn(ε) ≥ −2 �π 6 Φ−1(ε) The function −2� π 6Φ−1(ε) is asymptotically equivalent to 2 √ 6 √ − ln ε as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' By the main diagonal of a cube (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1)n we mean a segment from the origin to (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' , 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' We would be considering hyperplanes orthogonal to the main diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' For each point on the main diagonal we could consider the area of the corresponding orthogonal hyperplane section of (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' This gives rise to a probability distribution on the segment from (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' , 0) to (1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' , 1), whose length is √n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' We will take the midpoint of this segment as the origin, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' we have a probability distribution on [− √n 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' + √n 2 ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The variance of the uniform distribution on the segment [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1] is equal to σ2 = � 1 0 � x − 1 2 �2 dx = 2 � 1 2 0 x2dx = 21 3 1 23 = 1 12 And our distribution on [− √n 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' + √n 2 ] could be produced by n random variables X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' , Xn uniformly distributed on [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1] as √n �X1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' + Xn n − 1 2 � 45 So by central limit theorem as n goes to infinity our distribution converges to a normal distribution N (0, σ2), whose probability density function would be 1 σ √ 2πe− 1 2 x2 σ2 = � 6 πe−6x2 Pick a number a such that �π 6 Φ−1(ε) < −a < 0 And consider the part of the unit cube (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1)n whose orthogonal projection on the main diagonal lies inside [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2 √n − a], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' a certain hyperplane section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' As n goes to infinity the volume of this region would converge to the value of the cumulative distribution function of N (0, σ2) at −a, which is equal to � −a −∞ � 6 πe−6x2dx = � −a −∞ e −π �√ 6 π x �2 d �� 6 πx � = � −√ 6 π a −∞ e−πx2dx = Φ � − � 6 πa � > Φ �� 6 π �π 6 Φ−1(ε) � = ε Thus for large enough n a part of volume at least ε is going to be cut off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' By symmetry the same is true for an orthogonal hyperplane section of our cube corresponding to [ 1 2 √n + a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' √n] on the main diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' So for n large enough we get two subsets of volume at least ε at a distance 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' But a was chosen as an arbitrary number smaller than − �π 6 Φ−1(ε), from which the statement of the theorem follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' When Kn are the unit-volume simplexes we have lim inf n→∞ dn(ε) ≥ − √ 2 e ln(2ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The volume of the unit simplex ∆n defined by ∆n = {(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' , xn) ∈ Rn + | x1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' + xn = 1} 46 is equal to n√n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' If we set ωn = � n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' n√n � 1 n−1 ∼ n e , then ωn∆n will be a regular unit-volume simplex, whose side length is √ 2ωn Now consider any number ε′ such that ε < ε′ < 1 2 Let P and Q be two vertices of ωn∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Hyperplane passing through the midpoint of the side PQ and orthogonal to it divides our simplex into two parts of equal volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' We set α = (2ε′) 1 n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Consider the image of the part containing P after a homotethy with cen- ter at P and coefficient α, the resulting subset of our simplex will be of volume 1 2αn−1 = ε′ > ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Analogously, we construct a subset of volume ε′ corresponding to the vertex Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Note that we have two homotethies with coefficient α applied to the halves of PQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The distance between our subsets will be equal to √ 2ωn (1 − α) = √ 2ωn � 1 − (2ε′) 1 n−1 � Now we take limit lim n→∞ √ 2ωn � 1 − (2ε′) 1 n−1 � = lim n→∞ √ 2 e n � 1 − (2ε′) 1 n−1 � = lim n→∞ − √ 2 e n n − 1 (2ε′)0 − (2ε′) 1 n−1 0 − 1 n−1 = − √ 2 e d dt(2ε′)t��� t=0 = − √ 2 e ln(2ε′) In other words,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' for each n we have two subsets of ωn∆n of volume at least ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' and the distance between them tends to − √ 2 e ln(2ε′) 47 as n goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' But ε′ was chosen as an arbitrary number from (ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2), from which the statement of the theorem follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' When Kn are the unit-volume ℓp balls for p ∈ [1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 2] we have lim inf n→∞ dn(ε) ≥ −2Ψ−1 p (ε), where function −2Ψ−1 p (ε)(see Appendix C) is asymptotically equivalent to 1 e 1 pΓ � 1 + 1 p �(− ln ε) 1 p as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' In the unit-volume ℓp ball ωnℓn p, where ωn = Γ � 1 + n p � 1 n 2Γ � 1 + 1 p � ∼ n 1 p 2Γ � 1 + 1 p � (pe) 1 p , consider the segment from (−ωn, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' , 0) to (ωn, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' , 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' We will be con- sidering the hyperplanes orthogonal to this segment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' hyperplanes defined by X1 = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' By Vn(t) denote the function that measures the volume of the part given by X1 ≥ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' We use notation from Appendix C, where it was established that func- tions Vn(x) uniformly converge to Ψp(−x) on (−∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Pick an arbitrary number a such that Ψ−1 p (ε) < −a < 0 Then for all sufficiently large n we shall have Vn(a) > ε Because of symmetry, for all sufficiently large n we would have two bodies in ωnℓn p of volume at least ε at a distance at least 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' And, since the choice of a above was arbitrary, we would have lim inf n→∞ dn(ε) ≥ −2Ψ−1 p (ε) 48 Assume that we have a family Kn of bounded unit-volume centrally sym- metric bodies that are not necessarily convex, where the origin will be the center of symmetry for each Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' By µn denote the uniform probability mea- sure on Kn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Fix ε ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Since Kn is bounded, it has a finite diameter, thus for each Kn we could consider dn(ε) – the supremum of all possible distances between two subsets of Kn, whose volume is ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Turns out that a simple averaging argument gives us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' For every ε ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2) lim inf n→∞ dn(ε) ≥ −2 1 √eΦ−1(ε), where function −2 1 √eΦ−1(ε) is asymptotically equivalent to −2 1 √πe √ − ln ε as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Consider a number d such that 1 √eΦ−1(ε) < −d < 0 We would like to show that for all sufficiently large n there would be a direction defined by a unit vector u on the sphere Sn−1 such that the part of our body Kn where ⟨x, u⟩ ≤ −d would be of volume at least ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' By central symmetry the part where ⟨x, u⟩ ≥ d would be of the same volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' But then we will be having two subsets of Kn of volume at least ε at a distance at least 2d, where d was chosen as an arbitrary number lesser than − 1 √eΦ−1(ε), from which the statement of the theorem would follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' We are looking for a direction u ∈ Sn−1 with µn(⟨x, u⟩ ≤ −d) ≥ ε 49 If the average value of µn(⟨x, u⟩ ≤ −d) is greater than ε, then such a direction will exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' By ϑn denote the uniform probability measure on Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' We would like to have � Sn−1 µn(⟨x, u⟩ ≤ −d)dϑn ≥ ε (36) Volume of the hyperplane cut defined by ⟨x, u⟩ ≤ −d is the integral of the indicator function [⟨x, u⟩ ≤ −d] on Kn, which allows to rewrite (36) as � Sn−1 � Kn [⟨x, u⟩ ≤ −d]dµndϑn ≥ ε We may switch the order of integration � Kn �� Sn−1[⟨x, u⟩ ≤ −d]dϑn � dµn ≥ ε (37) We would like to consider the integrand of (37) for x ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' First, we rewrite [⟨x, u⟩ ≤ −d] as �√n � x ∥x∥2 , u � ≤ − √nd ∥x∥2 � Now we could think of u as of a random vector on the unit sphere Sn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Then � x ∥x∥2 , u � corresponds to the projection of u on the diameter from − x ∥x∥2 to + x ∥x∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' And thus by theorem 1 from [20] as n goes to infinity the distribution of random variable √n � x ∥x∥2 , u � converges in total variation to the standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Note that the value of � Sn−1 �√n � x ∥x∥2 , u � ≤ − √nd ∥x∥2 � dϑn only depends on − √nd ∥x∥2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' So we define the function Ψn(x) = � Sn−1 �√n � x ∥x∥2 , u � ≤ x � dϑn 50 As was remarked above, for every x lim n→∞ Ψn(x) = 1 √ 2π � x −∞ e− 1 2x2dx Since functions Ψn are monotonic, we should also have lim n→∞ Ψn(xn) = 1 √ 2π � x −∞ e− 1 2 x2dx for any sequence xn that tends to x as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' We rewrite (37) as � Kn\\{0} Ψn � − √nd ∥x∥2 � dµn ≥ ε (38) The function Ψn is non-decreasing, thus ∥x∥2 ≥ r > 0 ⇒ Ψn � − √nd ∥x∥2 � ≥ Ψn � − √nd r � So for (38) condition � Kn\\Bn r Ψn � − √nd r � dµn ≥ ε (39) would be sufficient, we may go further and require (1 − V (Bn r ))Ψn � − √nd r � ≥ ε, (40) where V (Bn r ) is the volume of the n-ball with radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The volume of the unit n-ball Bn 1 is √πn Γ �n 2 + 1 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The unit volume corresponds to the radius ωn = Γ �n 2 + 1 � 1 n √π ∼ � n 2πe 51 Pick a number α ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1) such that ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='α 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='√eΦ−1(ε) < −d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='(41) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='and set ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='r = αωn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='Inequality (40) turns into ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='(1 − αn)Ψn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='√nd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='αωn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='≥ ε ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='(42) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='Note that ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='lim ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='n→∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='√nd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='αωn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='= lim ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='n→∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2πe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='√nd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2πed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='Now take limit of the left side of (42) and apply condition (41) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='lim ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='n→∞(1 − αn)Ψn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='√nd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='αωn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2πed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='e− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2 x2dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2πed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2π x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2πx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� − ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='√ed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='e−πx2dx = Φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='√ed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='> Φ(Φ−1(ε)) = ε ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='Thus for all sufficiently large n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='(1 − αn)Ψn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='√nd ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='αωn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='> ε ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' consequently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' � Sn−1 µn(⟨x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' u⟩ ≤ −d)dϑn ≥ ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' (43) which means that for all sufficiently large n the desired direction u ∈ Sn−1 with property µn(⟨x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' u⟩ ≤ −d) ≥ ε exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 52 Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 is an immediate corollary of the Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Roughly speaking, the results established in [12], [11] tell us that generally hyperplane sections of convex bodies across arbitrary directions lead to a gaussian distribution, suggesting that the asymptotic behavior of dn(ε) different from Φ−1(ε) is probably caused by a few degenerate directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' For example, Theorems 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='4 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='5 tell us that the asymptotic behavior of dn(ε) for simplexes corresponds to a function − ln ε, but one could also note that the simplex ∆n is unusually «stretched» in n directions corresponding to its corners, and that almost all of its volume by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='3 is concentrated in the euclidean ball of a diameter much smaller than the diameter of ∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 53 7 Discrete isoperimetric problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' In this section instead of considering the euclidean distance between two subsets of volume ε we will be considering the Manhattan dis- tance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' We can no longer say that for fixed ε this distance is bounded, but we will derive a result(Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='3) concerning its asymptotic behaviour in the case of the unit cube [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1]n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The key idea in the proof of it would be to replace the unit cube [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1]n with a lattice, for which the solution of our problem is already known(see Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' We consider the Manhattan distance d in the unit cube [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1]n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' We may ask a similar question: what is the largest Manhattan distance between two bodies of volume ε > 0 in the unit cube [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1]n?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Turns out this problem could be dealt with by discretization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Consider a lattice L = { 0 m, 1 m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' , m m}n inside [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1]n, two points x and y in L are adjacent whenever d(x, y) = 1 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' By t-boundary A(t) of a subset A ⊆ L we mean the set of all points of L that are at a distance at most t m from A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The latter concept is analagous to δ-enlargements, and one could think of |A(1) \\ A| as an analogue of the surface area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' This gives rise to the discrete isoperimetirc problem: how large the t-boundary of a set A ⊆ L with fixed size can be?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' This question was answered in [4] Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ([4, Corollary 9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Let A ⊂ [k]n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' For any t = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' , the t-boundary of A is at least as large as the t-boundary of the first |A| elements of [k]n in the simplicial order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Here [k]n is the lattice {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' , k − 1}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Simplicial order on [k]n is defined by setting x < y if either � xi < � yi, or � xi = � yi and for some j we have xj > yj and xi = yi for all i < j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' This discrete isoperimetric inequality leads to Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2 ([4, Corollary 10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' There are sets A, B ⊂ [k]n with |A| = r, |B| = s, and d(A, B) ≥ d iff the distance between the first r and the last s elements of the simplicial order on [k]n is at least d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' From this discrete version of the problem considered in the first pararaph we can derive Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' If by dn(ε) we denote the largest Manhattan distance between two bodies of volume ε ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2) in the unit cube [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1]n, then lim n→∞ dn(ε) √n = −2 �π 6Φ−1(ε) 54 The function −2� π 6Φ−1(ε) is asymptotically equivalent to 2 √ 6 √ − ln ε as ε → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Pick a number a such that −a < �π 6 Φ−1(ε) < 0 If by Vn and Wn we denote the subsets of [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1]n defined by inequalities � xi ≤ 1 2 √n − a and � xi ≥ 1 2 √n + a, respectively, then by applying the central limit theorem just as we did in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='3 we shall get lim n→∞ µ(Vn) = lim n→∞ µ(Wn) = � −a −∞ � 6 πe−6x2dx = Φ � − � 6 πa � < Φ �� 6 π �π 6 Φ−1(ε) � = ε So there is a number N such that for all n > N the volumes of Vn and Wn are going to be lesser than ε, and it could be noted that the Manhattan distance between them is equal to 2a√n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' From now on we assume that n > N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Now consider two bodies A and B inside [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1]n of volume ε and a lat- tice Lm = { 0 m, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' , m m}n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The discretization is justified by the fact that the distance does not decrease when we restrict our attention to the lattice Lm d(A, B) ≤ d(A ∩ Lm, B ∩ Lm) Since µ(Vn) < µ(A) and µ(Wn) < µ(B), for all m large enough we shall have |Vn ∩ Lm| ≤ |A ∩ Lm| |Wm ∩ Lm| ≤ |B ∩ Lm| And by Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2 this implies d(A ∩ Lm, B ∩ Lm) ≤ d(Vn ∩ Lm, Wn ∩ Lm), because Vn ∩ Lm and Wn ∩ Lm are the first |Vn ∩ Lm| and the last |Wn ∩ Lm| elements of the simplicial order on Lm, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 55 We conclude that for all m large enough d(A, B) ≤ d(Vn ∩ Lm, Wn ∩ Lm) Also lim m→∞ d(Vn ∩ Lm, Wn ∩ Lm) = d(Vn, Wn) = 2a√n Thus for all n > N d(A, B) ≤ 2a√n, but a was chosen here as an arbitrary number greater than −� π 6Φ−1(ε), from which lim sup n→∞ dn(ε) √n ≤ −2 �π 6 Φ−1(ε) (44) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' If we assume that �π 6 Φ−1(ε) < −a < 0, then as it was already shown in the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='3 for all n large enough µ(Vn) = µ(Wn) > ε But the Manhattan distance between Vn and Wn is 2a√n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' From this observation we can conclude lim inf n→∞ dn(ε) √n ≥ −2 �π 6 Φ−1(ε), which together with inequality (44) gives us the statement of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' There are also other variations of the discrete isoperimet- ric problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' For example, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2 has an analogue for the non-negative orthant of the integer lattice Zn +(see [22] and [4, Theorem 4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 56 8 Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Upper bounds on the distance between subsets in unit-volume ℓp balls with p ∈ [1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 2] ∪ {+∞} have been established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' A case of p ∈ (2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' +∞) remains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Since asymptotically our estimates for unit-volume euclidean balls(p = 2) and for unit cubes(p = +∞) are the same, we expect to have a similar asymptotic behavior for all p ∈ [2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' +∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' As was remarked before both cases of p = 2 and p = +∞ could be approached by providing a Lipschitz map that transforms Gaussian measure into the uniform measure on the corresponding ℓp balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Perhaps, the same approach might work out for p ∈ (2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' On the page 4 of [3] it was remarked that the uniform measure on ℓn p balls with p ∈ [2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' +∞] «can be obtained from the canonical Gaussian measure as Lipschitz transform».' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Although, we are not sure about the exact meaning and implications of this statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' In Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='6 we established a sort of a general lower bound regarding our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' But is it possible to find a general upper bound on the distance between two subsets of volume ε ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2) in a unit-volume convex body?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Clearly, we are going to have to consider only some certain «good» convex bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' For example, in a convex body stretched far in a particular direc- tion two subsets of fixed volume could be at an arbitrarily large distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' One such notion of a «good» convex body is related to the isotropic position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' But this means that we are looking for general estimates on the isoperimetric problem in isotropic convex bodies, this appears to be a complicated open question(Kannan-Lov´asz-Simonovits conjecture, see [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Furthermore, even the relation between the volume and the isotropic constant of a convex body seems to be at the core of another open problem(isotropic constant conjec- ture, see [5], [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' In paper [6] a very good lower bound related to the KLS conjecture was proven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' There is a lot of material on isotropic convex bodies, for example, [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Log-concave probability measures generalize uniform measures on convex bodies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' So one might try to consider the problem in a more general setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Gaussian measures are log-concave, which means that in a more general setting our estimates may be sharp for some specific distributions in the one- dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' In paper [15] some general estimates on the specific version of the isoperimetric problem were proven(see, for example, theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 57 References [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Bobkov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' “Isoperimetric and Analytic Inequalities for Log-Concave Probability Measures”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' In: The Annals of Probability 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='4 (1999), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1903 –1921.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1214/aop/1022874820.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1214/aop/1022874820.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Bobkov and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Houdr´e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' “Isoperimetric constants for product probability measures”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' In: The Annals of Probability 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 (1997), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 184 –205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1214/aop/1024404284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1214/aop/1024404284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' [3] Sergey G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Bobkov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' “On Isoperimetric Constants for Log-Concave Prob- ability Distributions”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' In: Geometric Aspects of Functional Analysis: Is- rael Seminar 2004–2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' by Vitali D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Milman and Gideon Schecht- man.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Berlin, Heidelberg: Springer Berlin Heidelberg, 2007, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 81–88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' isbn: 978-3-540-72053-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1007/978-3-540-72053-9_4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1007/978-3-540-72053-9_4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' [4] B´ela Bollob´as and Imre Leader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' “Compressions and isoperimetric in- equalities”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' In: Journal of Combinatorial Theory, Series A 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 (1991), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 47–62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' issn: 0097-3165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' doi: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1016/0097-3165(91)90021-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' url: https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='com/science/article/pii/0097316591900218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' [5] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Bourgain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' “On the distribution of polynomials on high dimensional convex sets”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' In: Geometric Aspects of Functional Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' by Jo- ram Lindenstrauss and Vitali D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Milman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Berlin, Heidelberg: Springer Berlin Heidelberg, 1991, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 127–137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' isbn: 978-3-540-47355-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' [6] Yuansi Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' “An Almost Constant Lower Bound of the Isoperimet- ric Coefficient in the KLS Conjecture”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' In: Geometric and Functional Analysis 31 (Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1007/s00039-021-00558-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' [7] Sean Dar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' “Remarks on Bourgain’s Problem on Slicing of Convex Bod- ies”.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' issn: 00029939, 10886826.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' url: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='jstor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='org/stable/2048262 (visited on 12/07/2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' [18] Gideon Schechtman and Joel Zinn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' “Concentration on the ℓn p ball”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' In: May 2007, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 245–256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' isbn: 978-3-540-41070-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1007/BFb0107218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' [19] Sasha Sodin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' “An isoperimetric inequality on the ℓp balls”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' In: Annales de l’I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Probabilit´es et statistiques 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2 (2008), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 362– 373.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' url: http://eudml.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='org/doc/77974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' [20] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Stam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' “Limit Theorems for Uniform Distributions on Spheres in High-Dimensional Euclidean Spaces”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' In: Journal of Applied Probability 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 (1982), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 221–228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' issn: 00219002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' url: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='jstor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='org/stable/3213932 (visited on 11/28/2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 59 [21] Marta Strzelecka.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' “Isoperimetric problem for exponential measure on the plane with ℓ1-metric”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' In: Positivity (June 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1007/s11117-017-0476-y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' [22] Da-Lun Wang and Ping Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' “Extremal Configurations on a Discrete Torus and a Generalization of the Generalized Macaulay Theorem”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' In: SIAM Journal on Applied Mathematics 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 (1977), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 55–59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' issn: 00361399.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' url: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='jstor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='org/stable/2100602 (visited on 01/27/2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' [23] Xianfu Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' “Volumes of Generalized Unit Balls”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' In: Mathematics Magazine 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='5 (2005), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 390–395.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' issn: 0025570X, 19300980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' url: http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='jstor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='org/stable/30044198 (visited on 11/24/2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 60 A Asymptotic behavior of Φ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='We want to show that ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='lim ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='ε→0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='√π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='− ln ε ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−Φ−1(ε) = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='(45) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='We begin with the following observation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='e−πx2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−2πx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='= e−πx2 + e−πx2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2πx2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='By integrating both parts from −∞ to a < 0 we get ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='e−πa2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−2πa = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='e−πx2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−2πx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='e−πx2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2πx2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='(46) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='From this we can derive the following upper bound on Φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='Φ(a) ≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='e−πx2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2πx2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='dx = e−πa2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−2πa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='Now we apply −Φ−1 to both sides ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−a ≥ −Φ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='e−πa2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−2πa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='Keep in mind that a < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' By putting e−πa2 −2πa instead of ε into the expression from (45) we arrive at 1 √π � πa2 + ln(−2πa) −Φ−1( e−πa2 −2πa ) ≥ 1 √π � πa2 + ln(−2πa) −a But as a goes to −∞ e−πa2 −2πa → 0 and 1 √π � πa2 + ln(−2πa) −a → 1 So lim inf ε→0 1 √π √ − ln ε −Φ−1(ε) ≥ 1 (47) 61 if defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='From (46) we can also derive a lower bound on Φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='Φ(a) ≥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2πa2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='�−1 � a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='e−πx2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2πx2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='dx = e−πa2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−2πa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2πa2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='�−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='Again we apply −Φ−1 to both sides and get ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−a ≤ −Φ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='e−πa2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−2πa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2πa2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='�−1� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='By putting e−πa2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−2πa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2πa2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='�−1 instead of ε into the expression from (45) we ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='arrive at ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='√π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='πa2 + ln(−2πa) + ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2πa2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−Φ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='e−πa2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−2πa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2πa2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='�−1� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='√π ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='πa2 + ln(−2πa) + ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2πa2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='But as a goes to −∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='e−πa2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−2πa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2πa2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='�−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='→ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 √π � πa2 + ln(−2πa) + ln � 1 + 1 2πa2 � −a → 1 Thus lim sup ε→0 1 √π √ − ln ε −Φ−1(ε) ≤ 1 (48) Together (47) and (48) give lim ε→0 1 √π √ − ln ε −Φ−1(ε) = 1 62 B Function x(− log x)1−1 p is increasing on (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' To show that function x(− log x)1− 1 p is increasing on (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2] for p ∈ [1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 2] we simply take the derivative d dx � x(− log x)1− 1 p � = (− log x)1− 1 p + x d dx(− log x)1− 1 p = (− log x)1− 1 p + xd(− log x)1− 1 p d(− log x) d(− log x) dx = (− log x)1− 1 p + � 1 − 1 p � x(− log x)− 1 p � −1 x � = (− log x)1− 1 p − � 1 − 1 p � (− log x)− 1 p = (− log x)− 1 p � (− log x) − � 1 − 1 p �� Clearly, (− log x)− 1 p > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' On the half-interval (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2] we should have (− log x) − � 1 − 1 p � ≥ � − log 1 2 � − 1 2 = log 2 − 1 2 > 0, which means that for all x ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1 2] d dx � x(− log x)1− 1 p � > 0 C Functions Vn(x) and Sn(x) in limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' For p ∈ [1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' ∞) consider the unit ℓn p ball defined by |x1|p + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' + |xn|p = 1 It has volume � 2Γ � 1 + 1 p ��n Γ � 1 + n p � 63 Let ωn = Γ � 1 + n p � 1 n 2Γ � 1 + 1 p � ∼ n 1 p 2Γ � 1 + 1 p � (pe) 1 p (49) For a non-negative number x by Sn(x) denote the volume of the section x1 = x of the unit-volume body ωnℓn p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' For x > ωn we assume Sn(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' For x ≤ ωn the ℓn−1 p ball corresponding to this section is defined by equation |x2|p + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' + |xn|p = ωp n − xp It has volume Sn(x) = (ωp n − xp) n−1 p � 2Γ � 1 + 1 p ��n−1 Γ � 1 + n−1 p � = � 1 − xp ωp n � n−1 p 1 ωn ωn n � 2Γ � 1 + 1 p ��n−1 Γ � 1 + n−1 p � = � 1 − xp ωp n � n−1 p 1 ωn Γ � 1 + n p � � 2Γ � 1 + 1 p ��n � 2Γ � 1 + 1 p ��n−1 Γ � 1 + n−1 p � = � 1 − xp ωp n � n−1 p Γ � 1 + n p � 2ωnΓ � 1 + 1 p � Γ � 1 + n−1 p � Since lim x→∞ Γ(x + α) Γ(x)xα = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' we shall have lim n→∞ Γ � 1 + n p � 2ωnΓ � 1 + 1 p � Γ � 1 + n−1 p � = lim n→∞ � ωn2Γ � 1 + 1 p ��−1 Γ � 1 + n p � Γ � 1 + n−1 p � = lim n→∞ \uf8eb \uf8ed n 1 p 2Γ � 1 + 1 p � (pe) 1 p 2Γ � 1 + 1 p �\uf8f6 \uf8f8 −1 �n p � 1 p = e 1 p 64 Also lim n→∞ n pωp n = lim n→∞ n \uf8eb \uf8edp n 2pΓ � 1 + 1 p �p pe \uf8f6 \uf8f8 −1 = 2pΓ � 1 + 1 p �p e Thus on the segment [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' D] the sequence of functions � 1 − xp ωp n � n−1 p = � 1 − xp ωp n � � 1 − xp n pωp n n p � n p uniformly converges to e−xp2pΓ(1+ 1 p) pe = e − � 2Γ(1+ 1 p)e 1 p x �p as n tends to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Thus on the segment [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' D] functions Sn(x) uniformly converge to ψp(x) = e 1 p e−xp2pΓ(1+ 1 p) pe = e 1 p e − � 2Γ(1+ 1 p)e 1 p x �p For an arbitrary ε > 0 pick a number D such that ψp(D) < ε, then for all sufficiently large n Sn(D) < 2ε And, since both functions Sn(x) and ψp(x) are non-increasing, we will have |ψp(x) − Sn(x)| < 3ε for every x ≥ D for all sufficiently large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' This implies that functions Sn(x) uniformly converge to ψp(x) on the whole [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' For a non-negative x by Vn(x) denote the volume of the part of ωnℓn p defined by x1 ≥ x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' For x > ωn we assume that Vn(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Note that Vn(x) = 1 2 − � d 0 Sn(t)dt 65 And, since Sn(t) uniformly converge to ψp(x) on [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' D], we shall have that Vn(x) uniformly converge to Λp(x) = 1 2 − � d 0 ψp(t)dt on [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' D].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' Also � ∞ 0 ψp(x)dx = � ∞ 0 e 1 pe − � 2Γ(1+ 1 p)e 1 p x �p dx = 1 2Γ � 1 + 1 p � � ∞ 0 e − � 2Γ(1+ 1 p)e 1 p x �p \uf8eb \uf8ed 1 2Γ � 1 + 1 p � e 1 p dx \uf8f6 \uf8f8 = 1 2Γ � 1 + 1 p � � ∞ 0 e−xpdx = 1 2Γ � 1 + 1 p � � ∞ 0 e−tdt 1 p = 1 2pΓ � 1 + 1 p � � ∞ 0 t 1 p −1e−tdt = Γ � 1 p � 2pΓ � 1 + 1 p � = 1 2 For an arbitrary ε there is a number D such that Λp(D) < ε Since Vn(D) converge to Λp(D),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' we shall have Vn(D) < 2ε for all sufficiently large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' And, since both functions Vn(x) and Λp(x) are non-increasing, |Λp(x) − Vn(x)| < 3ε for every x ≥ D for all sufficiently large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' This means that functions Vn(x) uniformly converge to Λp(x) = � ∞ x φp(t)dt on the whole [0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 66 For Section 3 note that ψ2(x) = √ee−πex2 = Ψ′(−x) � ∞ x √ee−πet2dt = � ∞ x e−π(√et)2d(√et) = � ∞ √ex e−πx2dx = Ψ(−x) For convenience we introduce new functions defined on (−∞;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' +∞) φp(x) = e−cp|x|p cp = 2pΓ � 1 + 1 p �p Φp(a) = � a −∞ e−cp|x|pdx Ψp(a) = � a −∞ φp(x)dx The latter function is the reflection of Λp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='We would like to show that ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='lim ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='ε→0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p (− ln ε) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−Φ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p (ε) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='= 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='(50) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='Note that ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='e−cpxp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='pcpxp−1 = −pcpxp−1e−cpxp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='pcpxp−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='− e−cpxpp(p − 1)cpxp−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p2c2px2(p−1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='= −e−cpxp � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 + p − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='cp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='xp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='Let ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='A = p − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='cp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='We conclude that for a negative x we shall have ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� e−cp|x|p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='pcp|x|p−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='�′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='= e−cp|x|p � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 + A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='|x|p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='Integrating both sides from −∞ to a < 0 gives us ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='e−cp|x|p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='pcp|x|p−1 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='e−cp|x|p � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 + A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='|x|p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='dx ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='(51) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='67 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='From this we can derive an upper bound on Φp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='Φp(a) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='e−cp|x|pdx ≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='e−cp|x|p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='pcp|x|p−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='Applying −Φ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='to both sides should give us ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−a ≥ −Φ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� e−cp|x|p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='pcp|x|p−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='If we put ε = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='e−cp|x|p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='pcp|x|p−1 into the expression from (50),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' we get c − 1 p p (cp|x|p + ln(pcp|x|p−1)) 1 p −Φ−1 p � e−cp|x|p pcp|x|p−1 � ≥ c − 1 p p (cp|x|p + ln(pcp|x|p−1)) 1 p −a But as a goes to −∞ e−cp|x|p pcp|x|p−1 → 0 and c − 1 p p (cp|x|p + ln(pcp|x|p−1)) 1 p −a → 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='from which we conclude ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='lim inf ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='ε→0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p (− ln ε) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−Φ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p (ε) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='≥ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='(52) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='Equality (51) also leads to a lower bound on Φp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='Φp(a) ≥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 + A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='|a|p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='�−1 e−cp|x|p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='pcp|x|p−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='By applying −Φ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='to both sides again we get ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−a ≤ −Φ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 + A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='|a|p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='�−1 e−cp|x|p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='pcp|x|p−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='68 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='By putting ε = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='|a|p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='�−1 e−cp|x|p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='pcp|x|p−1 into the expression from (50) we arrive ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='at ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='cp|x|p + ln(pcp|x|p−1) + ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='|a|p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='�� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−Φ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='|a|p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='�−1 e−cp|x|p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='pcp|x|p−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='cp|x|p + ln(pcp|x|p−1) + ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='|a|p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='�� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='As a goes to −∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 + A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='|a|p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='�−1 e−cp|x|p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='pcp|x|p−1 → 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='cp|x|p + ln(pcp|x|p−1) + ln ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='|a|p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='�� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='→ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='And we conclude ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='lim sup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='ε→0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p (− ln ε) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−Φ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p (ε) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='≤ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='Together with (52) this leads to ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='lim ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='ε→0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='c ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p (− ln ε) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='−Φ−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p (ε) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='= 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='If by Ψp we denote the function ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='Ψp(a) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='pe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='2Γ(1+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p)e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='�p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='dx = Φp(e ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content='p a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' then we shall have −Ψ−1 p (ε) ∼ 1 2e 1 p Γ � 1 + 1 p �(− ln ε) 1 p 69 D Average distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' The radius of the unit-volume euclidean n-ball is ωn = Γ �n 2 + 1 � 1 n √π ∼ � n 2πe (53) For an arbitrary α ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1) define rn = α � n 2πe (54) By (53) we have lim n→∞ rn ωn = α < 1 If by Vn we denote the volume of the n-ball of radius rn, then lim n→∞ Vn = 0 (55) Now consider an arbitrary unit-volume convex body K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' For each point p ∈ K the set of points of K that are at a distance at most rn from p is of volume not greater than Vn, so the average distance d(K) between points in K should be at least d(K) ≥ (1 − Vn)rn And by (54), (55) (1 − Vn)rn ∼ α � n 2πe Since the choice of α ∈ (0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} +page_content=' 1) above was arbitrary, for a family of unit- volume convex bodies Kn we should have lim inf n→∞ d(Kn) � n 2πe ≥ 1 70' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9FRT4oBgHgl3EQfJTdB/content/2301.13495v1.pdf'} diff --git a/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf b/PdE1T4oBgHgl3EQfaQSn/content/2301.03160v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..2855d87e451c0b25289b9dfc4eed7c09bb409a97 --- /dev/null +++ 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100644 index 0000000000000000000000000000000000000000..f4f41a50bc12cdb295f3db1f7bfd01dbef4536f0 --- /dev/null +++ b/QNAyT4oBgHgl3EQftvli/content/tmp_files/2301.00600v1.pdf.txt @@ -0,0 +1,981 @@ +Fluctuations and arctic curve in the +Aztec diamond +Bryan Debin1, Jean-Franc¸ois de Kemmeter2, Philippe Ruelle1 +2 Department of Mathematics and Namur Institute for Complex Systems (naXys), University of Namur, Rue de Bruxelles 61, Namur, B-5000, +Belgium +jean-francois.dekemmeter@unamur.be +1 Institut de Recherche en Math´ematique et Physique, Universit´e catholique de Louvain, Louvain-la-Neuve, B-1348, Belgium +January 3, 2023 +Abstract +Domino tilings of Aztec diamonds are known to exhibit an arctic phenomenon, namely a separation between +frozen regions (in which all the dominoes have the same orientation) and a central disordered region (where +dominoes are found without any apparent order). This separation was proved to converge, under a suitable +rescaling, to the Airy process whose 1-point distribution is the Tracy-Widom distribution. +In this work, we +conjecture, by means of numerical analysis, that the boundary between the frozen and disordered regions, +converges, for the same rescaling, to the Airy line ensemble, a generalisation of the Airy process. +Introduction +A variety of patterns can be observed in Nature. They +often result from the interactions among microscopic +units. An example is the adsorption of some molecules +on graphite [2]. When adsorbed, these molecules bind to +each other such that the angle formed between any two +neighboring molecules is either 60◦ or 120◦. As revealed +by scanneling tunneling microscope, +the resulting +entropically stabilized configurations are equivalent to +rhombus tilings. Tilings and more specifically domino +tilings will be at the heart of this article. Tiling models +are also closely related to vertex models, the latter being +first introduced by Linus Pauling to explain the residual +entropy of ice at zero temperature [14]. In those models, +boundary conditions can strongly influence the bulk +properties of the system, a feature recently highlighted +experimentally in a colloïdal artificial ice [16]. In the +following, we introduce the notion of domino tiling and +discuss the impact of boundary conditions through the +celebrated Aztec diamond. +m +n +(a) +(b) +Figure 1: (a) A rectangulair domain of size 4 × 3 tileable by +dominoes. (b) A domino, be its orientation vertical or +horizontal, is the union of two unit squares. +Domino tilings of a rectangle +Imagine a tiler who wishes to tile a rectangular domain +of dimension m × n (m, n ∈ N). He has at its disposal +rectangular tiles of dimension 1× 2, here and now named +dominoes, see Figure 1. +Before he gets down to work, the tiler would like to +answer the following question: is the domain tileable +and, if so, how many distinct tilings are there ? The +domain is said tileable if there exists at least one tiling, +namely a configuration for which any point of the +1 +arXiv:2301.00600v1 [cond-mat.stat-mech] 2 Jan 2023 + +Z2,n +≡ +n − 1 +Z2,n−1 ++ +n − 2 +Z2,n−2 +Figure 2: The number Z2,n of domino tilings of a 2 × n rectangle can +be decomposed into two sums, depending on the orientation +of the domino that covers the leftmost unit squares. This +leads to the Fibonacci recurrence relation Z2,n = Z2,n−1 + +Z2,n−2. +domain is covered by exactly one domino, such that no +domino crosses the boundary of the domain. Let Zm,n +be the number of tilings of such domain. A necessary +(and actually sufficient) condition for this domain to +be tileable, i.e. Zm,n ≥ 1, is that m · n must be even +(and non-zero). As an example, let us consider the case +m = 2, for which an explicit formula can be readily +obtained. The first few terms of the sequence +� +Z2,n +� +n are +0, 1, 2, 3, 5, 8, · · · . The reader might have recognised the +Fibonacci sequence. Indeed, as depicted in Figure 2, the +terms Z2,n satisfy the following recurrence relation: +Z2,n = Z2,n−1 + Z2,n−2, +Z2,0 = 0 , Z2,1 = 1. +(1) +Solving this recurrence relation leads to the explicit +formula for the number of domino tilings of a 2 × n +rectangle: +Z2,n = +1 +√ +5 +� +1 + +√ +5 +2 +�n +− 1 +√ +5 +� +1 − +√ +5 +2 +�n +. +(2) +An explicit formula also exists for general values of m +and n, athough its derivation is much more involved +than the case m = 2. It was proved that [13, 18]: +Zm,n = +⌈m/2⌉ +∏ +j=1 +⌈n/2⌉ +∏ +k=1 +� +4 cos2 +πj +m + 1 + 4 cos2 +πk +n + 1 +� +. +(3) +The number of tilings grows quite rapidly with m and +n, provided m, n are not both odd integers. For example, +if m = n, the above formula gives Z2,2 = 2, Z4,4 = +36, Z6,6 = 6728, Z8,8 = 12988816. +Figure 3 shows a +configuration, for m = n = 100, randomly chosen among +all the tilings. Let us for now make abstraction of the +colours attributed to the dominoes. +As can be seen, any macroscopic portion of this +domain contains, on average, the same fraction of vertical +Figure 3: Configuration of a square of size n = m = 100, randomly +sampled, using the Janvresse algorithm [10]. +dominoes than horizontal ones. +One might wonder +whether similar observations hold true for any other +tileable domain. The answer to this question is no. Take +for instance the domain given in Figure 4 (a), whose +dimension is parametrised by n. +The domain is tileable but the constraints induced +by the boundary are so strong that there is only one +configuration, the one for which all the dominoes are +placed horizontally. +Aztec +diamond +and +arctic +phenomenon +The example shown in Figure 4 (a) is in some sense +pathological and not so interesting per se. A much more +interesting situation is obtained by slightly modifying +the domain, through the introduction of a row of +unit squares, as shown in Figure 4 (b). +This slight +modification has drastic consequences as we shall see in +the following. This domain is known as the Aztec diamond +(AD in the following) and was first introduced in [8, 9]. +Formally, it is the set of unit squares whose centres +(i, j) are such that |i| + |j| ≤ n, with the origin (0, 0) +coinciding with the centre of the AD. Notice already that +2 + +n +(a) +n +•O +(b) +Figure 4: (a) The unique tiling of the domain consists exclusively of +horizontal dominoes. By adding a row of 2n unit squares +to this domain, we obtain the Aztec diamond (AD) of order +n shown in right (b). The AD is the set of unit squares +whose centres (i, j) satisfy the inequality |i| + |j| ≤ n +with the origin O taken to be the centre of the AD. +Figure 5: The AD of order 2 can be tiled in 8 distinct ways. +the domain is symmetric under a quarter-turn rotation. +For n = 2, there are eight configurations shown in Figure +5. +Let +An +(n +∈ +N0) be the number of distinct +configurations of an AD of order n. It was proved that: +An = 2n(n+1)/2 +(4) +Although remarkable by its simplicity, none of the +many known proofs is elementary. Maybe even more +surprising, tilings of an AD exhibit a peculiar behaviour +as n increases, as can be seen in Figure 6 which shows +configurations sampled at random for orders n = 10, n = +100 and n = 1000. +We observe that the four corners of the AD look like +brickwalls: each of them contains exclusively horizontal +or vertical dominoes. In contrast, far away from the +boundaries, horizontal and vertical dominoes alternate +in a disordered fashion. As the order of the AD increases, +the separation between the four frozen corners and the +central disordered region becomes sharper. In the limit +n → +∞, this separation is a circle, known as the arctic +curve of the model. It means that, with probability 1, +each of the four regions outside the circle is covered by +the same type of dominoes. This may be heuristically +understood as follows. +Suppose that the unit square centred at (−n + +1/2, −1/2) is covered by an horizontal domino, see +Figure 7 (a). This forces all the dominoes adjacent to the +northwest and southwest boundaries to be horizontal too. +The number of configurations satisfying this contraint is +exactly An−1; hence, the probability p to observe one +of them becomes negligible as n gets larger since it +decreases exponentially with n: +p = An−1 +An += 2−n −−−−→ +n→+∞ 0. +(5) +In other words, with probability 1, the unit square +centred at (−n + 1/2, −1/2) will be covered by a vertical +domino. Because the AD is invariant under a quarter +tour rotation, the unit square centred at (n − 1/2, −1/2) +will also be covered, almost surely, by a vertical domino +while the unit squares centred at (−1/2, n − 1/2) and +(−1/2, −n + 1/2) will be covered by an horizontal +domino. Let us push the reasoning a little farther to +grasp the genesis of the arctic phenomenon, see Figure +7 (b). The probability to have four horizontal dominoes +with their left unit square centred at (−n + 3/2, −j + +1/2), with j = −2, −1, 0, 1, also vanishes in the limit +n → +∞, since the probability to observe a configuration +satisfying these requirements is: +An−1 − An−2 +An +−−−−→ +n→+∞ 0. +(6) +Hence, with probability 1, there must also be (at least) +one vertical domino covering a unit square centred at +(−n + 3/2, −j + 1/2) for some j ∈ {−2, −1, 0, 1}. This +heuristic suggests the emergence of brickwall patterns +within each corner. Maybe unexpectedely, these frozen +regions cover a non-negligible fraction of the AD (about +21.5%), even in the limit n → +∞. +Non-intersecting lattice paths +In order to further characterise this arctic phenomenon, +it is convenient to introduce an equivalent description +in terms of non-intersecting lattice paths. The latter is +obtained by first considering a checkerboard coloring of +the domain, see Figure 8 (a). +3 + +Figure 6: Configurations of an AD of order 10, 100 and 1000 (from left to right), generated by the shuffling algorithm. +(a) +(b) +Figure 7: (a) If, initially, the unit square centred at (−n + +1/2, −1/2) is covered by an horizontal domino (in +grey), then all the dominoes adjacent to the northwest +and southwest boundaries must also be horizontal (pink +dominoes). The number of such tilings is An−1. (b) if +the grey dominoes are fixed, then all the other dominoes +adjacent to the northwest ans southwest boundaries must +also be horizontal. +The number of configurations is +An−1 − An−2. Indeed, with those fixed dominoes, we are +left with an AD of order n − 1 whose two leftmost squares +are excluded or equivalently covered by a vertical domino. +Hence the number of configuration is An−1 from which +we must subtract the number of configurations (of an AD +of order n − 1) with two horizontal dominoes covering its +leftmost part, which is An−2. +(a) +S +N +W +E +(b) +Figure 8: (a) Checkerboard coloring of the AD such that the unit +square centred at (−n + 1/2, −1/2) is filled. (b) There +are four distinct types of dominoes, labelled by N, S, W and +E, that account for their orientation and the position of the +hatched unit square. Each of them, except the green one, is +associated with an elementary step (shown in magenta). +This enables to distinguish four types of dominoes, +labelled by a capital letter, according to the position of +the hatched unit square, see Figure 8 (b). +Then, we +draw on each domino, except for the N-dominoes which +remain empty, a line segment as follows: the S-domino +carries a horizontal (2, 0) line segment, the W-domino +a diagonal (1, 1) line segment and the E-domino a +diagonal (1, −1) line segment. When the segments are +actually drawn on the dominoes as prescribed, they form +continuous paths which go across the diamond from the +southwest boundary to the southeast boundary without +intersecting. This procedure links, bijectively, each tiling +of an AD of order n to a set of n non-intersecting paths +([11], section 2.1). For instance, Figure 9 (a) shows a +configuration of order 6 along with its bijection in terms +of non-intersecting lattice paths. +As suggested by Figures 6 and 9 (b), the frozen north +4 + +P1 +Q1 +P2 +Q2 +P3 +Q3 +P4 +Q4 +P5 +Q5 +P6 +Q6 +(a) +74 +Chapter 3. Simulations +when n → ∞. We will discuss shortly the precise zoom to perform in +order to obtain a non-trivial result. +X1 +n(T) +X2 +n(T) +X3 +n(T) +... +| +0 +τ +| +1 +| +−n +T +| +0 +| +n +Figure 3.4: Left: Non-intersecting Brownian motions with the same starting +and ending points, respectively at τ =0 and τ =1. Adapted from [115]. Right: +Tiling of an Aztec diamond, in its NILP description. The uppermost paths, +seen as functions of the abscissa T ∈[−n, n], are noted Xi +n(T) with i=1, 2, · · · +indexing the paths from top to bottom. +The reference frame is such that +X1 +n(±n)=−1/2. +When n → ∞ the NILP around (0, +1 +√ +2) locally resemble non-intersecting +Brownian bridges around the maximum of their limit shape (see the grey +circles in Figure 3.4). This apparently rough resemblance turns out to +be quite deep as some aspects of the statistics of these two processes +have been rigorously shown to be exactly identical, once appropriate +rescalings (or zooms) are performed. More precisely, the Airy process +describes the fluctuations of the (suitably rescaled) uppermost Brownian +bridge mentioned above [116] and Johansson characterized the statistics +of the uppermost path as follows: +Theorem 3.2.1. (Johansson [112]) Let X1 +n(T) be the vertical position +of the uppermost path of a (uniform) AD at a given lattice abscissa T +(see Figure 3.4) and A2(t) the Airy process. Then, +X1 +n(2−1/6n2/3t) − +n +√ +2 +2−5/6n1/3 +→ A2(t) − t2, +(3.4) +(b) +Figure 9: (a) A configuration of an AD of order 6, along with its bijection in terms of non-intersecting lattice paths. (b) The same, for an +AD of order 50. The vertical position of the kth uppermost path is denoted by Xkn(T) for −n + k − 1 ≤ T ≤ n − k + 1. +region is made up exclusively of N-dominoes and hence +void of paths. In the disordered region, it was proved +that the level curves for the density of N-dominoes +(namely the set of points for which the density of +N-dominoes is the same) are (incomplete) ellipses [4], as +substantiated in Figure 10: the closer to the north region +and the larger the density of N-dominoes. +Tracy-Widom +distribution +and +Gaussian Unitary ensemble +Let Xk +n(T) denote the vertical position of the kth topmost +path at abscissa T (−n + k − 1 ≤ T ≤ n − k + 1), for +k = 1, · · · , n. The uppermost path X1 +n(T) separates the +central disordered region from the frozen north region. +Configurations shown in Figure 11 and 11 (a) suggest +the following convergence in probability: +X1 +n(T) +n +P +−−−−→ +n→+∞ 1 + T +n +∀ T +n ∈ [−1, −1/2[ +X1 +n(T) +n +P +−−−−→ +n→+∞ +� +1 +2 − +� T +n +�2 +∀ T +n ∈ [−1/2, 1/2[ +X1 +n(T) +n +P +−−−−→ +n→+∞ 1 − T +n +∀ T +n ∈]1/2, 1] +(7) +1.0 +0.5 +0.0 +0.5 +1.0 +x +1.0 +0.5 +0.0 +0.5 +1.0 +y +0.5 +0.25 +0.125 +0.75 +0.2 +0.4 +0.6 +0.8 +1.0 +Figure 10: Density of N-dominoes, obtained from a sample of +100 000 configurations of order n = 500. +Lengths +were divided by n. The arctic circle is shown in black. +The predicted density level curves (dashed curves) give +the set of points (i.e. +ellipses) for which the density +of N-dominoes is equal to 0.125, 0.25, 0.5, 0.75. These +curves agree with the numerical results shown in color. +5 + +In other words, as n +→ ++∞ and after dividing +all the lengths by n, the uppermost path converges +almost surely to a straight line segment from (−1, 0) +to (−1/2, 1/2), a quarter circle from (−1/2, 1/2) to +(1/2, 1/2) and again a straight line segment from +(1/2, 1/2) to (1, 0). +In particular, for T = 0, we have: +X1 +n(0) +n +P +−−−−→ +n→+∞ +1 +√ +2 +(8) +This is however not the whole story. If we generate a +vast collection of tilings and report the vertical position +X1 +n(0) for each of them, then we obtain a set of points +whose distance from +n +√ +2 is proportional, on average, to +n1/3, see Figure 11. In other words: +E +� +X1 +n(0) − n +√ +2 +� +∼ n1/3 as n → +∞, +(9) +where the expected value E is taken over the whole +set of tilings of AD of order n. +The next step is to +determine which probability distribution ξ (as far as it +exists) governs the fluctuations of the properly rescaled +vertical position X1 +n(0): +lim +n→+∞ P +� X1 +n(0) − +n +√ +2 +n1/3 +≤ s +� += ξ(s), +(10) +Let us decompose X1 +n(0) as follows: +X1 +n(0) = Xn +n(0) + +n−1 +∑ +k=1 +Yk, +Yk = Xk +n(0) − Xk+1 +n +(0). +(11) +If the increments Yk were independent and identically +distributed random variables, then ξ would be the +normal distribution (provided a scaling n1/2 for the +fluctuations is used), by virtue of the central limit +theorem. +Here, however, the increments are not +independent and identically distributed. Indeed, based +on heuristic arguments (see section 3.2 in [6]), we have +with probability 1 that Xk +n(0) − Xk+1 +n +(0) = O(n1/3) for +k = O(1) while Xk +n(0) − Xk+1 +n +(0) = 1 almost surely in the +south frozen region, see Figure 9 (b). Hence, we should +not expect ξ to be a Gaussian distribution. Actually, the +distribution ξ is explicitly known and is given by [12]: +ξ(s) = F1 +2 (25/6s), +(12) +with F1 +2 the Tracy-Widom cumulative distribution1, see +Figure 12. The Tracy-Widom was first introduced in +random matrix theory to describe the fluctuations of the +largest eigenvalue of matrices belonging to the Gaussian +Unitary Ensemble [19]. +This ensemble is the set of +n × n hermitian matrices H whose entries are Gaussian +variables distributed as follows: +Hkk ∼ N(0, 1/2), +Hkl ∼ N(0, 1/4) + iN(0, 1/4) +(1 ≤ k < l ≤ n). +(13) +The joint probability density function P(λ1, λ2, · · · , λn) +of the ordered eigenvalues λ1 ≥ λ2 ≥ · · · ≥ λn is given +by: +P(λ1, λ2, · · · , λn) = 1 +Zn +∏ +1≤i 0, that we mea- +sure in units of thermal energy per elementary charge +arXiv:2301.00610v1 [physics.chem-ph] 2 Jan 2023 + +2 ++ +FIG. 1. An ideal electric double layer capacitor (EDLC). The +total amount of salt is fixed and, in the linear regime, it defines +the thickness of the double layer λD. +kBT/e. +The plates charge up and ionic concentration +profiles change in response to this, obeying the Poisson- +Nernst-Planck theory [35]. For two ionic species of va- +lences ±q± and diffusion coefficients D±, this relates the +electric potential φ(z, t) to the ion densities n±(z, t): +∂n± +∂t += D± +∂ +∂z +� +± q±e +kBT n± +∂φ +∂z + ∂n± +∂z +� +(1) +−∂2φ +∂z2 = +ρ +ε0εr +. +(2) +Here, ρ(z, t) = q+en+(z, t) − q−en−(z, t) is the electric +charge density. Eq. (1) is a continuity equation, whose +current has a drift and a diffusion term. Its equilibrium +zero-current state retrieves the Poisson-Boltzmann dis- +tribution [35]. Eq. (2) is the Poisson equation. +We +solve +numerically +Eqs. +(1)-(2) +via +a +flux- +conservative finite-difference integration scheme, +de- +scribed in [36]. Boundary conditions are the desired po- +tential difference and the vanishing of the ionic current at +the electrodes. We work in the canonical ensemble, with +no salt reservoir. We collect data for a range of applied +voltages and salt concentrations spanning, respectively, +5 and 10 orders of magnitude. We quantify the initial +densities n0 +± = n±(z, 0) = +� L +−L n±(z, t) dz/(2L) through +the dimensionless quantity λD/L, where λD = [(q2 ++n0 ++ + +q2 +−n0 +−)e2/(kBTε0εr)]−1/2 and defines the Debye screening +length. The salt concentration is n0 = n0 ++/q+ = n0 +−/q−. +We analyse the time evolution of the surface charge den- +sity σ(t) on the electrodes, (equal and opposite on the two +sides, as the current entering the generator equals the ex- +iting one) and of the ionic densities n±(z, t) through the +proxy ρ(±L, t) to reduce dimensionality. We characterise +the relaxation to the equilibrium values σeq = σ(t → ∞) +and ρ± +eq = ρ(±L, t → ∞) by inspecting all curves for +exponential or linear relaxation rates. +For the symmetric electrolyte case (D+ = D− = D +and q+ = q−), relaxation times τ are summarised in +Fig. 2 and are used to formulate the regime diagram of +FIG. 2. Exponential relaxation times τ extracted from linear +fits of log(σ(t)) vs t, as a function of dimensionless voltage +v. +For given λD and v, two different relaxation processes +are often seen in σ(t) (see Fig. 3): filled symbols indicate +the early-time process, whereas empty symbol the late-time +process, when present. For v ≤ 1 and λD/L ≥ 1, the relax- +ation is purely diffusive and takes place on a scale 2L. For +v ≤ 1 and λD/L ≪ 1, the double layer relaxes at early times +on a time LλD/D, that extends into the nonlinear regime +as (LλD/D) cosh(v/2) (dotted curves). This is followed by a +slower diffusive relaxation over a length L, signaling deple- +tion (empty symbols). For v ≫ 1, collective ion migration +causes full depletion: this early-time process is not shown +here because it is non-exponential (the gray dashed line how- +ever shows its time scale for the unscreened regime of Fig. 3, +where the process is linear). At late times, a fast diffusive +relaxation follows (empty symbols), signaling ion rearrange- +ment inside counterionic double layers of thickness µnen (the +Gouy-Chapman length). +Fig. 3a. For small v, the system is treatable analytically +[28, 29] and presents an infinite series of exponential re- +laxation times. For λD/L > 1, they all scale as ∼ L2/D, +the largest being 4L2/(π2D): this is the dominant time +scale and the only one clearly visible (Fig. 3b) and it +is a signature of ions diffusing over a length 2L toward +the oppositely charged plate. For λD/L < 1, subdom- +inant times are of order λ2 +D/D while the dominant one +scales as LλD/D (Fig. 3c). The exact expression of re- +laxation times in the linear regime was obtained by [29] +and we present an alternative derivation in [36]. Inter- +estingly, already from the linear regime, at small λD/L, +a sign of nonlinearity appears, termed depletion: due +to asymmetric accumulation in the electric double lay- +ers (EDL), the salt concentration decreases in the mid- +dle of the capacitor. +Indeed, at equilibrium, for each +species, the EDL next to the oppositely charged elec- +trode is more populated than the other EDL is depop- +ulated. After most of the EDL has built up, a neutral +excess of ions remains around z = 0 and diffuses away in +a time L2/(π2D) ≃ 10−1L2/D. This diffusion, occurring +on a length L, is visible in Fig. 2 (v < 1, empty symbols, +representing late-time relaxation) and in the long-time + +3 +a +b +c +d +e +f +g +FIG. 3. a) Regime diagram for the symmetric electrolyte case (D+ = D− = D, q+ = q− = 1). Five different regimes are +separated by the boundary lines discussed in the text. b-g) In black, the relative difference between the instantaneous electrode +charge density σ(t) and its equilibrium value σeq = σ(t → ∞). In addition, in e and g, the gray curves show the relative +difference between volume charge density ρ(t) at contact with the electrode and its equilibrium value ρeq. Time t has units of +L2/D. In crimson, the scaling behavior of relaxation times as extracted from linear fits and confirmed analytically [36]. +slope of Fig. 3c. +In Fig. 3a, v > 1 defines the nonlinear regime. Deple- +tion, quantified through the depletion fraction n±(0, t → +∞)/n0 +± and indicated by shadings of blue, dramatically +affects this regime. However, at small λD/L the ion con- +centration is large enough to make depletion a second +order effect and a non-depleted, purely nonlinear regime +is visible. +Such a window is delimited by the condi- +tion 2 (λD/L) sinh(v/2) ≪ 1 (dark-blue dashed curve), +matching the numerical calculations. +This is obtained +through the Grahame equation, relating potential and +electrode charge [35, 37]. We impose that the latter is +much less than the charge of all oppositely charged ions +in the system: σeq ≪ 2n0L (see [36]). +Relaxation, in +Fig. 3d, is governed at short times by a new exponential +time scale for EDL formation, increasing with v as shown +in Fig. 2 (symbols on dotted line). A RC-circuit equiv- +alence shows that this time scale is (λDL/D) cosh(v/2) +(dotted lines) and reflects the increased charge, i.e. ca- +pacitance, of the nonlinear EDL. At late times, as in the +linear regime, depletion manifests as a diffusive relax- +ation rate at late times, corresponding to the relaxation +of the neutral excess of ions. +As v increases further into the nonlinear regime, the +system rapidly transitions to a fully depleted final state +(depletion fraction ≃ 1, i.e. practically no ions in the +bulk) and the physics changes drastically. To understand +it, we focus first on the top-right part of Fig. 3a. For +such strong voltages, ionic screening has a relatively small +effect on the electrode charge, so that ions are pulled +at constant velocity toward the oppositely charged wall. +Charge density in the EDL grows linearly in time, as +shown in Fig. 3e (gray). +Also, since the applied volt- +age results from the sum of the electric fields due to the +electrodes’ charge and to the ionic charge, the latter are +linearly related. Because the non-neutral bulk portion +increases linearly with time, the electric field due to the +solution is parabolic in time, and so is σ (black). Once +the two purely counterionic EDLs are formed, a final ex- +ponential relaxation occurs (Fig. 3f). +The only mean- +ingful length scale is now the Gouy-Chapman length, +the distance over which an isolated counterion can be +dragged away from the electrode, with an energy bud- +get kBT [35, 37]; it sets the extension of the double- +layer in a salt-free regime. +Since one electrode carries +a larger charge than its counterions, this half-system is +not electroneutral and the right Gouy-Chapman length +reads µnen = 2ε0εrkBT/(e2qσres), where nen stands for +non-electroneutral and σres = σeq − 2n0L is the part +of electrode charge not neutralised by counterions [36]. +As σres ∝∼ v, the late relaxation time is µ2 +nen/D ∝ v−2 +(dashed black line in Fig. 2). +We have just assumed that the ion dynamics do not +perturb much the externally imposed electric field in the +EDL (yet it sensibly affects electrode charge relaxation, +as seen in Fig. 3e). This only occurs if σeq ∝∼ v is ≫ 2n0L, +which defines the region above the pink line in Fig. 3a. +Below such line, ions partially screen the applied field, by +an amount that varies during relaxation. In this regime +the early-time relaxation dynamics is neither linear, nor + +4 +quadratic or exponential (Fig. 3g). However, it ends as +abruptly as in the unscreened regime and, at late times, +the purely counterionic EDL relaxes exponentially over +the Gouy-Chapman length µnen (inset). +The general asymmetric case is described in Fig. 4. To +obtain clear separations between the relevant time scales, +we choose D+/D− = 1/10, q+ : q− = 1 : 2, and call pos- +itive ions slow and negative ions fast; other choices lead +to analogous results [36]. Several differences arise com- +pared to the symmetric case described so far, making the +phenomenology even richer. +In the linear, large λD/L +regime, the two ionic species are completely decoupled. +Each of the infinitely many relaxation modes from the +symmetric case splits into two, so that half of them are +proportional to L2/D+ and half to L2/D−; the slowest +mode concerns the diffusion of the slowest species over +a length 2L, so its characteristic time is 4L2/(π2Dslow), +with Dslow = D+ (Fig. 4b). Valences do not affect re- +laxation times, but they play a role in determining the +weight of each mode [36]. +In constrast, at small λD/L (Fig. 4c) valences mat- +ter: as the bulk conductivities due to the two species +are additive and each of them is proportional to q±D±, +a simple RC-circuit analogy shows that the relevant dif- +fusivity is Dave = (q+D+ + q−D−)/(q+ + q−), and the +EDL forms on a timescale λDL/Dave [36]. At late times, +relaxation is due to the neutral excess diffusion gov- +erned instead by the Nernst-Hartley diffusivity DNH = +(q+ + q−)D+D−/(q+D+ + q−D−) [38], relevant also in +the context of impedance spectroscopy [39–41]. Indeed, +in relaxing the neutral excess by diffusion, positive and +negative ions must move together, with the slow species +slowing down the fast and the fast pulling the slow: this +is reflected by a friction (inverse diffusivity) which is the +average of the frictions of the two species. In the ion- +symmetric case, relaxation of the neutral excess occurs +over a length L, from the exact centre to the electrodes, +effectively forbidding a diffusive mode with wavelength +2L; ionic asymmetry relaxes this constraint, so that the +late-time characteristic time is now 4L2/(π2DNH). This +neutral excess relaxation is a purely linear phenomenon +and is not due to depletion (that still happens, but on a +faster, hidden scale here). As the fast species tends to re- +lax according to the instantaneous distribution assumed +by the slow species, an overcrowding of carriers occurs at +the electrode of same charge as the slow ions: these move +away slowly, so that the fast ions, in the attempt to equi- +librate the EDL, arrive in larger amounts than needed for +equilibrum. Eventually, as the slow ions gradually move +away, the overabundant fast ones also leave the EDL in +the observed relaxation of the neutral excess by diffusion. +Macroscopic depletion, at large λD/L, now occurs at +different voltages for the two species. +This is shown +by the mismatch between colour gradient and hatch- +ing patterns in Fig. 4a. +The linear regime shrinks to +the region v < q−1 +max = q−1 +− . +In the intermediate re- +gion q−1 +max < v < q−1 +min, sharply defined for strong valence +asymmetry, the depletion mismatch results in a rapid re- +laxation of the higher-charge (depleted) species, followed +by a linear-regime-like relaxation of the smaller-charge +(non-depleted) species [36]. At small λD/L, where non- +linear features emerge before bulk depletion, this effect +disappears and the two species are equally depleted at +given v. The analytic expression for the onset of deple- +tion (dashed blue line) is given in [36]: as depletion is +measured at equilibrium, it depends on q± but not on +D±. +The non-depleted nonlinear regime (Fig. 4d) confirms +the importance of Dave and DNH. In particular, during +early-time EDL build-up, the cosh factor accounting for +increased EDL capacitance in the symmetric case is re- +placed by ˜c(v), whose cumbersome expression we work +out in [36]. For v ≫ 1, asymmetry always enhances ca- +pacitance, thereby increasing the relaxation time com- +pared to the 1 : 1 case. +Finally, +in the fully depleted, +unscreened regime +(Fig. 4e), the situation is analogous to the symmetric +case, but the two trains of positive and negative ions +are now dragged by the electric field at different veloci- +ties. This results in ρ relaxing with different time scales +at the two electrodes (dashed and dotted gray). +The +electrode charge (black) is piecewise quadratic. At late +times, the only visible relaxation is due to the counte- +rions of the slower EDL, each with its own µnen±. The +characteristic time is then the largest one between the +times µ2 +nen±/D± = 4L2/(D±q±v2), which in Fig. 4f is +given by the slower and lower-charge species. Such re- +laxation process is also the last one to happen in the +partially screened regime (Fig. 4g), while, again, early- +time behavior is neither exponential nor linear. +In summary, we identified the dominant relaxation +processes within mean-field and their often counter- +intuitive characteristic times, for the whole parameter +space. +This work provides a long-missed, easy-access +frame of comparison for all theories intended to in- +corporate steric effects and ionic correlations or aimed +at describing ionic liquids. +While some of these have +pointed out the limits of mean-field, many have already +confirmed its relevance for real physical systems [17]. +Understanding the relaxation dynamics paves the way +to optimisation strategies and design of supercapacitors. +Acknowledgements +This work has received fund- +ing from the European Union’s Horizon 2020 research +and innovation program under the Marie Sk�lodowska- +Curie grant agreements Nos. 674979-NANOTRANS and +101034413. This project received funding from the Euro- +pean Research Council under the European Union’s Hori- +zon 2020 research and innovation program (Grant Agree- +ment No. 863473). +B.R. acknowledges financial sup- +port from the French Agence Nationale de la Recherche +(ANR) under Grant No. ANR-21-CE29-0021-02 (DIA- + +5 +a +b +c +d +e +f +g +FIG. 4. a) Regime diagram for the completely asymmetric case (D+/D− = 1/10, q+ = 1, q− = 2). Depletion of positive ions +(blue tones) is distinct from that of negative, more charged, ions (hatch patterns). b-g) As in Fig. 3. In e and g, dotted and +dashed grey lines represent the charge densities at the negative and the positive electrodes, respectively; in the symmetric case, +these were equal. Time t has units of L2/D+. +DEM). +[1] M. Salanne, B. Rotenberg, K. Naoi, K. Kaneko, P.-L. +Taberna, C. P. Grey, B. Dunn, and P. Simon, Efficient +Storage Mechanisms for Building Better Supercapacitors, +Nature Energy 1, 16070 (2016). +[2] P. Simon and Y. 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Robinson and R. H. Stokes, Electrolyte solutions: +Second Revised Edition (Dover Publications, 2002) pp. +286–292. +[39] G. Barbero and L. R. Evangelista, Adsorption Phenom- +ena and Anchoring Energy in Nematic Liquid Crystals +(CRC Press, 2005). +[40] G. Barbero and I. Lelidis, Evidence of the ambipolar dif- +fusion in the impedance spectroscopy of an electrolytic +cell, Physical Review E 76, 051501 (2007). +[41] G. Barbero, F. Batalioto, and A. M. Figueiredo Neto, +Theory of small-signal ac response of a dielectric liquid +containing two groups of ions, Applied Physics Letters +92, 172908 (2008). + diff --git a/StAyT4oBgHgl3EQfuPmz/content/tmp_files/load_file.txt b/StAyT4oBgHgl3EQfuPmz/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..971b9da0f4311231a9cf3820027f010123b366ce --- /dev/null +++ b/StAyT4oBgHgl3EQfuPmz/content/tmp_files/load_file.txt @@ -0,0 +1,448 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf,len=447 +page_content='Charging dynamics of electric double layer nanocapacitors in mean-field Ivan Palaia Institute of Science and Technology Austria, 3400 Klosterneuburg, Austria Adelchi J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Asta Younited, 75009 Paris, France Patrick B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Warren Hartree Centre,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Science and Technology Facilities Council (STFC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Sci-Tech Daresbury,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Warrington WA4 4AD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' United Kingdom Benjamin Rotenberg Sorbonne Universit´e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Physico-chimie des ´Electrolytes et Nanosyst`emes Interfaciaux,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' PHENIX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 75005 Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' France and R´eseau sur le Stockage Electrochimique de l’Energie (RS2E),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' FR CNRS 3459,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 80039 Amiens Cedex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' France Emmanuel Trizac Universit´e Paris-Saclay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' LPTMS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 91405 Orsay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' France (Dated: January 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 2023) An electric double layer capacitor (EDLC) stores energy by modulating the spatial distribution of ions in the electrolytic solution that it contains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' We determine the mean-field time scales for planar EDLC relaxation to equilibrium, after a potential difference is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' We tackle first the fully symmetric case, where positive and negative ionic species have same valence and diffusivity, and then the general, more complex, asymmetric case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Depending on applied voltage and salt concentration, different regimes appear, revealing a remarkably rich phenomenology relevant for nanocapacitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Two conductive surfaces separated by an ionic solu- tion form an electric double layer capacitor (EDLC), that stores electrostatic energy by modulating the distribu- tion of charged species in solution [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Nanoporous conductive materials offer an optimized contact between the electrolyte and the electrode, where charge storage occurs, leading to specific capacities as large as 100 F/g of material [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Their ability to store and release charge much faster than in batteries, which involve electrochem- ical reactions [4], allows their use in applications requir- ing high powers, from the recovery of breaking energy to electrical public transportation means covering short distances and recharging during stops [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' A promising use of so-called supercapacitors concerns the extraction of ”blue energy” from fresh and salty water, or conversely the desalination of water, using cycles of charge and dis- charge of capacitors [6–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Finally, it is now possible to use electrodes in nanocapacitors and nanofluidic devices to study extremely small volumes of electrolytes [11–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Predicting the charging dynamics of EDLCs is essen- tial, because tuning the related characteristic time allows to maximize efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' For both aqueous solutions and ionic liquids, the effects of ion correlations and size on the charging dynamics have been described by density func- tional theories [14–17], lattice models [18] and molecular simulations [19–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' More fundamental studies rely on mean-field continuum models of the electrolyte [28, 29], in planar and non-planar geometries [30–32], which can also be simulated using lattice-based models to capture electrokinetic couplings [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' In this context, particu- larly well studied is the linear regime (small applied volt- age) for a symmetric electrolyte (cations and anions with same valence and diffusivity) [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' The large-voltage regime remains however elusive and the asymmetric elec- trolyte case is unexplored, even though it is in fact the rule rather than the exception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' In addition, the non- linear effects emerging from the fact that the number of ions enclosed in real capacitors is fixed (rather than their chemical potential) are scarcely known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Here, we fill these gaps and identify how the time scales governing relaxation to equilibrium depend on the key parameters of the system: applied voltage, ion concentration and dis- tance between electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' We find an unexpectedly rich regime diagram, highlighting the importance of finite sys- tem size and ion asymmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Our model system is represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' The plates, distant 2L from each other, are ideal conductors whose dielectric mismatch with the solution we neglect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' To allow analytical and computational treatment, water is supposed to have constant permittivity ε0εr and correla- tions between ions are neglected – a condition known as weak coupling [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' The system is at equilibrium at zero potential difference (2v = 0) for times t < 0: the two ionic species, positive and negative, are homogeneously distributed, the solution is locally neutral everywhere and the plates are uncharged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' At time t = 0 we instanta- neously switch on the potential 2v > 0, that we mea- sure in units of thermal energy per elementary charge arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content='00610v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content='chem-ph] 2 Jan 2023 2 + FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' An ideal electric double layer capacitor (EDLC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' The total amount of salt is fixed and, in the linear regime, it defines the thickness of the double layer λD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' kBT/e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' The plates charge up and ionic concentration profiles change in response to this, obeying the Poisson- Nernst-Planck theory [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' For two ionic species of va- lences ±q± and diffusion coefficients D±, this relates the electric potential φ(z, t) to the ion densities n±(z, t): ∂n± ∂t = D± ∂ ∂z � ± q±e kBT n± ∂φ ∂z + ∂n± ∂z � (1) −∂2φ ∂z2 = ρ ε0εr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' (2) Here, ρ(z, t) = q+en+(z, t) − q−en−(z, t) is the electric charge density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' (1) is a continuity equation, whose current has a drift and a diffusion term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Its equilibrium zero-current state retrieves the Poisson-Boltzmann dis- tribution [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' (2) is the Poisson equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' We solve numerically Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' (1)-(2) via a flux- conservative finite-difference integration scheme, de- scribed in [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Boundary conditions are the desired po- tential difference and the vanishing of the ionic current at the electrodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' We work in the canonical ensemble, with no salt reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' We collect data for a range of applied voltages and salt concentrations spanning, respectively, 5 and 10 orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' We quantify the initial densities n0 ± = n±(z, 0) = � L −L n±(z, t) dz/(2L) through the dimensionless quantity λD/L, where λD = [(q2 +n0 + + q2 −n0 −)e2/(kBTε0εr)]−1/2 and defines the Debye screening length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' The salt concentration is n0 = n0 +/q+ = n0 −/q−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' We analyse the time evolution of the surface charge den- sity σ(t) on the electrodes, (equal and opposite on the two sides, as the current entering the generator equals the ex- iting one) and of the ionic densities n±(z, t) through the proxy ρ(±L, t) to reduce dimensionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' We characterise the relaxation to the equilibrium values σeq = σ(t → ∞) and ρ± eq = ρ(±L, t → ∞) by inspecting all curves for exponential or linear relaxation rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' For the symmetric electrolyte case (D+ = D− = D and q+ = q−), relaxation times τ are summarised in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 2 and are used to formulate the regime diagram of FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Exponential relaxation times τ extracted from linear fits of log(σ(t)) vs t, as a function of dimensionless voltage v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' For given λD and v, two different relaxation processes are often seen in σ(t) (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 3): filled symbols indicate the early-time process, whereas empty symbol the late-time process, when present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' For v ≤ 1 and λD/L ≥ 1, the relax- ation is purely diffusive and takes place on a scale 2L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' For v ≤ 1 and λD/L ≪ 1, the double layer relaxes at early times on a time LλD/D, that extends into the nonlinear regime as (LλD/D) cosh(v/2) (dotted curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' This is followed by a slower diffusive relaxation over a length L, signaling deple- tion (empty symbols).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' For v ≫ 1, collective ion migration causes full depletion: this early-time process is not shown here because it is non-exponential (the gray dashed line how- ever shows its time scale for the unscreened regime of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 3, where the process is linear).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' At late times, a fast diffusive relaxation follows (empty symbols), signaling ion rearrange- ment inside counterionic double layers of thickness µnen (the Gouy-Chapman length).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' For small v, the system is treatable analytically [28, 29] and presents an infinite series of exponential re- laxation times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' For λD/L > 1, they all scale as ∼ L2/D, the largest being 4L2/(π2D): this is the dominant time scale and the only one clearly visible (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 3b) and it is a signature of ions diffusing over a length 2L toward the oppositely charged plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' For λD/L < 1, subdom- inant times are of order λ2 D/D while the dominant one scales as LλD/D (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 3c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' The exact expression of re- laxation times in the linear regime was obtained by [29] and we present an alternative derivation in [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Inter- estingly, already from the linear regime, at small λD/L, a sign of nonlinearity appears, termed depletion: due to asymmetric accumulation in the electric double lay- ers (EDL), the salt concentration decreases in the mid- dle of the capacitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Indeed, at equilibrium, for each species, the EDL next to the oppositely charged elec- trode is more populated than the other EDL is depop- ulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' After most of the EDL has built up, a neutral excess of ions remains around z = 0 and diffuses away in a time L2/(π2D) ≃ 10−1L2/D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' This diffusion, occurring on a length L, is visible in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 2 (v < 1, empty symbols, representing late-time relaxation) and in the long-time 3 a b c d e f g FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' a) Regime diagram for the symmetric electrolyte case (D+ = D− = D, q+ = q− = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Five different regimes are separated by the boundary lines discussed in the text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' b-g) In black, the relative difference between the instantaneous electrode charge density σ(t) and its equilibrium value σeq = σ(t → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' In addition, in e and g, the gray curves show the relative difference between volume charge density ρ(t) at contact with the electrode and its equilibrium value ρeq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Time t has units of L2/D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' In crimson, the scaling behavior of relaxation times as extracted from linear fits and confirmed analytically [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' slope of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 3a, v > 1 defines the nonlinear regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Deple- tion, quantified through the depletion fraction n±(0, t → ∞)/n0 ± and indicated by shadings of blue, dramatically affects this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' However, at small λD/L the ion con- centration is large enough to make depletion a second order effect and a non-depleted, purely nonlinear regime is visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Such a window is delimited by the condi- tion 2 (λD/L) sinh(v/2) ≪ 1 (dark-blue dashed curve), matching the numerical calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' This is obtained through the Grahame equation, relating potential and electrode charge [35, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' We impose that the latter is much less than the charge of all oppositely charged ions in the system: σeq ≪ 2n0L (see [36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Relaxation, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 3d, is governed at short times by a new exponential time scale for EDL formation, increasing with v as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 2 (symbols on dotted line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' A RC-circuit equiv- alence shows that this time scale is (λDL/D) cosh(v/2) (dotted lines) and reflects the increased charge, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' ca- pacitance, of the nonlinear EDL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' At late times, as in the linear regime, depletion manifests as a diffusive relax- ation rate at late times, corresponding to the relaxation of the neutral excess of ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' As v increases further into the nonlinear regime, the system rapidly transitions to a fully depleted final state (depletion fraction ≃ 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' practically no ions in the bulk) and the physics changes drastically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' To understand it, we focus first on the top-right part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' For such strong voltages, ionic screening has a relatively small effect on the electrode charge, so that ions are pulled at constant velocity toward the oppositely charged wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Charge density in the EDL grows linearly in time, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 3e (gray).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Also, since the applied volt- age results from the sum of the electric fields due to the electrodes’ charge and to the ionic charge, the latter are linearly related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Because the non-neutral bulk portion increases linearly with time, the electric field due to the solution is parabolic in time, and so is σ (black).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Once the two purely counterionic EDLs are formed, a final ex- ponential relaxation occurs (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 3f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' The only mean- ingful length scale is now the Gouy-Chapman length, the distance over which an isolated counterion can be dragged away from the electrode, with an energy bud- get kBT [35, 37];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' it sets the extension of the double- layer in a salt-free regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Since one electrode carries a larger charge than its counterions, this half-system is not electroneutral and the right Gouy-Chapman length reads µnen = 2ε0εrkBT/(e2qσres), where nen stands for non-electroneutral and σres = σeq − 2n0L is the part of electrode charge not neutralised by counterions [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' As σres ∝∼ v, the late relaxation time is µ2 nen/D ∝ v−2 (dashed black line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' We have just assumed that the ion dynamics do not perturb much the externally imposed electric field in the EDL (yet it sensibly affects electrode charge relaxation, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 3e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' This only occurs if σeq ∝∼ v is ≫ 2n0L, which defines the region above the pink line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Below such line, ions partially screen the applied field, by an amount that varies during relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' In this regime the early-time relaxation dynamics is neither linear, nor 4 quadratic or exponential (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 3g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' However, it ends as abruptly as in the unscreened regime and, at late times, the purely counterionic EDL relaxes exponentially over the Gouy-Chapman length µnen (inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' The general asymmetric case is described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' To obtain clear separations between the relevant time scales, we choose D+/D− = 1/10, q+ : q− = 1 : 2, and call pos- itive ions slow and negative ions fast;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' other choices lead to analogous results [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Several differences arise com- pared to the symmetric case described so far, making the phenomenology even richer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' In the linear, large λD/L regime, the two ionic species are completely decoupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Each of the infinitely many relaxation modes from the symmetric case splits into two, so that half of them are proportional to L2/D+ and half to L2/D−;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' the slowest mode concerns the diffusion of the slowest species over a length 2L, so its characteristic time is 4L2/(π2Dslow), with Dslow = D+ (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Valences do not affect re- laxation times, but they play a role in determining the weight of each mode [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' In constrast, at small λD/L (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 4c) valences mat- ter: as the bulk conductivities due to the two species are additive and each of them is proportional to q±D±, a simple RC-circuit analogy shows that the relevant dif- fusivity is Dave = (q+D+ + q−D−)/(q+ + q−), and the EDL forms on a timescale λDL/Dave [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' At late times, relaxation is due to the neutral excess diffusion gov- erned instead by the Nernst-Hartley diffusivity DNH = (q+ + q−)D+D−/(q+D+ + q−D−) [38], relevant also in the context of impedance spectroscopy [39–41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Indeed, in relaxing the neutral excess by diffusion, positive and negative ions must move together, with the slow species slowing down the fast and the fast pulling the slow: this is reflected by a friction (inverse diffusivity) which is the average of the frictions of the two species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' In the ion- symmetric case, relaxation of the neutral excess occurs over a length L, from the exact centre to the electrodes, effectively forbidding a diffusive mode with wavelength 2L;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' ionic asymmetry relaxes this constraint, so that the late-time characteristic time is now 4L2/(π2DNH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' This neutral excess relaxation is a purely linear phenomenon and is not due to depletion (that still happens, but on a faster, hidden scale here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' As the fast species tends to re- lax according to the instantaneous distribution assumed by the slow species, an overcrowding of carriers occurs at the electrode of same charge as the slow ions: these move away slowly, so that the fast ions, in the attempt to equi- librate the EDL, arrive in larger amounts than needed for equilibrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Eventually, as the slow ions gradually move away, the overabundant fast ones also leave the EDL in the observed relaxation of the neutral excess by diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Macroscopic depletion, at large λD/L, now occurs at different voltages for the two species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' This is shown by the mismatch between colour gradient and hatch- ing patterns in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' The linear regime shrinks to the region v < q−1 max = q−1 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' In the intermediate re- gion q−1 max < v < q−1 min, sharply defined for strong valence asymmetry, the depletion mismatch results in a rapid re- laxation of the higher-charge (depleted) species, followed by a linear-regime-like relaxation of the smaller-charge (non-depleted) species [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' At small λD/L, where non- linear features emerge before bulk depletion, this effect disappears and the two species are equally depleted at given v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' The analytic expression for the onset of deple- tion (dashed blue line) is given in [36]: as depletion is measured at equilibrium, it depends on q± but not on D±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' The non-depleted nonlinear regime (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 4d) confirms the importance of Dave and DNH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' In particular, during early-time EDL build-up, the cosh factor accounting for increased EDL capacitance in the symmetric case is re- placed by ˜c(v), whose cumbersome expression we work out in [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' For v ≫ 1, asymmetry always enhances ca- pacitance, thereby increasing the relaxation time com- pared to the 1 : 1 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Finally, in the fully depleted, unscreened regime (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 4e), the situation is analogous to the symmetric case, but the two trains of positive and negative ions are now dragged by the electric field at different veloci- ties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' This results in ρ relaxing with different time scales at the two electrodes (dashed and dotted gray).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' The electrode charge (black) is piecewise quadratic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' At late times, the only visible relaxation is due to the counte- rions of the slower EDL, each with its own µnen±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' The characteristic time is then the largest one between the times µ2 nen±/D± = 4L2/(D±q±v2), which in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 4f is given by the slower and lower-charge species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Such re- laxation process is also the last one to happen in the partially screened regime (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 4g), while, again, early- time behavior is neither exponential nor linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' In summary, we identified the dominant relaxation processes within mean-field and their often counter- intuitive characteristic times, for the whole parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' This work provides a long-missed, easy-access frame of comparison for all theories intended to in- corporate steric effects and ionic correlations or aimed at describing ionic liquids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' While some of these have pointed out the limits of mean-field, many have already confirmed its relevance for real physical systems [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Understanding the relaxation dynamics paves the way to optimisation strategies and design of supercapacitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Acknowledgements This work has received fund- ing from the European Union’s Horizon 2020 research and innovation program under the Marie Sk�lodowska- Curie grant agreements Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 674979-NANOTRANS and 101034413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' This project received funding from the Euro- pean Research Council under the European Union’s Hori- zon 2020 research and innovation program (Grant Agree- ment No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 863473).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' acknowledges financial sup- port from the French Agence Nationale de la Recherche (ANR) under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' ANR-21-CE29-0021-02 (DIA- 5 a b c d e f g FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' a) Regime diagram for the completely asymmetric case (D+/D− = 1/10, q+ = 1, q− = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' Depletion of positive ions (blue tones) is distinct from that of negative, more charged, ions (hatch patterns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' b-g) As in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' In e and g, dotted and dashed grey lines represent the charge densities at the negative and the positive electrodes, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} +page_content=' in the symmetric case, these were 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/StAyT4oBgHgl3EQfuPmz/content/2301.00610v1.pdf'} diff --git a/V9AyT4oBgHgl3EQfu_l-/vector_store/index.faiss b/V9AyT4oBgHgl3EQfu_l-/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..e3e7ce4fb654b2486dbc987e8bdcfd291ac15fe1 --- /dev/null +++ b/V9AyT4oBgHgl3EQfu_l-/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b49d81a42b53d13a64455af98f2ab080b983772175b20ea30fe1d677279b8db9 +size 2162733 diff --git a/V9AyT4oBgHgl3EQfu_l-/vector_store/index.pkl b/V9AyT4oBgHgl3EQfu_l-/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..786490c13b31e55714877234def88ee234be0c67 --- /dev/null +++ b/V9AyT4oBgHgl3EQfu_l-/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a3617f33e90c27680732c6999018dbefba19487b94e5b802d3ea340a9f364c98 +size 99455 diff --git 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Mahichi1, ∗ and Alireza Amani +1, † +1Department of Physics, Ayatollah Amoli Branch, Islamic Azad University, Amol, Iran +(Dated: January 13, 2023) +In this paper, we study the viscous f(T, B) gravity model as a source of dark energy, +and the Extended Bose-Einstein Condensate (EBEC) as a source of dark matter, in a flat- +FRW metric. In the presence of bulk viscosity, we obtain Friedmann equations and write +two continuity equations of dark energy and dark matter by interacting them. Using the +generalized Gross-Pitaeveskii equation, we earn Equation of State (EoS) of dark matter by +EBEC regime as pm = αρm + βρ2 +m in which the both of terms are respectively introduced +as normal dark matter and dark matter halo. The innovation of the work is that we can +simultaneously describe the nature of the dark parts of the universe with the viscous f(T, B) +gravity and the EBEC regime, which leads to a deep understanding of the different epochs +of the universe from early to late times. In what follows, the energy density and the pressure +of dark energy are reconstructed in terms of the redshift parameter, and then we fit the +obtained results with 53 supernova data from the Hubble data constraints. Next, we plot +the cosmological parameters in terms of the redshift parameter and conclude that the current +universe is in an accelerated phase. Finally, we analyze the stability and instability of the +current model with the sound speed parameter as well as we draw the density parameter +values for dark energy in terms of the redshift parameter. +PACS numbers: 98.80.-k; 98.80.Es; 04.50.Kd; 95.35.+d; 03.75.Nt +Keywords: +Dark energy, Dark matter, Extended Bose-Einstein condensate, The viscous f(T, B) +gravity. +∗ elhammahichi@gmail.com +† Corresponding Author: al.amani@iau.ac.ir + +2 +I. +INTRODUCTION +One of the greatest concerns of human beings is to know about the universe, which requires +knowing the contents of the universe, such as dark energy, dark matter, and so on. From the very +beginning, many efforts were made to this end, including Friedmann and his famous equations, +although Hubble’s contribution to this issue is also important because Hubble-Lumiter’s law con- +firmed that the universe is expanding at a constant speed. This velocity is proportional to the +distance of the galaxies from the Earth. Because of the importance of knowing the universe, in re- +cent decades, by measuring cosmic distances and observing cosmic objects beyond the Milky Way, +they have found that the universe is expanding rapidly. Therefore, the expansion of the universe +at a non-constant speed was discovered and confirmed by Ia-type supernovae, cosmic microwave +background radiation, and large-scale structures in the universe [1–4]. The cause of this event +is a mysterious and unknown force that is greater than the force of gravity but in the opposite +direction, which is called dark energy. Since the universe is considered a thermodynamic system, +and its evolution depends only on the internal contents of the universe, therefore, due to the lack +of matter-energy transfer from outside to inside, must be attributed to a strong negative pressure +that acts as a repulsion that leads to the expansion of the universe. With these explanations, the +problem of dark energy can be interpreted in the fundamental theoretical framework of string the- +ory and quantum gravity. Extensive studies on dark energy with models such as the cosmological +constant [5, 6], scalar fields [7–22], modified gravity [23–31], holography and agegraphic [32–39], +bouncing theory [40–44], teleparallel gravity [45–49] and braneworld models [50–52]. +Among these mentioned models, teleparallel gravity is considered to be one of the suitable +candidates for dark energy, which was first introduced by Einstein for the union between elec- +tromagnetism and gravity. This theory states that space-time is associated with a curvature-free +linear relationship by a metric tensor field called tetrad-vector field dynamics. In that case, a +tetrad field is naturally used to define a linear Weitzenb¨ock connection, which represents a tor- +sion connection without curvature. But Levi-Civita connection is used in the geometry of general +relativity to show the curvature connection without torsion. Therefore, by converting the tetrad +components and metric components to each other, we can convert teleparallel gravity model to +the general relativity model and vice versa. This means, the curvature term in general relativity +changes to a torsion term in the teleparallel model and vice versa, which confirms the claim that +teleparallel gravity and general relativity are equivalent. This equivalent is caused the relation +R = −T + 2∇µT µ in which R, T, and T µ are curvature scalar, torsion scalar, and torsion vector, + +3 +respectively. Nevertheless, the corresponding field equations are clearly covariant and then the +theory has a local Lorentz invariant [53]. By converting T to f(T), we expand teleparallel gravity +to modified teleparallel gravity, where f(T) is the function of the torsion scalar. Also, by introduc- +ing B = 2∇µT µ, we can redevelop the modified teleparallel gravity in the form f(T, B) gravity in +which B is the boundary term. Thus, model f(T, B) gravity can simultaneously covers both the +models of f(T) gravity and f(R) gravity with help of relation R = −T + B, i.e., it provides an +equivalence relation between torsion and curvature. [54, 55]. +To study the universe more realistically, we consider it as an anisotropic fluid with viscosity, +because the bulk viscosity creates internal friction that converts the kinetic energy of particles into +heat, an idea that could help to describe the recent acceleration of the universe. Therefore, bulk +viscosity play a critical role in determining the dissipative effect of fluids. [56–60]. +Since in physics it is believed that the nature of everything unknown to us is called dark or +black, so the unknown parts of the universe are called dark energy and dark matter. +Hence, +we turn our attention to another dark part of the universe, i.e., dark matter. +The nature of +dark matter can be baryonic, this means, astronomical objects such as massive and compact +haloes are composed of ordinary matter, but their electromagnetic radiation is negligible. Also, +dark matter can be non-baryonic, that is, it is composed of hypothetical and real particles, in +which axions and Weakly Interacting Massive Particles (WIMPs) are hypothetical particles, and +Bose-Einstein condensate (BEC) is a state of matter created when particles called bosons are +cooled to near absolute zero [61–79]. Hence dark matter can be described as a non-relativistic, +Newtonian Bose-Einstein gravitational condensate gas, whose density and pressure are related by +a barotropic equation of state [79]. For this purpose, we intend to relate dark matter and Bose- +Einstein condensate (BEC). As we know, Bose-Einstein statistics are a concept that formed the +basis of mathematics and enabled us to distinguish particular particles from each other. +BEC +is a substance in which dilute boson gas is cooled to very low temperatures. Due to this very +low temperature, phase transfer occurs and most of the boson gases occupy the lowest quantum +state, and the macroscopic quantum phenomenon appears at this point. +Cold bosons fall on +each other, and superparticles appear that behave like microwaves. +This means that the dark +matter is assumed to be a bosonic gas below the critical temperature that forms BEC [74–79]. By +using the generalized Gross-Pitaevskii equation, the EoS of dark matter as a barotropic fluid is +obtained, i.e., matter pressure, pm, is only a function of matter energy density, ρm. Note that the +corresponding EoS has been introduced as a normal dark matter [74]. Also, in the same of Ref. +[74], by considering the dark matter halo in a quantum ground state, the EoS of dark matter was + +4 +obtained as pm ∝ ρ2 +m. After that, we extended the BEC model to the Extended Bose-Einstein +Condensation (EBEC) model as cited in Ref. [80]. This means that dark matter is simultaneously +considered in both cases of the normal dark matter and dark matter halo. In fact, EBEC regime +helps us to understand more accurately the various epochs of the universe. +In general, dark matter and dark energy have the nature of attraction and repulsion, respectively. +This means that the attractive force in dark matter pulls matter inside, but the repulsive force of +dark energy pushes it outside. Therefore, in this study, we attribute the viscous f(T, B) gravity +model to dark energy and the EBEC model to dark matter. +This assessment motivates us to +investigate the nature of the largest cosmic scales by viscous f(T, B) gravity, and to study the +nature of individual galaxies by EBEC. +This paper is organized as follows: +In Sec. II, we present a foundation of f(T, B) gravity in the flat-FRW metric by bulk viscosity +fluid. +In Sec. +III, we will explore EBEC as dark matter, and write the dark matter EoS as +the virial expansion. +In Sec. +IV, we reconstruct the f(T, B) gravity in terms of the redshift +parameter, and fit the corresponding system by the Hubble data constraints. +In Sec. +V, we +obtain the relevant cosmological parameters in terms of the redshift parameter and then plot the +corresponding graphs. In Sec. VI, we analyse the stability and instability of the present model by +the sound speed parameter. Finally, in Sec. VII, we provide a summary of the current job. +II. +FOUNDATION OF f(T, B) THEORY +In this Sec., we intend to explore the modified teleparallel gravity entitled f(T, B) model. The +geometry of the teleparallel theory is raised as the dynamical variables entitled the components of +tetrad field ea(xµ), which are introduced in the form of an orthonormal basis for the tangent space +at each point xµ of the space-time manifold. The corresponding metric in terms of tetrad field is +written as gµν = ηABeA +µeB +ν that ηAB = diag(−1, 1, 1, 1) and det(eAµ) = e = √−g, in which the +Greek letters and the Latin alphabets run over from 0 to 3, and denote space-time components +and tangent space components, respectively. In that case, we write down f(T, B) model as the +following action +S = +� +d4x e +�f(T, B) +κ2 ++ Lm +� +, +(1) +where e and Lm are determinant of tetrad components and lagrangian of matter, respectively, as +well as we have κ2 = 8πG and B = 2∂µ(eT µ)/e, in which T µ as the torsion vector that can be + +5 +defined by Tµ = T ν +νµ. We obtain Einstein field equation by making the variation of the action with +respect to tetrad field as +2eδλ +ν ∇µ∇µ∂Bf − 2e∇λ∇ν∂Bf + eB∂Bfδλ +ν + 4e (∂µ∂Bf + ∂µ∂T f) Sνµλ ++4ea +ν∂µ +� +eSaµλ� +∂T f − 4e∂T fT σµνSσλµ − efδλ +ν = 16πeT λ +ν , +(2) +where T λ +ν = ea +νT λ +a is the energy-momentum tensor of the matter. In the present work, we consider +the flat-Friedmann-Robertson-Walker (FRW) metric as +ds2 = −dt2 + a2(t) +� +dx2 + dy2 + dz2� +, +(3) +where a(t) is the scale factor. +The components of tetrad field are eaµ = diag(1, a, a, a). +The +torsion scalar, the torsion tensor, the asymmetry tensor, and the contorsion tensor are written, +respectively, as +T = SρµνT ρµν, +(4) +T ρµν = eAρ � +∂µeAν − ∂νeAµ +� +, +(5) +Sµν +ρ += 1 +2 +� +Kµν +ρ ++ δµ +ρ T αν +α +− δν +ρT αµ +α +� +, +(6) +Kµν +ρ += − 1 +2 (T µνρ − T νµ +ρ +− T µν +ρ ) . +(7) +Now we can obtain the torsion scalar and the boundary term, respectively, in the following form +T = 6H2, +(8) +B = 6 +� +˙H + 3H2� +, +(9) +where H = ˙a/a is Hubble parameter. It should be noted that standard action for general relativity +is included only with a term of Ricci scalar, R, but in the modified teleparallel action is contained +with terms of torsion scalar and boundary. +These two theories (general relativity theory and +modified teleparallel theory) have a difference only in a boundary term, i.e., R = B − T = 6 ˙H + +12H2. This issue shows us that the f(T, B) gravity is an appropriate alternative for f(R) gravity +[81, 82]. Therefore, We obtain Friedmann equations by inserting Eq. (3) into field equation (2) as +− 3H2 (3∂Bf + 2∂T f) + 3H∂B ˙f − 3 ˙H∂Bf + 1 +2f = κ2T 0 +0 , +(10a) +− +� +3H2 + ˙H +� +(3∂Bf + 2∂T f) − 2H∂T ˙f + ∂B ¨f + 1 +2 f = κ2T i +i , +(10b) +where indices 0 and i indicate components of time and space in time-space geometric. To have a +more realistic model for evolution of the universe, we consider the viscose fluid instead of perfect +fluid. Therefore, the energy–momentum tensor is written in the presence of the bulk viscosity as +T j +i = (ρtot + ptot + pbulk)uiuj − (ptot + pbulk) δj +i , +(11) + +6 +where ρtot, ptot, and pbulk = −3ξH are the total energy density, the total pressure, and the pressure +of bulk viscosity fluid inside the universe, respectively, in which ξ is a positive constant for bulk +viscosity. Also, the 4-velocity is uµ is ui = (+1,0,0,0) and have uiuj = 1. However, elements of +energy–momentum tensor are as +T 0 +0 = ρtot, +(12a) +T i +i = −ptot + 3ξH, i = 1, 2, 3. +(12b) +The total continuity equation in the presence of bulk viscosity yields +˙ρtot + 3H(ρtot + ptot) = 0, +(13) +where ptot = ptot − 3ξH. Now we consider that the contains of the inside universe contain both +matter and torsion, then we have +ρtot = ρm + ρTB, +(14a) +ptot = pm + pTB − 3ξH, +(14b) +where ρTB and pTB are the energy density and the pressure for dark energy, respectively. Therefore, +we can immediately write continuity equations of matter and torsion in the following form +˙ρm + 3H(ρm + pm) = Q, +(15a) +˙ρTB + 3H(ρTB + pTB − 3ξH) = −Q, +(15b) +where Q is an interaction term between the elements of the universe. When energy flow is trans- +ferred from dark matter to dark energy, the interaction term becomes a negative, and when energy +flow is transferred from dark energy to dark matter, the interaction term becomes a positive. From +the perspective of dimensionality, the Q term is equal to the product of the Hubble parameter and +the energy density. Therefore, we take Q = 3b2Hρm in which 0 < b < 1 is the intensity of energy +transfer [11, 18, 83]. +In that case, we write down ρTB and pTB by using of Eqs. (10), (12), and (14) as below +ρTB = +1 +κ2 +� +−3H2 (3∂Bf + 2∂T f) + 3H∂B ˙f − 3 ˙H∂Bf + 1 +2f +� +− ρm, +(16a) +pTB = −1 +κ2 +� +− +� +3H2 + ˙H +� +(3∂Bf + 2∂T f) − 2H∂T ˙f + ∂B ¨f + 1 +2f +� +− pm + 3ξH. +(16b) +The equation of state (EoS) for the dark energy is +ωTB = pTB +ρTB +, +(17) +where the EoS is dependent on the parameters of f(T, B) model and the viscous fluid. + +7 +III. +THE EXTENDED BOSE-EINSTEIN CONDENSATE AS DARK MATTER +The Bose-Einstein condensation, known as the ”fifth state of matter,” is a state of matter that +occurs when boson particles cool to near zero (minus 273.15 degrees Celsius). At such a low tem- +perature, the particles do not have enough energy to enter situations where their distinct quantum +properties overlap. Without the energy difference to separate the particles, they all merge into a +single quantum state to form superparticles that behave like a microwave, unlike ordinary particles. +The speed of light passing through the Bose-Einstein condensate is very slow. According to Bose +and Einstein’s experiments and observations, when atoms approach absolute zero temperature, +the waves expand and eventually overlap, so that their constituent elementary particles all merge +into a single quantum state, which this condition is called Bose-Einstein condensation. Here we +are going to describe one of the components of the universe called dark matter with the help of +Extended Bose-Einstein Condensation (EBEC). In fact, the described model can be similar to the +model of our universe. In order to obtain the EoS of dark matter, we take BEC description from +the generalized Gross-Pitaevskii equation that leads to developing it in cosmology [84, 85]. Thus, +EoS of dark matter is written as +pm = u0 ρ2 +m, +(18) +where u0 = 2πla/m3 +m, in which la is the s-wave scattering length (refer to Ref. [86–88] for more +details). As mentioned above, we study dark matter from the perspective of Bose-Einstein conden- +sation, and also use EBEC regime to better understand the various periods of the universe from +the early time to the late time. Then write the equation of state for dark matter as follows: +pm = αρm + βρ2 +m, +(19) +where α is introduced as single-body interaction, resulting from normal dark matter and β is +introduced as two-body interaction, resulting from dark matter halo. Because α and β plays an +important role in understanding the nature of dark matter in the universe, the values of α and β +indicate the both contribution of the normal dark matter and dark matter halo [80]. +By looking closely at Eqs. (19), we find that this equation is the EoS of the viral expansion, +which in fact helps us to write the pressure of dark matter as a power series in terms of the energy +density of dark matter. +pm = +� +aiρi +m, +(20) + +8 +where the first term, a1 = α, represents normal dark matter, the second term, a2 = β, represents +the dark matter halo in a quantum ground state, and the third term, a3, and higher terms, ai, +are related to the excited quantum system of the dark matter halo. Since we deal with forms +of barotropic and halo for dark matter in this work, so we only take the first two terms of the +virial expansion. In that case, we look forward to the above consequences, which will help us to +understand the early era to the late era of the universe. +In order to obtain the energy density of dark matter, we insert Eq. (19) into (15a) and have +ρm = ρ0 +c η a3η +0 − β +c η a3η − β , +(21) +where ρ0 = η/(c η a3η +0 − β), a0, and c are respectively the present energy density of dark matter, +the current scale factor, and an integral constant, in which η = α + 1 − b2. It should be noted +that this relationship shows that it depends on the scale factor, interacting term, and coefficients +of EBEC dark matter. +IV. +RECONSTRUCTION OF f(T, B) GRAVITY, AND THE HUBBLE DATA +CONSTRAINTS +In this section, we intend to reconstruct the corresponding Friedman equation according to the +redshift parameter so that we can solve the corresponding Friedman equations in the presence of +EBEC for f(T, B) gravity. For this purpose, we write the relationship between redshift parameter, +z, and the present scale factor, a0, as 1 + z = a0/a. Also, we can clearly write the relationship +between the time derivative and the redshift derivative in the form of d/dt = −H(1 + z) d/dz. In +order to solve the present system for interacting f(T, B) gravity in presence of EBEC, we introduce +the below relation +H2 = H2 +0E(z), +(22) +where E(z) is the parametrization function that is functional of the redshift parameter, and the +late time Hubble parameter is equal to H0 = 67.4 ± 0.5 km s−1 Mpc−1 [89]. In that case, torsion +scalar, T, and boundary term, B, are written in terms of redshift parameter as +T = 6H2 +0E(z), +(23) +B = 3H2 +0(1 + z)E′(z) − 18H2 +0E(z), +(24) +where the prime index displays the derivative with respect to redshift parameter. +The energy +density and the pressure of dark energy in Eqs. (16) are rewritten in terms of redshift parameter + +9 +as follows: +ρTB = +1 +κ2 +� +− 3H2 +0E (3∂Bf + 2∂T f) − 3H2 +0(1 + z)E∂Bf ′ + 3 +2H2 +0(1 + z)E′ ∂Bf ++ 1 +2f +� +− ρm, +(25a) +pTB = +1 +κ2 +� +H2 +0 +� +3E − 1 +2 (1 + z) E′� +(3∂Bf + 2∂T f) − 2H2 +0(1 + z)E ∂T f ′ − 1 +2H2 +0(1 + z)2E′ ∂Bf ′ +− 1 +2f − H2 +0(1 + z)E ∂Bf ′ − H2 +0(1 + z)2E ∂Bf ′′� +− αρm − βρ2 +m + 3ξH0 +√ +E. +(25b) +In order to solve the corresponding system, we use a model that is commonly used for simplicity +and ease of calculations. For this purpose, we consider a power-law model for the scale factor in +terms of cosmic time as follows: +a(t) = a0 +� t +t0 +�s +, +(26) +where a0 is the current scale factor, and s is a dimensionless positive coefficient as a correction +factor [90]. The Hubble parameter easily yields +H = s +t , +(27) +where in the late time we will have +t0 = s +H0 +, +(28) +where t0 is the age of the universe1. Also, we earn the Hubble parameter in terms of redshift +parameter as +H2(z) = H2 +0 (1 + z) +2 +s , +(29) +where +E(z) = (1 + z) +2 +s . +(30) +Note that Eq. (29) has only one free parameter, s, which can easily be examined with obser- +vational data. To demonstrate the accuracy of our model, we fit it with 53 supernova data for +Hubble parameter that shown in Tab. I, data sets collected from Refs. [91–109]. The estimate is +in the range of 0.07 ≤ z ≤ 2.36, in which the corresponding data sets are measured by techniques +of galaxy differential age or cosmic chronometer and radial BAO size methods. Since the present +work related to statistics estimation, then we take maximum likelihood analysis with chi-squared +1 Note that the age of the universe is t0 = +1 +H0 +� ∞ +0 +dz +(1+z) +√ +E which is the same as the result of Eq. (28) + +10 +value, χ2 +min, which is introduced as a very essential instrument for fitting of the cosmological data. +The chi-squared is +χ2 +min = +53 +� +i=1 +(Hobs(zi) − Hth(zi, H0))2 +σ2 +H(zi) +, +(31) +where Hobs and Hth display the observed value and the theoretical value of the Hubble parameter +data, respectively, and σH represents the standard error or uncertainty values in the observed +values. +After examining with the Hubble parameter data set and minimizing χ2 +min, we obtain +s = 0.95 by the best fitting. Also, we can earn the age of the universe since the Big Bang as +t0 = 13.78Gyr from Eq. (28). Therefore, the obtained age of the universe is compatible with +Λ-CDM model [89], which indicates the validity of the present model. +In the next section, we will investigate the obtained results from the Hubble parameter data +set constraints in the presence of EBEC for f(T, B) gravity. +V. +THE INTERACTION BETWEEN VISCOUS f(T, B) GRAVITY AND EBEC +In this section, we intend to explore the interaction between EBEC and f(T, B) gravity in the +presence of bulk viscosity. Therefore, in order to compute the present model, the form of function +f(T, B) plays a very essential role. To do this, we have to opt a viable cosmic model for f(T, B) +gravity that can provide information about the accelerating universe in the late era. One of these +options is mixed power-law model, which is written for the function f(T, B) as follows: +f(T, B) = λBmT n, +(32) +where λ, m, and n are constant coefficients. According to this proposal, the contribution of torsion +term or boundary term may occur separately or together. +Now by replacing Eqs. (21), (30), and (32) into Eqs. (25) we obtain the energy density and the +pressure of dark energy versus the redshift parameter in the following form +ρTB = 6m+nλδH2(m+n) +0 +(3s−1)m−1 +2κ2sm +(1 + z) +2 +s (m+n) − +η +cη( +a0 +1+z) +3η−β, +(33a) +pTB = 6m+n−1λδ(2m+2n−3s)H2(m+n) +0 +(3s−1)m−1 +κ2sm+1 +(1 + z) +2 +s (m+n) − +αη +cη( +a0 +1+z) +3η−β +− +βη2 +� +cη( +a0 +1+z) +3η−β +�2 + 3ξH0(1 + z) +1 +s , +(33b) +where δ = 3m + 2n + 3s − 2m2 − 2mn − 3ms − 6ns − 1. +In order to get an acceptable answer from the above calculations, at least one set of free +parameter values should be selected so that the results represent an expanding universe. This + +11 +TABLE I. The Hubble parameter data set in terms of redshift parameter and their uncertainty values, units +of H(z) and σH are km s−1 Mpc−1 +No. Redshift H(z) σH +Ref. No. Redshift H(z) +σH +Ref. +1. +0.07 +69.0 19.6 [91] +28. +0.510 +90.4 +1.9 +[99] +2. +0.09 +69.0 12.0 [92] +29. +0.52 +94.35 2.64 +[97] +3. +0.12 +68.6 26.2 [91] +30. +0.56 +93.34 +2.3 +[97] +4. +0.17 +83 +8 +[93] +31. +0.57 +92.9 7.855 [102] +5. +0.179 +75.0 +4.0 +[94] +32. +0.59 +98.48 3.18 +[97] +6. +0.199 +75.0 +5.0 +[94] +33. +0.593 +104.0 13.0 +[94] +7. +0.200 +72.9 29.6 [91] +34. +0.6 +87.9 +6.1 +[103] +8. +0.24 +79.69 3.32 [95] +35. +0.610 +97.3 +2.1 +[99] +9. +0.27 +77 +14 +[93] +36. +0.64 +98.02 2.98 +[97] +10. +0.280 +88.8 36.6 [91] +37. +0.680 +92.0 +8.0 +[94] +11. +0.30 +81.7 +5.0 +[96] +38. +0.73 +97.3 +7 +[103] +12. +0.31 +78.18 4.74 [97] +39. +0.781 +105.0 +12 +[94] +13. +0.34 +83.8 2.96 [95] +40. +0.875 +125 +17 +[94] +14. +0.35 +82.7 +9.1 +[98] +41. +0.880 +90.0 +40.0 [104] +15. +0.352 +83 +14 +[94] +42. +0.900 +117 +23.0 +[93] +16. +0.36 +79.94 3.38 [97] +43. +1.037 +154 +20 +[94] +17. +0.38 +81.5 +1.9 +[99] +44. +1.300 +168 +17 +[93] +18. +0.3802 +83.0 13.5 [100] 45. +1.363 +160 +33.6 [105] +19. +0.40 +82.04 2.03 [97] +46. +1.430 +177 +18 +[93] +20. +0.4004 +77 +10.2 [100] 47. +1.530 +140 +14 +[93] +21. +0.4247 +87.1 11.2 [100] 48. +1.750 +202 +40 +[93] +22. +0.43 +86.45 3.27 [95] +49. +1.965 +186.5 50.4 [105] +23. +0.44 +84.81 1.83 [97] +50. +2.30 +224 +8.6 +[106] +24. +0.4497 +92.8 12.9 [100] 51. +2.33 +224 +8 +[107] +25. +0.470 +89 +34 [101] 52. +2.340 +222 +7 +[108] +26. +0.4783 +80.9 +9.0 [100] 53. +2.360 +226 +8 +[109] +27. +0.48 +87.79 2.03 [97] +means that the free parameters play a very important role in this model. Therefore, our choice +is motivated by the fact that the energy density and pressure of dark energy become positive and +negative, respectively, and also lead to the crossing of the divided-phantom line in the variations +of EoS versus the redshift parameter. Therefore, a series of selected values are: λ = 75, c = 0.001, + +12 +b = 0.1, m = 1.75, n = −1.5, α = 3, β = 0.0012, a0 = 0.75, and ξ = 0.1, 5. Note that the free +parameters λ, c, b, m, n, and a0 are dimensionless, and ξ has units pa.s in SI system or M3 in +Planck systems. +Now we plot the variation of ρTB and pTB versus to redshift parameter as shown in Figs. 1. +Therefore, the variation of ρTB depends on the coefficients of F(T, B) gravity, the interacting +model, and EBEC dark matter, but, the variation of pTB is also dependent on bulk viscosity in +addition to the mentioned ones. Also, it is evident that the value of ρTB is more than zero, and +the value of pTB is less than zero for the present universe (z = 0). It should be noted that ρTB +and pTB have the same unit ML−1T −2 in SI system and or units M4 in Planck system. +FIG. 1. The energy density and the pressure of dark energy in terms of redshift parameter. +On the other hand, by inserting Eqs. (33) into Eq. (17), we can calculate EoS of dark energy, +ωTB, as +ωTB = +(2m+2n−3s) +3s +A(1 + z) +2 +s (m+n) − αρm − βρ2 +m + 3ξH0(1 + z) +1 +s +A(1 + z) +2 +s (m+n) − αρm +, +(34) +where A = 6m+nλδH2(m+n) +0 +(3s − 1)m−1/(2κ2sm). The variation of EoS of dark energy is plotted +versus redshift parameter as shown in Fig. 2. +As we know, EoS parameter plays an important role in modern cosmology which is a dimension- +less quantity. So that, this parameter is a very good description for different eras of the formation of +the universe, i.e, we will have eras of accelerated phase (quintessence phase), cosmological constant, +and phantom phase for −1 < ω < − 1 +3, ω = −1 and ω < −1, respectively. However the correspond- +ing graph 2 represents us that the formation of the universe enters the era of the accelerated phase + +13 +FIG. 2. EoS parameter of dark energy in terms of redshift parameter. +and crosses the divided-phantom line. Therefore, the present values of EoS for ξ = 0.1 and ξ = 5 +are −1.075 and −0.97, respectively. In that case, we can observe that the universe moves between +regions quintessence and phantom when the bulk viscosity coefficient varies. It is worth noting +that the values of the dark energy EoS parameter versus the redshift parameter vary as shown in +Fig. 2. Ref. [89] is a report for Planck 2018 results measuring the dark energy EoS parameter as +−1.03 ± 0.03 with Type Ia supernovae (SNe), and earlier in 2010, Ref. [110] provided the best fit +for the dark energy EoS parameter to Union2 data in flat-universe as −0.997+0.050 +−0.054 for statistical +uncertainty and −0.997+0.077 +−0.082 for statistical and systematic uncertainties. Also, Ref. [111] mea- +sured the dark energy EoS parameter as −1.026 ± 0.041 with the Pantheon Sample. According to +the above evidence, the universe is undergoing accelerated expansion, which is consistent with the +results obtained in Ref. [89, 110, 111] and even consistent with a cosmological constant. +VI. +STABILITY AND INSTABILITY ANALYSIS +In this section, we intend to explore the stability and the instability of the our model from a +thermodynamic point of view based on the sound speed parameter, c2 +s. From a thermodynamic +perspective, the universe is in an adiabatic system, in such a way that no energy or mass is +exchanged from the universe to the outside. In that case, the entropy perturbation is zero, then +we have +δpTB(S, ρTB) = ∂pTB +∂S δS + ∂pTB +∂ρTB +δρTB = c2 +sδρTB, +(35) + +14 +FIG. 3. The sound speed parameter of dark energy in terms of redshift parameter. +where c2 +s = ∂zpTB/∂zρTB, and symbol ∂z represents derivative with respect to redshift parameter. +Now the values of the sound speed parameter play a key role in describing the stability and +instability of the system. Therefore, the corresponding parameter has the conditions c2 +s > 0 and +c2 +s < 0 which respectively declare the stability and instability of the model. We draw the variation +of the sound speed parameter in terms of redshift parameter as shown in Fig. 3. We can see from +Fig. 3 for ξ = 0.1 that the formation of the universe begins instability phase and continues to the +stability phase at the late time. Also in ξ = 5, we see that the formation of the universe begins from +a stability phase and then at the threshold of the accelerated phase enters an instability phase, and +again return to stability phase in late time. Therefore, with the existence of viscous fluid gives rise +to the different stability analyses for the formation of the universe. At the end, it can be said that +the phases of stability and instability indicate that the energy density is or is not in a controllable +growth, respectively. +Now, on the other hand, we are looking for more complete explanations for this model. For this +purpose, we estimate the density parameter of dark energy, ΩTB = ρTB/ρc, where ρc = 3H2/κ2 +is the critical density. By substituting (33a) into ΩTB, we can plot the variation of the density +parameter in terms of redshift parameter as shown in Fig. 4. We can see from Fig. 4, that the +amount of ΩTB increases from the early universe to the late universe, in which the present amount +of the density parameter of dark energy is 0.69. According to the Planck 2018 results report [89], +the value of the dark energy density parameter with Type Ia supernovae (SNe) has been obtained +as 0.6847 ± 0.0073, so that it is compatible with our results. As a result, the contribution of dark + +15 +FIG. 4. The density parameter of dark energy in terms of redshift parameter. +energy is increasing, which indeed indicates that the fate of the universe will continue to expand. +VII. +CONCLUSION +In this paper, we studied the dark parts of the universe by f(T, B) gravity and EBEC. First +we explored the modified teleparallel gravity entitled f(T, B) model in a viscous fluid by the flat- +FRW universe. Then, by writing the Einstein equation, we earned the Friedmann equations in the +presence of bulk viscosity. Also, we considered an interaction between the contents of the universe, +i.e., dark matter and dark energy with term of Q = 3b2Hρm for a more realistic universe. Next, +the continuity equations, and EoS parameter of dark energy have been written in terms of matter +and torsion. The interesting point of f(T, B) gravity model is that boundary term, B relates to +the Ricci scalar B = R+T, i.e., f(T, B) gravity can covers the gravity models of f(T) based on the +Weitzenb¨ock connection and the f(R) based on the Levi-Civita connection. In fact, the f(T, B) +model covers both models simultaneously. +On the other hand, we studied EBEC as an appropriate alternative instead of the dark matter. +In fact, we describe BEC from the generalized Gross-Pitaeveskii equation and use EBEC regime to +write the EoS of dark matter. The interesting point of EBEC regime is that the EoS of dark matter +is adapted to the EoS of the virial expansion, which is written in terms of normal dark matter and +dark matter halo. Therefore, the elegance of the EBEC regime gives a better understanding of the +different epochs of the universe from the early time to late time. + +16 +In what follows, in order to examine the present model with the astronomical data, we took +the power-law for the scale factor and then wrote the Hubble parameter in terms of the redshift +parameter. Next, the corresponding Hubble parameter was fitted with 53 supernova data, so-called +the Hubble data constraints, and as a result, the age of the universe as t0 = 13.75 Gyr was found. +After that, we opted a viable cosmic model for f(T, B) gravity as mixed power-law model that +presented information on the late universe. In addition, we reconstructed the energy density, the +pressure, and the EoS of dark energy in terms of redshift parameter and some free parameters that +come from f(T, B) gravity, viscous fluid, interacting model, and EBEC regime. Note that choice +of free parameters is very sensitive to prove that the universe has an expanding acceleration. In +that case, we plotted ρTB, pTB, and ωTB in terms of redshift parameter for some acceptable free +parameters. So that, the present amounts of the EoS parameter are equal to −1.075 and −0.97 for +ξ = 0.1 and ξ = 5, respectively. Therefore, we concluded that the obtained results are completely +consistent with the observational data. +For a more thorough review, we analyzed the stability and instability of the present model by +the sound speed parameter. In that case, we drew the variation of the sound speed parameter +in terms of the redshift parameter with different values of bulk viscosity coefficient. The results +showed us that the stability conditions passed through a series of instability conditions and reached +a stability condition at the late time. This means that the energy density of dark energy is in a +controlled growth. Finally, in order to make the results of the present study consistent with the +observational data, we calculated the value of the dark energy density parameter with a value of +0.69, as shown in Fig. 4. As a further work, we propose the development of the present work for +axion models. +VIII. +ACKNOWLEDGMENT +We thank Prof. Iver H. Brevik for his useful comments and suggestions to improve our work. +[1] A. G. Riess, A. V. Filippenko, P. Challis, A. Clocchiatti, A. Diercks, P. M. Garnavich, R. L. Gilliland, +and et al, The Astronomical Journal 116, no. 3 (1998): 1009. +[2] S. Perlmutter, G. Aldering, G. Goldhaber, R. A. Knop, P. Nugent, P. G. Castro, S. Deustua, and et +al, The Astrophysical Journal 517, no. 2 (1999): 565. + +17 +[3] C. L. Bennett, M. Halpern, G. Hinshaw, N. Jarosik, A. Kogut, M. Limon, S. S. Meyer et al, The +Astrophysical Journal Supplement Series 148, no. 1 (2003): 1. +[4] M. Tegmark, M. A. Strauss, M. R. Blanton, K. Abazajian, S. Dodelson, H. Sandvik, X. 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Huber et al, The +Astrophysical Journal 859, no. 2 (2018): 101. + diff --git a/V9E4T4oBgHgl3EQfMwxD/content/tmp_files/load_file.txt b/V9E4T4oBgHgl3EQfMwxD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b92bbd9d367e29ad1bff6c012618906fe9b8140d --- /dev/null +++ b/V9E4T4oBgHgl3EQfMwxD/content/tmp_files/load_file.txt @@ -0,0 +1,1109 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf,len=1108 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='04950v1 [gr-qc] 12 Jan 2023 The interaction of extended Bose-Einstein condensate dark matter with viscous f(T, B) gravity E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Mahichi1, ∗ and Alireza Amani 1, † 1Department of Physics, Ayatollah Amoli Branch, Islamic Azad University, Amol, Iran (Dated: January 13, 2023) In this paper, we study the viscous f(T, B) gravity model as a source of dark energy, and the Extended Bose-Einstein Condensate (EBEC) as a source of dark matter, in a flat- FRW metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' In the presence of bulk viscosity, we obtain Friedmann equations and write two continuity equations of dark energy and dark matter by interacting them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Using the generalized Gross-Pitaeveskii equation, we earn Equation of State (EoS) of dark matter by EBEC regime as pm = αρm + βρ2 m in which the both of terms are respectively introduced as normal dark matter and dark matter halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' The innovation of the work is that we can simultaneously describe the nature of the dark parts of the universe with the viscous f(T, B) gravity and the EBEC regime, which leads to a deep understanding of the different epochs of the universe from early to late times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' In what follows, the energy density and the pressure of dark energy are reconstructed in terms of the redshift parameter, and then we fit the obtained results with 53 supernova data from the Hubble data constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Next, we plot the cosmological parameters in terms of the redshift parameter and conclude that the current universe is in an accelerated phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Finally, we analyze the stability and instability of the current model with the sound speed parameter as well as we draw the density parameter values for dark energy in terms of the redshift parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' PACS numbers: 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='-k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='Es;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' 04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='Kd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='+d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' 03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='Nt Keywords: Dark energy, Dark matter, Extended Bose-Einstein condensate, The viscous f(T, B) gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' ∗ elhammahichi@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='com † Corresponding Author: al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='amani@iau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='ir 2 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' INTRODUCTION One of the greatest concerns of human beings is to know about the universe, which requires knowing the contents of the universe, such as dark energy, dark matter, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' From the very beginning, many efforts were made to this end, including Friedmann and his famous equations, although Hubble’s contribution to this issue is also important because Hubble-Lumiter’s law con- firmed that the universe is expanding at a constant speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' This velocity is proportional to the distance of the galaxies from the Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Because of the importance of knowing the universe, in re- cent decades, by measuring cosmic distances and observing cosmic objects beyond the Milky Way, they have found that the universe is expanding rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Therefore, the expansion of the universe at a non-constant speed was discovered and confirmed by Ia-type supernovae, cosmic microwave background radiation, and large-scale structures in the universe [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' The cause of this event is a mysterious and unknown force that is greater than the force of gravity but in the opposite direction, which is called dark energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Since the universe is considered a thermodynamic system, and its evolution depends only on the internal contents of the universe, therefore, due to the lack of matter-energy transfer from outside to inside, must be attributed to a strong negative pressure that acts as a repulsion that leads to the expansion of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' With these explanations, the problem of dark energy can be interpreted in the fundamental theoretical framework of string the- ory and quantum gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Extensive studies on dark energy with models such as the cosmological constant [5, 6], scalar fields [7–22], modified gravity [23–31], holography and agegraphic [32–39], bouncing theory [40–44], teleparallel gravity [45–49] and braneworld models [50–52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Among these mentioned models, teleparallel gravity is considered to be one of the suitable candidates for dark energy, which was first introduced by Einstein for the union between elec- tromagnetism and gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' This theory states that space-time is associated with a curvature-free linear relationship by a metric tensor field called tetrad-vector field dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' In that case, a tetrad field is naturally used to define a linear Weitzenb¨ock connection, which represents a tor- sion connection without curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' But Levi-Civita connection is used in the geometry of general relativity to show the curvature connection without torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Therefore, by converting the tetrad components and metric components to each other, we can convert teleparallel gravity model to the general relativity model and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' This means, the curvature term in general relativity changes to a torsion term in the teleparallel model and vice versa, which confirms the claim that teleparallel gravity and general relativity are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' This equivalent is caused the relation R = −T + 2∇µT µ in which R, T, and T µ are curvature scalar, torsion scalar, and torsion vector, 3 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Nevertheless, the corresponding field equations are clearly covariant and then the theory has a local Lorentz invariant [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' By converting T to f(T), we expand teleparallel gravity to modified teleparallel gravity, where f(T) is the function of the torsion scalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Also, by introduc- ing B = 2∇µT µ, we can redevelop the modified teleparallel gravity in the form f(T, B) gravity in which B is the boundary term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Thus, model f(T, B) gravity can simultaneously covers both the models of f(T) gravity and f(R) gravity with help of relation R = −T + B, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=', it provides an equivalence relation between torsion and curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' [54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' To study the universe more realistically, we consider it as an anisotropic fluid with viscosity, because the bulk viscosity creates internal friction that converts the kinetic energy of particles into heat, an idea that could help to describe the recent acceleration of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Therefore, bulk viscosity play a critical role in determining the dissipative effect of fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' [56–60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Since in physics it is believed that the nature of everything unknown to us is called dark or black, so the unknown parts of the universe are called dark energy and dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Hence, we turn our attention to another dark part of the universe, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=', dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' The nature of dark matter can be baryonic, this means, astronomical objects such as massive and compact haloes are composed of ordinary matter, but their electromagnetic radiation is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Also, dark matter can be non-baryonic, that is, it is composed of hypothetical and real particles, in which axions and Weakly Interacting Massive Particles (WIMPs) are hypothetical particles, and Bose-Einstein condensate (BEC) is a state of matter created when particles called bosons are cooled to near absolute zero [61–79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Hence dark matter can be described as a non-relativistic, Newtonian Bose-Einstein gravitational condensate gas, whose density and pressure are related by a barotropic equation of state [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' For this purpose, we intend to relate dark matter and Bose- Einstein condensate (BEC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' As we know, Bose-Einstein statistics are a concept that formed the basis of mathematics and enabled us to distinguish particular particles from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' BEC is a substance in which dilute boson gas is cooled to very low temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Due to this very low temperature, phase transfer occurs and most of the boson gases occupy the lowest quantum state, and the macroscopic quantum phenomenon appears at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Cold bosons fall on each other, and superparticles appear that behave like microwaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' This means that the dark matter is assumed to be a bosonic gas below the critical temperature that forms BEC [74–79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' By using the generalized Gross-Pitaevskii equation, the EoS of dark matter as a barotropic fluid is obtained, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=', matter pressure, pm, is only a function of matter energy density, ρm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Note that the corresponding EoS has been introduced as a normal dark matter [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Also, in the same of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' [74], by considering the dark matter halo in a quantum ground state, the EoS of dark matter was 4 obtained as pm ∝ ρ2 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' After that, we extended the BEC model to the Extended Bose-Einstein Condensation (EBEC) model as cited in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' This means that dark matter is simultaneously considered in both cases of the normal dark matter and dark matter halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' In fact, EBEC regime helps us to understand more accurately the various epochs of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' In general, dark matter and dark energy have the nature of attraction and repulsion, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' This means that the attractive force in dark matter pulls matter inside, but the repulsive force of dark energy pushes it outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Therefore, in this study, we attribute the viscous f(T, B) gravity model to dark energy and the EBEC model to dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' This assessment motivates us to investigate the nature of the largest cosmic scales by viscous f(T, B) gravity, and to study the nature of individual galaxies by EBEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' This paper is organized as follows: In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' II, we present a foundation of f(T, B) gravity in the flat-FRW metric by bulk viscosity fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' III, we will explore EBEC as dark matter, and write the dark matter EoS as the virial expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' IV, we reconstruct the f(T, B) gravity in terms of the redshift parameter, and fit the corresponding system by the Hubble data constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' V, we obtain the relevant cosmological parameters in terms of the redshift parameter and then plot the corresponding graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' VI, we analyse the stability and instability of the present model by the sound speed parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Finally, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' VII, we provide a summary of the current job.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' FOUNDATION OF f(T, B) THEORY In this Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=', we intend to explore the modified teleparallel gravity entitled f(T, B) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' The geometry of the teleparallel theory is raised as the dynamical variables entitled the components of tetrad field ea(xµ), which are introduced in the form of an orthonormal basis for the tangent space at each point xµ of the space-time manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' The corresponding metric in terms of tetrad field is written as gµν = ηABeA µeB ν that ηAB = diag(−1, 1, 1, 1) and det(eAµ) = e = √−g, in which the Greek letters and the Latin alphabets run over from 0 to 3, and denote space-time components and tangent space components, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' In that case, we write down f(T, B) model as the following action S = � d4x e �f(T, B) κ2 + Lm � , (1) where e and Lm are determinant of tetrad components and lagrangian of matter, respectively, as well as we have κ2 = 8πG and B = 2∂µ(eT µ)/e, in which T µ as the torsion vector that can be 5 defined by Tµ = T ν νµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' We obtain Einstein field equation by making the variation of the action with respect to tetrad field as 2eδλ ν ∇µ∇µ∂Bf − 2e∇λ∇ν∂Bf + eB∂Bfδλ ν + 4e (∂µ∂Bf + ∂µ∂T f) Sνµλ +4ea ν∂µ � eSaµλ� ∂T f − 4e∂T fT σµνSσλµ − efδλ ν = 16πeT λ ν , (2) where T λ ν = ea νT λ a is the energy-momentum tensor of the matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' In the present work, we consider the flat-Friedmann-Robertson-Walker (FRW) metric as ds2 = −dt2 + a2(t) � dx2 + dy2 + dz2� , (3) where a(t) is the scale factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' The components of tetrad field are eaµ = diag(1, a, a, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' The torsion scalar, the torsion tensor, the asymmetry tensor, and the contorsion tensor are written, respectively, as T = SρµνT ρµν, (4) T ρµν = eAρ � ∂µeAν − ∂νeAµ � , (5) Sµν ρ = 1 2 � Kµν ρ + δµ ρ T αν α − δν ρT αµ α � , (6) Kµν ρ = − 1 2 (T µνρ − T νµ ρ − T µν ρ ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' (7) Now we can obtain the torsion scalar and the boundary term, respectively, in the following form T = 6H2, (8) B = 6 � ˙H + 3H2� , (9) where H = ˙a/a is Hubble parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' It should be noted that standard action for general relativity is included only with a term of Ricci scalar, R, but in the modified teleparallel action is contained with terms of torsion scalar and boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' These two theories (general relativity theory and modified teleparallel theory) have a difference only in a boundary term, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=', R = B − T = 6 ˙H + 12H2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' This issue shows us that the f(T, B) gravity is an appropriate alternative for f(R) gravity [81, 82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Therefore, We obtain Friedmann equations by inserting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' (3) into field equation (2) as − 3H2 (3∂Bf + 2∂T f) + 3H∂B ˙f − 3 ˙H∂Bf + 1 2f = κ2T 0 0 , (10a) − � 3H2 + ˙H � (3∂Bf + 2∂T f) − 2H∂T ˙f + ∂B ¨f + 1 2 f = κ2T i i , (10b) where indices 0 and i indicate components of time and space in time-space geometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' To have a more realistic model for evolution of the universe, we consider the viscose fluid instead of perfect fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Therefore, the energy–momentum tensor is written in the presence of the bulk viscosity as T j i = (ρtot + ptot + pbulk)uiuj − (ptot + pbulk) δj i , (11) 6 where ρtot, ptot, and pbulk = −3ξH are the total energy density, the total pressure, and the pressure of bulk viscosity fluid inside the universe, respectively, in which ξ is a positive constant for bulk viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Also, the 4-velocity is uµ is ui = (+1,0,0,0) and have uiuj = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' However, elements of energy–momentum tensor are as T 0 0 = ρtot, (12a) T i i = −ptot + 3ξH, i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' (12b) The total continuity equation in the presence of bulk viscosity yields ˙ρtot + 3H(ρtot + ptot) = 0, (13) where ptot = ptot − 3ξH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Now we consider that the contains of the inside universe contain both matter and torsion, then we have ρtot = ρm + ρTB, (14a) ptot = pm + pTB − 3ξH, (14b) where ρTB and pTB are the energy density and the pressure for dark energy, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Therefore, we can immediately write continuity equations of matter and torsion in the following form ˙ρm + 3H(ρm + pm) = Q, (15a) ˙ρTB + 3H(ρTB + pTB − 3ξH) = −Q, (15b) where Q is an interaction term between the elements of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' When energy flow is trans- ferred from dark matter to dark energy, the interaction term becomes a negative, and when energy flow is transferred from dark energy to dark matter, the interaction term becomes a positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' From the perspective of dimensionality, the Q term is equal to the product of the Hubble parameter and the energy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Therefore, we take Q = 3b2Hρm in which 0 < b < 1 is the intensity of energy transfer [11, 18, 83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' In that case, we write down ρTB and pTB by using of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' (10), (12), and (14) as below ρTB = 1 κ2 � −3H2 (3∂Bf + 2∂T f) + 3H∂B ˙f − 3 ˙H∂Bf + 1 2f � − ρm, (16a) pTB = −1 κ2 � − � 3H2 + ˙H � (3∂Bf + 2∂T f) − 2H∂T ˙f + ∂B ¨f + 1 2f � − pm + 3ξH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' (16b) The equation of state (EoS) for the dark energy is ωTB = pTB ρTB , (17) where the EoS is dependent on the parameters of f(T, B) model and the viscous fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' 7 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' THE EXTENDED BOSE-EINSTEIN CONDENSATE AS DARK MATTER The Bose-Einstein condensation, known as the ”fifth state of matter,” is a state of matter that occurs when boson particles cool to near zero (minus 273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='15 degrees Celsius).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' At such a low tem- perature, the particles do not have enough energy to enter situations where their distinct quantum properties overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Without the energy difference to separate the particles, they all merge into a single quantum state to form superparticles that behave like a microwave, unlike ordinary particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' The speed of light passing through the Bose-Einstein condensate is very slow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' According to Bose and Einstein’s experiments and observations, when atoms approach absolute zero temperature, the waves expand and eventually overlap, so that their constituent elementary particles all merge into a single quantum state, which this condition is called Bose-Einstein condensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Here we are going to describe one of the components of the universe called dark matter with the help of Extended Bose-Einstein Condensation (EBEC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' In fact, the described model can be similar to the model of our universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' In order to obtain the EoS of dark matter, we take BEC description from the generalized Gross-Pitaevskii equation that leads to developing it in cosmology [84, 85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Thus, EoS of dark matter is written as pm = u0 ρ2 m, (18) where u0 = 2πla/m3 m, in which la is the s-wave scattering length (refer to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' [86–88] for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' As mentioned above, we study dark matter from the perspective of Bose-Einstein conden- sation, and also use EBEC regime to better understand the various periods of the universe from the early time to the late time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Then write the equation of state for dark matter as follows: pm = αρm + βρ2 m, (19) where α is introduced as single-body interaction, resulting from normal dark matter and β is introduced as two-body interaction, resulting from dark matter halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Because α and β plays an important role in understanding the nature of dark matter in the universe, the values of α and β indicate the both contribution of the normal dark matter and dark matter halo [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' By looking closely at Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' (19), we find that this equation is the EoS of the viral expansion, which in fact helps us to write the pressure of dark matter as a power series in terms of the energy density of dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' pm = � aiρi m, (20) 8 where the first term, a1 = α, represents normal dark matter, the second term, a2 = β, represents the dark matter halo in a quantum ground state, and the third term, a3, and higher terms, ai, are related to the excited quantum system of the dark matter halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Since we deal with forms of barotropic and halo for dark matter in this work, so we only take the first two terms of the virial expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' In that case, we look forward to the above consequences, which will help us to understand the early era to the late era of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' In order to obtain the energy density of dark matter, we insert Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' (19) into (15a) and have ρm = ρ0 c η a3η 0 − β c η a3η − β , (21) where ρ0 = η/(c η a3η 0 − β), a0, and c are respectively the present energy density of dark matter, the current scale factor, and an integral constant, in which η = α + 1 − b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' It should be noted that this relationship shows that it depends on the scale factor, interacting term, and coefficients of EBEC dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' RECONSTRUCTION OF f(T, B) GRAVITY, AND THE HUBBLE DATA CONSTRAINTS In this section, we intend to reconstruct the corresponding Friedman equation according to the redshift parameter so that we can solve the corresponding Friedman equations in the presence of EBEC for f(T, B) gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' For this purpose, we write the relationship between redshift parameter, z, and the present scale factor, a0, as 1 + z = a0/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Also, we can clearly write the relationship between the time derivative and the redshift derivative in the form of d/dt = −H(1 + z) d/dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' In order to solve the present system for interacting f(T, B) gravity in presence of EBEC, we introduce the below relation H2 = H2 0E(z), (22) where E(z) is the parametrization function that is functional of the redshift parameter, and the late time Hubble parameter is equal to H0 = 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='5 km s−1 Mpc−1 [89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' In that case, torsion scalar, T, and boundary term, B, are written in terms of redshift parameter as T = 6H2 0E(z), (23) B = 3H2 0(1 + z)E′(z) − 18H2 0E(z), (24) where the prime index displays the derivative with respect to redshift parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' The energy density and the pressure of dark energy in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' (16) are rewritten in terms of redshift parameter 9 as follows: ρTB = 1 κ2 � − 3H2 0E (3∂Bf + 2∂T f) − 3H2 0(1 + z)E∂Bf ′ + 3 2H2 0(1 + z)E′ ∂Bf + 1 2f � − ρm, (25a) pTB = 1 κ2 � H2 0 � 3E − 1 2 (1 + z) E′� (3∂Bf + 2∂T f) − 2H2 0(1 + z)E ∂T f ′ − 1 2H2 0(1 + z)2E′ ∂Bf ′ − 1 2f − H2 0(1 + z)E ∂Bf ′ − H2 0(1 + z)2E ∂Bf ′′� − αρm − βρ2 m + 3ξH0 √ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' (25b) In order to solve the corresponding system, we use a model that is commonly used for simplicity and ease of calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' For this purpose, we consider a power-law model for the scale factor in terms of cosmic time as follows: a(t) = a0 � t t0 �s , (26) where a0 is the current scale factor, and s is a dimensionless positive coefficient as a correction factor [90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' The Hubble parameter easily yields H = s t , (27) where in the late time we will have t0 = s H0 , (28) where t0 is the age of the universe1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Also, we earn the Hubble parameter in terms of redshift parameter as H2(z) = H2 0 (1 + z) 2 s , (29) where E(z) = (1 + z) 2 s .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' (30) Note that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' (29) has only one free parameter, s, which can easily be examined with obser- vational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' To demonstrate the accuracy of our model, we fit it with 53 supernova data for Hubble parameter that shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' I, data sets collected from Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' [91–109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' The estimate is in the range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='07 ≤ z ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='36, in which the corresponding data sets are measured by techniques of galaxy differential age or cosmic chronometer and radial BAO size methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Since the present work related to statistics estimation, then we take maximum likelihood analysis with chi-squared 1 Note that the age of the universe is t0 = 1 H0 � ∞ 0 dz (1+z) √ E which is the same as the result of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' (28) 10 value, χ2 min, which is introduced as a very essential instrument for fitting of the cosmological data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' The chi-squared is χ2 min = 53 � i=1 (Hobs(zi) − Hth(zi, H0))2 σ2 H(zi) , (31) where Hobs and Hth display the observed value and the theoretical value of the Hubble parameter data, respectively, and σH represents the standard error or uncertainty values in the observed values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' After examining with the Hubble parameter data set and minimizing χ2 min, we obtain s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='95 by the best fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Also, we can earn the age of the universe since the Big Bang as t0 = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='78Gyr from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Therefore, the obtained age of the universe is compatible with Λ-CDM model [89], which indicates the validity of the present model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' In the next section, we will investigate the obtained results from the Hubble parameter data set constraints in the presence of EBEC for f(T, B) gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' THE INTERACTION BETWEEN VISCOUS f(T, B) GRAVITY AND EBEC In this section, we intend to explore the interaction between EBEC and f(T, B) gravity in the presence of bulk viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Therefore, in order to compute the present model, the form of function f(T, B) plays a very essential role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' To do this, we have to opt a viable cosmic model for f(T, B) gravity that can provide information about the accelerating universe in the late era.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' One of these options is mixed power-law model, which is written for the function f(T, B) as follows: f(T, B) = λBmT n, (32) where λ, m, and n are constant coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' According to this proposal, the contribution of torsion term or boundary term may occur separately or together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Now by replacing Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' (21), (30), and (32) into Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' (25) we obtain the energy density and the pressure of dark energy versus the redshift parameter in the following form ρTB = 6m+nλδH2(m+n) 0 (3s−1)m−1 2κ2sm (1 + z) 2 s (m+n) − η cη( a0 1+z) 3η−β, (33a) pTB = 6m+n−1λδ(2m+2n−3s)H2(m+n) 0 (3s−1)m−1 κ2sm+1 (1 + z) 2 s (m+n) − αη cη( a0 1+z) 3η−β − βη2 � cη( a0 1+z) 3η−β �2 + 3ξH0(1 + z) 1 s , (33b) where δ = 3m + 2n + 3s − 2m2 − 2mn − 3ms − 6ns − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' In order to get an acceptable answer from the above calculations, at least one set of free parameter values should be selected so that the results represent an expanding universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' This 11 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' The Hubble parameter data set in terms of redshift parameter and their uncertainty values, units of H(z) and σH are km s−1 Mpc−1 No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Redshift H(z) σH Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Redshift H(z) σH Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='07 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='0 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='6 [91] 28.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='0 [100] 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='360 226 8 [109] 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='48 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='79 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='03 [97] means that the free parameters play a very important role in this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Therefore, our choice is motivated by the fact that the energy density and pressure of dark energy become positive and negative, respectively, and also lead to the crossing of the divided-phantom line in the variations of EoS versus the redshift parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Therefore, a series of selected values are: λ = 75, c = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='001, 12 b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='1, m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='75, n = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='5, α = 3, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='0012, a0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='75, and ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='1, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Note that the free parameters λ, c, b, m, n, and a0 are dimensionless, and ξ has units pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='s in SI system or M3 in Planck systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Now we plot the variation of ρTB and pTB versus to redshift parameter as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Therefore, the variation of ρTB depends on the coefficients of F(T, B) gravity, the interacting model, and EBEC dark matter, but, the variation of pTB is also dependent on bulk viscosity in addition to the mentioned ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Also, it is evident that the value of ρTB is more than zero, and the value of pTB is less than zero for the present universe (z = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' It should be noted that ρTB and pTB have the same unit ML−1T −2 in SI system and or units M4 in Planck system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' The energy density and the pressure of dark energy in terms of redshift parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' On the other hand, by inserting Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' (33) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' (17), we can calculate EoS of dark energy, ωTB, as ωTB = (2m+2n−3s) 3s A(1 + z) 2 s (m+n) − αρm − βρ2 m + 3ξH0(1 + z) 1 s A(1 + z) 2 s (m+n) − αρm , (34) where A = 6m+nλδH2(m+n) 0 (3s − 1)m−1/(2κ2sm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' The variation of EoS of dark energy is plotted versus redshift parameter as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' As we know, EoS parameter plays an important role in modern cosmology which is a dimension- less quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' So that, this parameter is a very good description for different eras of the formation of the universe, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='e, we will have eras of accelerated phase (quintessence phase), cosmological constant, and phantom phase for −1 < ω < − 1 3, ω = −1 and ω < −1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' However the correspond- ing graph 2 represents us that the formation of the universe enters the era of the accelerated phase 13 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' EoS parameter of dark energy in terms of redshift parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' and crosses the divided-phantom line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Therefore, the present values of EoS for ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='1 and ξ = 5 are −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='075 and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='97, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' In that case, we can observe that the universe moves between regions quintessence and phantom when the bulk viscosity coefficient varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' It is worth noting that the values of the dark energy EoS parameter versus the redshift parameter vary as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' [89] is a report for Planck 2018 results measuring the dark energy EoS parameter as −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='03 with Type Ia supernovae (SNe), and earlier in 2010, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' [110] provided the best fit for the dark energy EoS parameter to Union2 data in flat-universe as −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='997+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='050 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='054 for statistical uncertainty and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='997+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='077 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='082 for statistical and systematic uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Also, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' [111] mea- sured the dark energy EoS parameter as −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='026 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='041 with the Pantheon Sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' According to the above evidence, the universe is undergoing accelerated expansion, which is consistent with the results obtained in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' [89, 110, 111] and even consistent with a cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' STABILITY AND INSTABILITY ANALYSIS In this section, we intend to explore the stability and the instability of the our model from a thermodynamic point of view based on the sound speed parameter, c2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' From a thermodynamic perspective, the universe is in an adiabatic system, in such a way that no energy or mass is exchanged from the universe to the outside.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' In that case, the entropy perturbation is zero, then we have δpTB(S, ρTB) = ∂pTB ∂S δS + ∂pTB ∂ρTB δρTB = c2 sδρTB, (35) 14 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' The sound speed parameter of dark energy in terms of redshift parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' where c2 s = ∂zpTB/∂zρTB, and symbol ∂z represents derivative with respect to redshift parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Now the values of the sound speed parameter play a key role in describing the stability and instability of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Therefore, the corresponding parameter has the conditions c2 s > 0 and c2 s < 0 which respectively declare the stability and instability of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' We draw the variation of the sound speed parameter in terms of redshift parameter as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' We can see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' 3 for ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='1 that the formation of the universe begins instability phase and continues to the stability phase at the late time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Also in ξ = 5, we see that the formation of the universe begins from a stability phase and then at the threshold of the accelerated phase enters an instability phase, and again return to stability phase in late time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Therefore, with the existence of viscous fluid gives rise to the different stability analyses for the formation of the universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' At the end, it can be said that the phases of stability and instability indicate that the energy density is or is not in a controllable growth, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Now, on the other hand, we are looking for more complete explanations for this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' For this purpose, we estimate the density parameter of dark energy, ΩTB = ρTB/ρc, where ρc = 3H2/κ2 is the critical density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' By substituting (33a) into ΩTB, we can plot the variation of the density parameter in terms of redshift parameter as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' We can see from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' 4, that the amount of ΩTB increases from the early universe to the late universe, in which the present amount of the density parameter of dark energy is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' According to the Planck 2018 results report [89], the value of the dark energy density parameter with Type Ia supernovae (SNe) has been obtained as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='6847 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='0073, so that it is compatible with our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' As a result, the contribution of dark 15 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' The density parameter of dark energy in terms of redshift parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' energy is increasing, which indeed indicates that the fate of the universe will continue to expand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' CONCLUSION In this paper, we studied the dark parts of the universe by f(T, B) gravity and EBEC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' First we explored the modified teleparallel gravity entitled f(T, B) model in a viscous fluid by the flat- FRW universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Then, by writing the Einstein equation, we earned the Friedmann equations in the presence of bulk viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Also, we considered an interaction between the contents of the universe, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=', dark matter and dark energy with term of Q = 3b2Hρm for a more realistic universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Next, the continuity equations, and EoS parameter of dark energy have been written in terms of matter and torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' The interesting point of f(T, B) gravity model is that boundary term, B relates to the Ricci scalar B = R+T, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=', f(T, B) gravity can covers the gravity models of f(T) based on the Weitzenb¨ock connection and the f(R) based on the Levi-Civita connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' In fact, the f(T, B) model covers both models simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' On the other hand, we studied EBEC as an appropriate alternative instead of the dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' In fact, we describe BEC from the generalized Gross-Pitaeveskii equation and use EBEC regime to write the EoS of dark matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' The interesting point of EBEC regime is that the EoS of dark matter is adapted to the EoS of the virial expansion, which is written in terms of normal dark matter and dark matter halo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Therefore, the elegance of the EBEC regime gives a better understanding of the different epochs of the universe from the early time to late time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' 16 In what follows, in order to examine the present model with the astronomical data, we took the power-law for the scale factor and then wrote the Hubble parameter in terms of the redshift parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Next, the corresponding Hubble parameter was fitted with 53 supernova data, so-called the Hubble data constraints, and as a result, the age of the universe as t0 = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='75 Gyr was found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' After that, we opted a viable cosmic model for f(T, B) gravity as mixed power-law model that presented information on the late universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' In addition, we reconstructed the energy density, the pressure, and the EoS of dark energy in terms of redshift parameter and some free parameters that come from f(T, B) gravity, viscous fluid, interacting model, and EBEC regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Note that choice of free parameters is very sensitive to prove that the universe has an expanding acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' In that case, we plotted ρTB, pTB, and ωTB in terms of redshift parameter for some acceptable free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' So that, the present amounts of the EoS parameter are equal to −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='075 and −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='97 for ξ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='1 and ξ = 5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Therefore, we concluded that the obtained results are completely consistent with the observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' For a more thorough review, we analyzed the stability and instability of the present model by the sound speed parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' In that case, we drew the variation of the sound speed parameter in terms of the redshift parameter with different values of bulk viscosity coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' The results showed us that the stability conditions passed through a series of instability conditions and reached a stability condition at the late time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' This means that the energy density of dark energy is in a controlled growth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Finally, in order to make the results of the present study consistent with the observational data, we calculated the value of the dark energy density parameter with a value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content='69, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' As a further work, we propose the development of the present work for axion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' ACKNOWLEDGMENT We thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Iver H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Brevik for his useful comments and suggestions to improve our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Riess, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/V9E4T4oBgHgl3EQfMwxD/content/2301.04950v1.pdf'} +page_content=' Filippenko, P.' metadata={'source': 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algorithm for differentially private mean estimation is to first clip the samples +to a bounded range and then add noise to their empirical mean. Clipping controls the sensitivity +and, hence, the variance of the noise that we add for privacy. But clipping also introduces statistical +bias. We prove that this tradeoff is inherent – no algorithm can simultaneously have low bias, low +variance, and low privacy loss for arbitrary distributions. +On the positive side, we show that unbiased mean estimation is possible under approximate +differential privacy if we assume that the distribution is symmetric. Furthermore, we show that, +even if we assume that the data is sampled from a Gaussian, unbiased mean estimation is impossible +under pure or concentrated differential privacy. +∗Authors are ordered alphabetically. +†g@csail.mit.edu. Cheriton School of Computer Science, University of Waterloo. +‡amouzaki@uwaterloo.ca. Cheriton School of Computer Science, University of Waterloo. +§matt.regehr@uwaterloo.ca. Cheriton School of Computer Science, University of Waterloo. +�vikrant.singhal@uwaterloo.ca. Cheriton School of Computer Science, University of Waterloo. +∥bias@thomas-steinke.net. Google Research, Brain Team. +∗∗jullman@ccs.neu.edu. Khoury College of Computer Sciences, Northeastern University. +1 + +Contents +1 +Introduction +3 +1.1 +Our Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +1.2 +Our Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +7 +1.3 +Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +9 +2 +Preliminaries +10 +2.1 +Differential Privacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +11 +2.2 +Non-Private Error of Mean Estimation +. . . . . . . . . . . . . . . . . . . . . . . . . . . +12 +3 +Bias-Variance-Privacy Trilemma for General-Purpose Estimators +14 +3.1 +Negative Result via Fingerprinting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +14 +3.2 +Negative Result via Amplification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +20 +4 +Low-Bias Estimators for General Distributions +24 +4.1 +Lemmata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +24 +4.2 +Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +26 +5 +Unbiased Estimators for Symmetric Distributions +30 +5.1 +Coarse Unbiased Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +30 +5.2 +Final Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +35 +6 +An Impossibility Result for Exponential Families Under Pure DP +40 +6.1 +Locally Unbiased Estimators Are Globally Unbiased . . . . . . . . . . . . . . . . . . . . +41 +6.2 +Pure DP Estimators Are Uniformly Bounded . . . . . . . . . . . . . . . . . . . . . . . . +43 +Acknowledgements +44 +References +44 +A Background on Complex Analysis +50 +B +Background on Measure Theory +51 +C Impossibility Result for Concentrated DP +52 +2 + +1 +Introduction +While the goal of statistical inference and machine learning is to learn about a population, rather than +the sample, most statistical and learning algorithms reveal lot of information that is specific to their +sample, raising concerns about the privacy of the individuals who contribute their data. In response, +differential privacy [DMNS06] has emerged as the standard framework for addressing these privacy +concerns. Informally, a differentially private algorithm guarantees that no attacker can infer much +more about any one individual in the sample than they could have inferred in a hypothetical world +where that person’s data was never collected. There is a rich literature providing differentially private +algorithms for various statistical inference and machine learning tasks, and many of these are now +deployed. +Adding the constraint of differential privacy to a statistical inference or machine learning task can, +and often does, incur an inherent cost (e.g., [BUV14, DSS+15, KV18, KLSU19]), and there has been a +large body of work aimed at pinning down these costs for a variety of tasks. The costs are typically +studied via the two-way tradeoff between privacy and error, where error is measured according to a +single loss function. However, in many applications, we have multiple desiderata for the estimator, not +all of which can be captured by a single loss function. +In this work, we study the statistical bias of differentially private estimators, which adds an extra +dimension to the tradeoff betweeen privacy and error. Estimators with little or no bias are desirable +because error due to variance can be averaged out by combining multiple estimates, whereas error +due to bias can be difficult to eliminate. Unfortunately, for most estimation tasks, the existing private +estimators with optimal error are significantly biased. +Our main contribution is to show that statistical bias is an inherent feature of private mean estima- +tion, by establishing a trilemma among bias, variance, and privacy for the fundamental task of mean +estimation. Estimating the mean of a distribution is both a ubiquitous task on its own, and a subroutine +in algorithms for more sophisticated tasks such as stochastic first-order optimization. We also identify +asymmetry as the primary cause of bias in private mean estimation by constructing unbiased private +estimators for symmetric distributions. +Bias in Private Mean Estimation. We study mean estimation for a univariate distribution – given +a set of 푛 independent samples 푋1, . . . , 푋푛 ∈ R from an unknown distribution 푃, estimate the mean +휇(푃) ≔ E푋←푃 [푋], subject to the constraint that the estimator ˆ휇(푋) be differentially private. Without +a privacy constraint, we could solve this problem using the empirical mean ¯푋 = 1 +푛 +�푛 +푖=1 푋푖, which is +both unbiased and provides optimal error bounds for all the settings we consider. Research on private +mean estimation has pinned down the optimal mean squared error (MSE) E[( ˆ휇(푋) − 휇)2] for a variety +of families of distributions, such as subgaussian distributions [KV18, BS19] and distributions satisfying +bounded moment conditions [BD14, KSU20], but these estimators can be very biased. +There are a variety of methods for optimal mean estimation, all of which introduce bias. To un- +derstand the source of bias, it is useful to review one common approach – the noisy clipped mean. +Specifically, consider the following algorithm 푀(푋). First, it clips the samples to some bounded range +[푎,푏], defined by +clip[푎,푏](푥) ≔ + + +푎 +푥 < 푎 +푥 +푎 ≤ 푥 ≤ 푏 +푏 +푏 < 푥 +. +Next, it computes the empirical mean of the clipped samples ˆ휇[푎,푏](푋) ≔ 1 +푛 +�푛 +푖=1 clip[푎,푏](푋푖). Finally, +it perturbs the clipped mean with random noise whose variance is calibrated to the sensitivity of the +3 + +clipped mean – i.e., the width of the clipping interval. Specifically, to ensure 휀-differential privacy +(휀-DP), we have +푀(푋) = 1 +푛 +푛 +� +푖=1 +clip[푎,푏](푋푖) + Lap +�푏 − 푎 +휀푛 +� +, +where Lap denotes the Laplace distribution, which has mean 0 and variance 2(푏−푎)2 +휀2푛2 . +Since the Laplace distribution has mean 0, we have E[푀(푋)] = E[clip[푎,푏](푋푖)], so the only step +that can introduce bias is the clipping. If we choose a large enough interval so that the support of the +distribution 푃 is contained in [푎,푏], then clipping has no effect, and the estimator is unbiased. However, +in this case, [푎,푏] might have to be very wide, resulting in a large variance. On the other hand, if we +reduce the variance by choosing a small interval [푎,푏], then we will have E[clip[푎,푏](푋푖)] ≠ E[푋푖] and +the estimator will be biased. Thus, we are faced with a non-trivial bias-variance-privacy tradeoff. The +exact form of the bias and the variance depends on what assumptions we make about 푃. In particular, +if we consider the class of distributions 푃 with bounded variance E푋←푃 [(푋 − 휇(푃))2] ≤ 1, then for any +훽 > 0, one can instantiate the noisy-clipped-mean estimator 푀 with an appropriate interval so that it +satisfies 휀-DP, has bias at most 훽, and has MSE +E +� +(푀(푋) − 휇(푃))2� +≤ 푂 +�1 +푛 + 훽2 + +1 +푛2 · 휀2 · 훽2 +� +. +(1) +We show that no private estimator with bias bounded by 훽 can achieve a smaller MSE. +1.1 +Our Results +Our first result is the following optimal lower bound on the MSE of any differentially private estimator +for the mean of an arbitrary distribution with bounded variance. +Theorem 1.1 (Bias-Variance-Privacy Tradeoff). Let 푀 : R푛 → R be an (휀,훿)-DP algorithm, for some +휀,훿 satisfying 0 < 훿 ≤ 휀2/200 ≤ 1. Suppose 푀 satisfies the following bounds on its bias 훽 and MSE 훼2: for +every distribution 푃 with +E +푋←푃[푋] = 휇 ∈ [0, 1]1 and E푋←푃 [(푋 − 휇)2] ≤ 1, +���� +E +푋←푃푛,푀[푀(푋) − 휇] +���� ≤ 훽 ≤ +1 +100 +and +E +푋←푃푛,푀 +� +(푀(푋) − 휇)2� +≤ 훼2. +Then +훼2 ≥ Ω +� +min +� +1 +푛2 · 휀2 · 훽2 , +1 +푛2 · 휀 · 훿1/2 +�� +. +(2) +To interpret the lower bound in Theorem 1.1 and compare it to the upper bound (1), it helps to start +by assuming 훿 is small – specifically, 훿 ≤ 훽4휀2 – so that the first term in the minimum dominates and +the bound simplifies to 훼2 ≥ Ω(1/푛2휀2훽2). This is the most interesting case, because (휀,훿)-DP is only a +meaningful privacy constraint when 훿 is quite small (see, e.g., [KS14]). Observe that the upper bound +(1) has two other terms, which are not reflected in Theorem 1.1’s lower bound (2). These terms are also +inherent, but for reasons unrelated to the privacy constraint. First, we also know that 훼2 ≥ Ω(1/푛), +1Since this theorem proves a lower bound, restricting the mean only strengthens the result. In particular, even if our +estimator is provided a coarse estimate of the mean, we still face the same bias-variance-privacy tradeoff. It is common to +consider coarse and fine private mean estimation separately, see Section 1.2 for more discussion. +4 + +which is a lower bound on the MSE of any mean estimator, even those that are not private (such as the +unperturbed empirical mean). Second, since +E +� +(푀(푋) − 휇)2� +���������������������������������� +MSE += E[푀(푋) − 휇]2 +���������������������������� +squared bias ++ E +� +(푀(푋) − E[푀(푋)])2� +�������������������������������������������������������� +variance +, +we have that 훼(푃)2 ≥ 훽(푃)2 for each distribution 푃, where 훽(푃) and 훼(푃)2 denote, respectively, the +bias and MSE of the estimator on that distribution 푃. Thus, if we set2 훽 = sup푃 훽(푃) and combine the +three lower bounds, we conclude that +훼2 ≥ Ω +�1 +푛 + 훽2 + +1 +푛2 · 휀2 · 훽2 +� +, +(3) +which matches the upper bound (1) up to constant factors. +However, there is also a corner case when 훿 ≫ 훽4휀2, where the second term in the minimum +of Theorem 1.1 dominates. For small enough 훽, this privacy guarantee is still meaningful. Thus, for +completeness, we address this corner case by constructing an estimator that nearly matches the lower +bound of Theorem 1.1 in most parameter regimes. +Theorem 1.2 (Tightness of Bias-Variance-PrivacyTradeoff). For all 휀, 훿, 훽 > 0 and푛 ∈ N, there exists an +(휀,훿)-DP algorithm 푀 : R푛 → R satisfying the following bias and accuracy properties. For any distribution +푃 on R with +E +푋←푃[푋] = 휇 ∈ [0, 1], +E +푋←푃 +� +(푋 − 휇)2� +≤ 1, and +E +푋←푃 +� +(푋 − 휇)4� +≤ 휓4, we have +���� +E +푋←푃푛,푀[푀(푋) − 휇] +���� ≤ 훽 +and +E +푋←푃푛,푀 +� +(푀(푋) − 휇)2� += 푂 +�1 +푛 + min +� +1 +푛2 · 휀2 · 훽2 + 훽2, +휓2 +푛3/2 · 휀 · 훿1/2 + +1 +푛2 · 휀2, +1 +푛 · 훿 +�� +. +The lower bound in Theorem 1.1 applies to private mean estimators that are accurate for the entire +class of distributions with bounded variance. Can we obtain unbiased private estimators by making +stronger assumptions on the distribution? Our subsequent results show that the answer depends on +what assumptions we are willing to make. Namely, we show that a generalization of the lower bound +in Theorem 1.1 holds even for the case of distributions with bounded higher moments, but we also +show that unbiased private mean estimation is possible for symmetric distributions. +Generalization to Higher Moment Bounds. If the distribution 푃 is supported on a bounded interval, +then unbiased private mean estimation is possible, as clipping to this support becomes an identity +operation. More generally, if 푃 is more tightly concentrated, then we can clip more aggressively and +obtain better bias-variance-privacy tradeoffs for the noisy clipped mean. +We consider the class of distributions that satisfy the stronger assumption E푋←푃 [|푋 − 휇|휆] ≤ 1 for +some 휆 > 2. For bias 훽 we can achieve MSE +E +� +(푀(푋) − 휇(푃))2� +≤ 푂 +� 1 +푛 + 훽2 + +1 +푛2 · 휀2 · 훽2/(휆−1) +� +. +2Theorem 1.1 permits us to set 훽 ≫ sup푃 훽(푃). Thus, we cannot conclude 훼2 ≥ 훽2 in the theorem. +5 + +Note that, although we can achieve a lower MSE for the same bias, this tradeoff is still qualitatively +similar to the case of Theorem 1.1 in the sense that any estimator with vanishing bias 훽 → 0 must have +high MSE. We prove an analogue of Theorem 1.1 showing that this tradeoff is tight for this class of +distributions, for every 휆 > 2, and conclude that bias remains an essential feature of private estimation +even under stronger concentration assumptions. +Symmetric Distributions. The reason the noisy clipped mean method leads to bias is because clipping +to the interval [푎,푏] might affect the distribution asymmetrically, introducing bias. Thus, it is natural +to consider whether we can achieve unbiased private estimation when the distribution is symmetric +around its mean, which holds for many families of distributions like Gaussians. If the distribution is +symmetric and we could clip to an interval [푎,푏] = [휇 − 푐, 휇 + 푐], then the clipped mean would be +unbiased, but this would require us to already know 휇. Nonetheless, we construct a private, unbiased +mean estimator for any symmetric distribution. +Theorem 1.3 (Unbiased Private Mean Estimation for Symmetric Distributions). For all 휀, 훿 > 0, 휆 > 2, +and 푛 ≥ 푂(log(1/훿)/휀) with 훿 ≤ 1/푛, there exists an (휀,훿)-DP algorithm 푀 : R푛 → R satisfying the +following: Let 푃 be a symmetric distribution on R (i.e., there exists 휇 ∈ R so that 푋 − 휇 and 휇 − 푋 are +identically distributed) satisfying +E +푋←푃 +� +|푋 − 휇|휆� +≤ 1. If 푋 ← 푃푛, then E[푀(푋)] = 휇, and +E +� +(푀(푋) − 휇)2� +≤ 푂 +� 1 +푛 + +1 +(푛 · 휀)2−2/휆 + 훿 · 휇2 +푛 +� +. +Note that the MSE in the theorem has a dependance on 휇, which can be unbounded. However, +if we assume that we know some 푟 such that |휇| ≤ 푟, then we can remove this term by setting 훿 ≤ +1/푂(푛푟2). Furthermore, if the distribution 푃 is Gaussian (or subgaussian), then the central moments +satisfy E푋←푃 [|푋 − 휇|휆] ≤ 푂(log휆)휆/2 for all 휆. In particular, for the special case of Gaussians with +bounded mean, we can set 휆 = Θ(log푛) and the guarantee of our algorithm simplifies to +휇2 ≤ 1/훿 +=⇒ +E +푋←N(휇,1)푛,푀 +� +(푀(푋) − 휇)2� +≤ 푂 +� 1 +푛 + log log푛 +푛2 · 휀2 +� +. +This matches what is possible without the unbiasedness constraint [KV18, KLSU19, KSU20]. +We note that, unlike the noisy clipped mean method, and many other methods for private mean +estimation, the estimator of Theorem 1.3 only satisfies (휀,훿)-DP for 훿 > 0. This is fundamental to the +techniques we use; our estimator cannot be made to satisfy (휀, 0)-DP. We show that this is inherent +by proving that every unbiased mean estimator even for the restricted classes of distributions like +Gaussians cannot satisfy (휀, 0)-DP. +Theorem 1.4 (Impossibilty of Unbiased Estimators under Pure DP). Let 푀 : R푛 → R be a randomized +algorithm. Assume that 푀 satisfies the following guarantee of unbiased estimation for Gaussian data: +there is a nonempty open interval (푎,푏) such that for every 휇 ∈ (푎,푏), +E +푋←N(휇,1)푛,푀[푀(푋)] = 휇 +and +E +푋←N(휇,1)푛,푀[|푀(푋) − 휇|] < ∞. +Then 푀 does not satisfy (휀, 0)-DP for any 휀 < ∞. +We also extend this impossibility result beyond Gaussians to exponential families and from pure +DP to concentrated DP [DR16, BS16]. +6 + +1.2 +Our Techniques +We provide two different methods for proving lower bounds on the MSE of low-bias private estimators, +such as Theorem 1.1 and its generalization to higher moments. While this is redundant, we hope that +offering multiple perspectives can provide deeper insight. +Negative Results via the Fingerprinting Method (Theorem 1.1). +The fingerprinting method +[BUV14, DSS+15] (alternatively called “tracing attacks” or “membership-inference attacks”) is the main +approach for proving optimal lower bounds on the error of differentially private estimation. Our proof +of Theorem 1.1 is based on a refinement of this method that separately accounts for the bias and mean +squared error of the estimator, and thus allows for us to prove tradeoffs between these two parameters. +To give intuition for the argument, we consider the case where 푀 is an unbiased estimator, in +which case the argument is similar to the proof of the Cramér-Rao bound. Assume that we have a +suitable family of distributions 푃휇 with mean E푋←푃휇 [푋] = 휇 and an unbiased estimator 푀 such that +E푋←푃푛휇 [푀(푋)] = 휇. As in the proof of the Cramér-Rao bound, we take the derivative of the unbiased- +ness constraint, which gives +1 = d +d휇 +� +E +푋←푃푛휇 ,푀[푀(푋)] +� += +푛 +� +푖=1 +E +푋←푃푛휇 ,푀 +� +푀(푋) · d +d휇 log푃휇(푋푖) +� +, +where 푃휇(푥) denotes the probability mass or density function of 푃휇 evaluated at푥. The (휀,훿)-differential +privacy guarantee of 푀 says that 푀(푋) and푋푖 are close to being independent where휀 and훿 quantify the +distance from independence. Moreover, a straightforward calculation shows that E푋←푃휇 [ d +d휇 log푃휇(푋)] = +0. Thus, for all 푖 ∈ [푛], we have +E +푋←푃푛휇 ,푀 +� +푀(푋) · d +d휇 log푃휇(푋푖) +� +≈휀,훿 +E +푋←푃푛휇 ,푀[푀(푋)] · +E +푋←푃푛휇 +� d +d휇 log 푃휇(푋푖) +� += 0. +Intuitively, this leads to the contradiction +1 = +푛 +� +푖=1 +E +푋←푃푛휇 ,푀 +� +푀(푋) · d +d휇 log 푃휇(푋푖) +� +≈휀,훿 +푛 +� +푖=1 +0. +To make this argument precise, we must exactly quantify the approximation ≈휀,훿, which depends both +on the privacy parameters 휀 and 훿, as well as on the variances of 푀(푋) and of +d +d휇 log 푃휇(푋푖). The +variance of 푀(푋) is the quantity that we are trying to bound. The variance of +d +d휇 log 푃휇(푋푖) (which +is known as the Fisher information) is something we control by choosing the distribution 푃휇 to be a +distribution supported on two points. +The above proof sketch applies to the unbiased case (훽 = 0). The general case (훽 > 0) introduces +some additional complications to the proof. In particular, we cannot simply consider a single fixed value +of the mean parameter 휇, as we must rule out the pathological algorithm that ignores its input sample +and outputs 휇, which has somehow been hardcoded into the algorithm. This pathological algorithm +trivially satisfies privacy and is unbiased for the single distribution 푃휇. To rule out this algorithm, we +consider a distribution over the parameter 휇 and average over this distribution, where the distribution’s +support is wider than the allowable bias 훽. While we can no longer assume that +1 = d +d휇 +� +E +푋←푃푛휇 ,푀[푀(푋)] +� +, +we can still argue that the derivative must be ≥ Ω(1) on average over the choice of 휇. +7 + +Negative Results via Amplification (Theorem 1.1 Revisited). +We present an alternativeapproach +for proving MSE lower bounds on low-bias private estimators. While this approach gives slightly +weaker bounds than Theorem 1.1, it exploits less structure of the problem, and thus may be easier +to adapt to other settings. This method is a proof by contradiction: We start by assuming the existence +of a private estimator and we show that running such an algorithm on independent datasets and av- +eraging the results would violate previous lower bounds on the mean squared error of private mean +estimators [KSU20]. +We start with an (휀,훿)-DP private estimator 푀 : R푛 → R that takes 푛 samples from some dis- +tribution and estimates its mean with bias 훽 and variance 휎2. Then we construct a new estimator +퐴푚 that takes 푛푚 samples, randomly splits them into 푚 blocks of 푛 samples each, runs 푀 on each +block, and averages the outputs. This averaging won’t reduce the bias but will reduce the variance +by a factor of 푚. Thus, the MSE of 퐴푚 is 훽2 + 휎2/푚. Moreover, privacy amplification by shuffling +[EFM+19, CSU+19, BBGN19, FMT22, FMT23] shows that 퐴푚 is (휀′, 훿 ′)-DP for 휀′ = � +푂(휀 ·푚−1/2). To com- +plete the proof, we can apply any lower bound on the MSE of the private estimator 퐴푚. In particular, +if we consider the class of distributions with bounded variance, then we can use the lower bound of +Kamath, Singhal, and Ullman [KSU20], which shows that the MSE of 퐴푚 is 훽2 + 휎2/푚 ≥ Ω(1/푛푚휀′) ≥ +�Ω(1/푛푚1/2휀). Setting 푚 appropriately yields a lower bound on 휎 that roughly matches Theorem 1.1. +General-Purpose Low-Bias Mean Estimation (Theorem 1.2). +The estimator we construct in the +proof of Theorem 1.2 combines two well-known techniques: the noisy-clipped-mean that we already +discussed and the so-called name-and-shame algorithm. As discussed above, the noisy-clipped-mean +satisfies pure DP, but it leads to a tradeoff between bias and variance. +Name-and-shame is a pathological algorithm, which with probability 훿푛, outputs one random sam- +ple in the dataset without any privacy protection, and otherwise outputs nothing. This algorithm sat- +isfies (0,훿)-DP and can be used as the basis for an unbiased estimator. Specifically, given a sample +푋 ← 푃푛, with probability 훿푛, output 푋퐼/훿푛 for a uniformly random 퐼 ∈ [푛], and otherwise output 0. +This estimator is unbiased, but the variance scales with 1/훿푛, which is impractically large for reasonable +values of 훿. +Intuitively, our estimator breaks the distribution into the tail portion (far from the mean) and the +head portion (close to the mean), and uses the noisy-clipped mean on the head and then uses name- +and-shame on the tail to correct the bias. That is, suppose we estimate 1 +푛 +�푛 +푖=1 clip[푎,푏](푋푖) with noise +addition. Then we can estimate the tail 1 +푛 +�푛 +푖=1(푋푖 − clip[푎,푏](푋푖))) using name-and-shame. Combining +these two estimates yields an unbiased estimator, such that the variance of name-and-shame is greatly +reduced because 푋푖 − clip[푎,푏](푋푖) = 0 with high probability. +Unbiased Mean Estimation for Symmetric Distributions (Theorem 1.3). +Our unbiased private +estimator for symmetric distributions is based on the estimator of Karwa and Vadhan [KV18], with +some modifications to ensure unbiasedness, so we begin by reviewing the key ideas in their estimator. +Their estimator has three steps: First, obtain a coarse estimate �휇. Second, use this coarse estimate to +compute a clipped mean �휇 ≔ +1 +푛 +�푛 +푖=1 clip[�휇−푐,�휇+푐] (푋푖). Finally, add noise to this fine estimate �휇, so +푀(푋) = �휇 + Lap� 2푐 +푛휀 +�. +The coarse estimate only needs to satisfy a minimal accuracy guarantee; roughly, we require |�휇 − +휇(푃)| ≤ 푂(휎) with high probability, where 휎2 is the variance of 푃. We can compute such an estimate +using a histogram where we split the real line into intervals of length 푂(휎) and pick an interval that +contains many samples. With high probability, there is at least one interval containing many samples +and the midpoint of any such interval is a good coarse estimate. We can privately select such an interval +8 + +with high probability – even though there are infinitely many intervals to choose from – using standard +techniques in DP [KKMN09, Vad17]. +Intuitively, we modify the algorithm of Karwa and Vadhan so that the symmetry of the distribution +푃 is preserved in our estimates �휇 and �휇 of the mean. The only part of the algorithm that breaks the +symmetry is when we split the real line into intervals. We fix this by adding a random offset to the +intervals. That is, up to scaling, our intervals are of the form {[ℓ + 푇, ℓ + 1 + 푇) : ℓ ∈ Z}, where +푇 ∈ [−1/2, +1/2) is uniformly random. Note that the distribution of this set is equivariant under +translation, so we can proceed with the analysis as if we had translated it to be symmetric around the +unknown mean. +The key observation for the analysis is that, if the coarse estimate �휇 has a symmetric distribution +with center 휇, then the clipping does not introduce bias. This is because the clipping is equally likely to +introduce positive bias or negative bias and this averages out. There are two additional technicalities +in the algorithm: First, we must ensure that the coarse estimate �휇 is independent by using independent +samples for the two stages. Second, the coarse estimation procedure may fail to produce an estimate �휇 +because no interval contains enough samples. In this case, we fall back on a version of the name-and- +shame algorithm that is unbiased but has high variance 푂(휇2/훿). Since the probabiltiy of needing this +fallback will be 푂(훿2), this case does not contribute much to the overall variance. +Negative Result for Pure DP Unbiased Mean Estimation (Theorem 1.4). +Suppose, for the sake +of contradiction, that 푀 is an 휀-DP unbiased estimator for 휇 ∈ (푎,푏) given samples from N (휇, 1). +Consider the function +푔(휇) ≔ +E +푋←N(휇,1)푛,푀[푀(푋)]. +We first observe that, if 푀 is unbiased for 휇 ∈ (푎,푏), then it must be unbiased for all 휇 ∈ R. +Intuitively, this holds because of the smoothing induced by the Gaussian distribution – if 푀(푋) were +biased when the data is drawn from N (휇∗, 1) for some 휇∗, then the same data could occur under any +other distribution N (휇, 1) with some tiny but non-zero probability, and thus 푀(푋) would also be biased +for 휇 ∈ (푎,푏), as well. Formally, we show 푔 is analytic. Thus, its value on the whole real line can be +determined from the interval (푎,푏). +Second we observe that, if 푀(푋) has bounded mean absolute deviation E푋←N(휇∗,1)푛 [|푀(푋) − 휇∗|] +for some 휇∗, then E[|푀(푥)|] must be uniformly bounded for all inputs 푥. Namely, +|푔(휇)| ≤ +E +푋←N(휇,1)푛[|푀(푋)|] ≤ |휇∗| + +E +푋←N(휇,1)푛[|푀(푋) − 휇∗|] ≤ |휇∗| + 푒휀푛 +E +푋←N(휇∗,1)푛[|푀(푋) − 휇∗|] +for all 휇 and some fixed 휇∗ that does not depend on 휇. This follows from the strong group privacy +property of (휀, 0)-DP algorithms – changing the entire sample 푋 ← N (휇∗, 1)푛 to any other sample 푥 +can only change the distribution of 푀(푋) by a pointwise multiplicative factor of 푒휀푛. +The first observation shows that 푔(휇) = 휇 on the entire real line 휇 ∈ R, and the second observation +shows that 푔(휇) is uniformly bounded on the entire real line. This is a contradiction. +This result extends beyond the Gaussian distribution to exponential families. Furthermore, it ex- +tends from pure (휀, 0)-DP to concentrated DP [DR16, BS16] and other variants of differential privacy +that satisfy a strong group privacy property. However, it does not apply to approximate (휀,훿)-DP be- +cause the group privacy guarantee underpinning the second observation breaks when 훿 > 0. +1.3 +Related Work +Unbiased estimators have long been a topic of interest in statistics. For example, topics such as the min- +imum variance unbiased estimator (MVUE) and the best linear unbiased estimator (BLUE) are textbook. +9 + +A number of celebrated results derive properties of estimators with low or no bias, often proving certain +estimators are optimal within this class. Some examples include the Gauss-Markov theorem [Gau23, +Mar00], the Lehman-Scheffé theorem [LS11, LS12], and the Cramèr-Rao bound [Cra99, Rao92]. These +results often focus on unbiased estimators for mathematicalconvenience: it is easier to prove optimality +within this restricted class than for general estimators. +Within the context of differential privacy, relatively little work has considered the bias of private +estimators separately from their overall mean squared error. A number of works [DJW13, BD14, KV18, +KLSU19, KSU20] bound the bias of the clipped mean, though only to the ends of trying to minimize +the overall error of the estimator. Amin, Kulesza, Munoz, and Vassilvitskii [AKMV19] examine bias- +variance tradeoffs of a similar procedure in the context of private empirical risk minimization. Ka- +math, Liu, and Zhang [KLZ22] employ the mean estimation approach of Kamath, Singhal, and Ull- +man [KSU20] as an oracle for stochastic first-order optimization, but, due to specifics of their setting, +employ a different balance between bias and noise. They raise the question of whether unbiased al- +gorithms for mean estimation exist. Barrientos, Williams, Snoke, and Bowen [BWSB21b, BWSB21a] +empirically measure the bias induced by various mean estimation algorithms. Zhu, van Hentenryck, +and Fioretto [ZVHF21] study bias due to post-processing of differentially private statistics. Evans, King, +and Thakurta [EK21, EKST22] and Covington, He, Honaker, and Kamath [CHHK21] give methods for +unbiased private estimation, though these rely upon strong assumptions or caveat their unbiasedness +guarantees (e.g., guaranteeing a statistic is unbiased only with high probability). +Beyond considerations of bias, private statistical estimation has been a topic of much recent in- +terest. Mean estimation is perhaps the most fundamental question in this space, enjoying significant +attention (see, e.g., [BD14, KV18, BS19, KLSU19, KSU20, WXDX20, DFM+20, BDKU20, CWZ21, BGS+21, +HLY21, LKKO21, LKO22, KLZ22, HKM22, KMV22, TCK+22, DHK23]). Most relevant to our work are +those which focus on estimation in settings with bounds on only the low-order central moments of +the underlying distribution [BD14, KSU20, HKM22], as the bias introduced due to clipping is more sig- +nificant. Other related problems involve private covariance or density estimation [BKSW19, AAK21, +KMS+22b, AL22, AKT+22, HKMN22]. Beyond these settings, other works have examined statistical +estimation under privacy constraints for mixtures of Gaussians [KSSU19, AAL21, CCd+23], graphical +models [ZKKW20], discrete distributions [DHS15], median estimation [AMB19, TVGZ20, RC21, RJC22, +BEMZ22], and more. Several recent works explore connections between private and robust estima- +tion [LKKO21, HKM22, GH22, LKO22, KMV22, AKT+22, HKMN22, CCd+23] and between privacy and +generalization [HU14, DFH+15, SU15, BNS+16a, RRST16, FS17]. Emerging directions of interest include +guaranteeing privacy when one person may contribute multiple samples [LSY+20, LSA+21, GRST22], +a combination of local and central DP for different users [ADK19], or estimation with access to some +public data [BKS22]. See [KU20] for more coverage of recent work on private statistical estimation. +2 +Preliminaries +Here, we provide a brief background on certain families of distributions and on DP, along with a few +notations. We defer the information on the other tools (in complex analysis and measure theory) that +are used in this work to Appendices A and B, respectively. +• We denote the set of all distributions over R by Δ(R). +• We denote that 푋 is a random variable drawn from a distribution 푃 by 푋 ← 푃. +• We denote that two random variables 푋 and 푌 have the same distribution by 푋 ∼ 푌. +10 + +• Let 푃,푄 be distributionsover a set X. Then for휀,훿 ≥ 0, we say that푃 and푄 are (휀,훿)-indistinguishable +(denoted by 푃 ∼휀,훿 푄), if for all measurable 푆 ⊆ X, +푒−휀 · +� +P +푋←푄[푋 ∈ 푆] − 훿 +� +≤ +P +푋←푃[푋 ∈ 푆] ≤ 푒휀 · +P +푋←푄[푋 ∈ 푆] + 훿. +• For any distribution 푃 over a set X and 푆 ⊆ X, we say 푃 (푆) ≔ +P +푋←푃[푋 ∈ 푆]. +• For a distribution 푃 over 푈 ⊆ R, we denote the product distribution over 푈 푑, with each marginal +being 푃, by 푃푑. +• Let 푀 : R푛 → R be an estimator that takes 푛 samples 푥 = (푥1, . . . ,푥푛) i.i.d. from a distribution +푃 ∈ Δ(R) (with 휇(푃) ≔ +E +푋←푃[푋]) as input, and outputs an estimate 푀(푥) of 휇(푃). Let 휇푀 ≔ +E +푋←푃푛,푀[푀(푋)]. Then we define the following. +– Bias of 푀: 훽 ≔ |휇푀 − 휇(푃)|. +– Variance of 푀: 휎2 ≔ +E +푋←푃푛,푀 +� +(푀(푋) − 휇푀)2� +. +– Mean Squared Error of 푀: 훼2 ≔ +E +푋←푃푛,푀 +� +(푀(푋) − 휇(푃))2� +. +The standard bias-variance decomposition of mean squared error gives 훼2 = 훽2 + 휎2. +• Throughout this manuscript, for any 푥 ∈ R, we use the following notation to denote clipping of +푥 to within an interval [푎,푏] ⊆ R: clip[푎,푏](푥) ≔ min{max{푥, 푎},푏}. +2.1 +Differential Privacy +A dataset 푥 = (푥1, . . . ,푥푛) ∈ X푛 is a collection of elements from a data universe X. +Definition 2.1 (Differential Privacy (DP) [DMNS06]). We say that two datasets 푥, 푥 ′ ∈ X푛 are neigh- +boring (denoted by, 푥 ∼ 푥 ′) if they differ in at most one entry. For 휀,훿 ≥ 0, a randomized algorithm +퐴 : X푛 → Y satisfies (휀,훿)-differential privacy if for every neighboring pair of datasets 푥 ∼ 푥 ′ ∈ X푛, +∀푌 ⊆ Y, +P[퐴(푥) ∈ 푌] ≤ 푒휀 · P[퐴(푥 ′) ∈ 푌] + 훿. +This property is called pure DP (or 휀-DP) when 훿 = 0, and approximate DP when 훿 > 0. +This definition is closed under post-processing. +Lemma 2.2 (Post-Processing [DMNS06]). If 푀 : X푛 → Y is (휀,훿)-DP and 푃 : Y → Z is any random- +ized function, then the algorithm 푃 ◦ 푀 is (휀,훿)-DP. +It is also closed under pre-processing that respects the neighboring relationship. +Lemma 2.3 (Pre-Processing). If 푀 : X푛 → Y is (휀,훿)-DP and 푃 : X푛 → X푛 is any randomized function +such that 푥 ∼ 푥 ′ implies 푃 (푥) ∼ 푃 (푥 ′) with probability 1, then the algorithm 푀 ◦ 푃 is (휀,훿)-DP. +Proof. For any 푌 ⊆ Y and 푥 ∼ 푥 ′ ∈ X푛, 푃 (푥) ∼ 푃 (푥 ′) with probability 1, so +P[(퐴 ◦ 푃)(푥) ∈ 푌] = E +푃 [P[퐴(푃 (푥)) ∈ 푌]] ≤ E +푃 [푒휀P[퐴(푃 (푥 ′)) ∈ 푌] + 훿] = 푒휀P[(퐴 ◦ 푃)(푥 ′) ∈ 푌] + 훿. +We have our result. +□ +11 + +The following result quantifies the privacy guaranteed by a DP algorithm for a group of individuals +within a dataset. +Lemma 2.4 (Group Privacy [DMNS06, DR14]). Let 퐴 : X푛 → Y be (휀,훿)-DP. Then for any integer +푘 ∈ {0, . . . ,푛}, subset 푌 ⊆ Y, and pairs of datasets 푥, 푥 ′ ∈ X푛 differing in 푘 entries, +P[퐴(푥) ∈ 푌] ≤ 푒푘휀 · P[퐴(푥 ′) ∈ 푌] + 푒푘휀 − 1 +푒휀 − 1 · 훿. +Common Differentially Private Mechanisms +The following defines the sensitivity of a function +to changing a single point within a dataset in the worst case. +Definition 2.5 (ℓ1-Sensitivity). Let 푓 : X푛 → R be a function, its ℓ1-sensitivity is +Δ푓 ,1 ≔ +sup +푥∼푥′∈X푛|푓 (푥) − 푓 (푥 ′)|. +Our estimators will use the well-known Laplace mechanism to ensure DP. +Lemma 2.6 (Laplace Mechanism). Let 푓 : X푛 → R be a function with ℓ1-sensitivity Δ푓 ,1. Then the +Laplace mechanism +푀(푋) ≔ 푓 (푋) + Lap(Δ푓 ,1/휀) +satisfies 휀-DP. Furthermore, +P +� +|푀(푋) − 푓 (푋)| ≥ Δ푓 ,1 · log(1/훽) +휀 +� +≤ 훽. +Our estimators will also make use of DP mechanisms for computing noisy histograms. In particular, +we will use well-known approximate DP algorithms that release a histogram in such a way that the +maximum error on every bucket depends only on the privacy parameters, and is independent of the +number of buckets. +Lemma 2.7 (DP Histograms [KKMN09, BNS16b, Vad17]). Let 푥 = (푥1, . . . ,푥푛) be a multiset of samples +from some (possibly infinite) data universe 푈 . Let Ω be a partition of 푈 into a (possibly infinite) collection +of disjoint subsets (the “bins” of the histogram) and define the vector ℎ(푥) ∈ RΩ as ℎ(푥)휔 = |푥 ∩휔|. There +is an (휀,훿)-DP algorithm 푀 : 푈 푛 → RΩ with the following guarantees. +• With probability at least 1 − 훽, max +휔∈Ω |푀(푥)휔 − ℎ(푥)휔 | = 푂(log(1/훿훽)/휀), and +• the algorithm runs in time poly(푛, log(1/휀훽)). +2.2 +Non-Private Error of Mean Estimation +Given independent samples 푋1, · · · , 푋푛 ∈ R from an unknown distribution 푃, the empirical mean +ˆ휇(푋) ≔ +1 +푛 +�푛 +푖 푋푖 is an unbiased estimator of the distribution mean 휇(푃) ≔ +E +푋←푃[푋] and its mean +squared error is +훼2 ≔ +E +푋←푃푛 +� +( ˆ휇(푋) − 휇(푃))2� += +E +푋←푃 +� +(푋 − 휇(푃))2� +푛 += 푂(1/푛). +This mean squared error is asymptotically optimal in a minimax sense and is optimal for the univariate +Gaussian case 푃 = N (휇, 휎2). +We have the following well-known result which shows that the empirical mean is asymptotically +optimal even for the simple case of Bernoulli data. +12 + +Proposition 2.8. Let 푀 : {0, 1}푛 → R be an estimator with the following property. +∀푝 ∈ [0, 1] +E +푋←Bernoulli(푝)푛 +� +(푀(푋) − 푝)2� +≤ 훼2. +Then 훼2 ≥ +1 +6(푛+2) . +The empirical mean attains MSE +E +푋←Bernoulli(푝)푛 +� +( ˆ휇(푋) − 푝)2� += 푝 (1−푝) +푛 +. However, this is not the +minimax optimal estimator of the mean of a Bernoulli distribution, rather it is the biased estimator +ˇ휇(푋) ≔ +1 +푛 + √푛 +�√푛 +2 + +푛 +� +푖 +푋푖 +� +, +which has MSE +E +푋←Bernoulli(푝)푛 +� +( ˇ휇(푋) − 푝)2� += +1 +4(√푛 + 1)2 +for all 푝 ∈ [0, 1] [HL50]. +Proof of Proposition 2.8. Let 푃 ∈ [0, 1] be uniform and, conditioned on 푃, let 푋 ← Bernoulli(푃)푛 be 푛 +independent bits, each with conditional expectation 푃. Note that the marginal distribution of �푛 +푖 푋푖 is +uniform on {0, 1, · · · ,푛}. +Given 푋 = 푥, the conditional distribution of 푃 is +푃|푋=푥 ∼ Beta +� +1 + +푛 +� +푖 +푥푖, 1 + +푛 +� +푖 +(1 − 푥푖) +� +. +In terms of mean squared error, the best estimator of 푃 is simply the mean of this conditional dis- +tribution. That is, the function 푓 : {0, 1}푛 → R that minimizes E +푃,푋 +� +(푃 − 푓 (푋))2� +is the conditional +expectation 푓 (푥) = E[푃 | 푋 = 푥]. Indeed, this is the definition of conditional expectation in the general +measure theoretic setting. Consequently, the best possible mean squared error of an estimator of 푃 +given 푋 is the variance of this conditional distribution 푃|푋. +The distribution Beta(푎,푏) has mean +푎 +푎+푏 and variance +푎푏 +(푎+푏)2(푎+푏+1) . Now we have +훼2 ≥ +E +푃←[0,1],푋←Bernoulli(푃)푛 +� +(푀(푋) − 푃)2� +≥ E +푋 +� +E +푃 +�� +E +푃 [푃 | 푋] − 푃 +�2 +| 푋 +�� += +E +푃←[0,1],푋←Bernoulli(푃)푛 +��1 + �푛 +푖 푋푖 +2 + 푛 +− 푃 +�2� += +E +푃←[0,1],푋←Bernoulli(푃)푛 +� (1 + �푛 +푖 푋푖)(1 + �푛 +푖 (1 − 푋푖)) +(푛 + 2)2(푛 + 3) +� += +1 +푛 + 1 +푛 +� +푘=0 +(1 + 푘)(1 + 푛 − 푘) +(푛 + 2)2(푛 + 3) += +1 +(푛 + 1)(푛 + 2)2(푛 + 3) +푛 +� +푘=0 +(1 + 푛) + 푛 · 푘 − 푘2 += +1 +(푛 + 1)(푛 + 2)2(푛 + 3) +� +(1 + 푛) · (푛 + 1) + 푛 · 푛(푛 + 1) +2 +− 푛(푛 + 1)(2푛 + 1) +6 +� +13 + += 6(푛 + 1)2 + 3푛2(푛 + 1) − 푛(푛 + 1)(2푛 + 1) +6(푛 + 1)(푛 + 2)2(푛 + 3) += 6(푛 + 1) + 3푛2 − 푛(2푛 + 1) +6(푛 + 2)2(푛 + 3) += +5푛 + 6 + 푛2 +6(푛 + 2)2(푛 + 3) = +(푛 + 2)(푛 + 3) +6(푛 + 2)2(푛 + 3) = +1 +6(푛 + 2) . +This completes the proof. +□ +If we change the distribution of 푃 ∈ [0, 1] from uniform to Beta(√푛/2, √푛/2) in the above proof, +then we obtain the stronger conclusion 훼2 ≥ +1 +4(√푛+1)2 , which is exactly optimal. However, this requires +a more complicated calculation. +3 +Bias-Variance-Privacy Trilemma for General-Purpose Estimators +We now prove our main negative result. Informally, we show that if an algorithm is differentially private +and has low bias, then it must have high error. There are, of course, other parameters that arise in the +analysis, such as bounds on the tails of the unknown distribution 푃. +We provide two different proofs, which give slightly different results. The first proof directly applies +the fingerprinting technique for lower bounds on differentially private estimation [BUV14], while the +second proof is a “black-box” reduction. +3.1 +Negative Result via Fingerprinting +We begin by stating our general result and, before giving the proof, we provide some remarks and +corollaries to help interpret the result. +Theorem 3.1 (Bias-Variance-Privacy Tradeoff). Let 휀,훿, 훽, 훼,휏 ≥ 0 and 휆 > 1. Let 푀 : R푛 → R be an +(휀,훿)-DP algorithm that satisfies the following bias and accuracy properties. For any distribution 푃 on R +with 휇(푃) ≔ +E +푋←푃[푋] ∈ [0, 1] and +E +푋←푃 +� +|푋 − 휇(푃)|휆� +≤ 1, we have the following: +���� +E +푋←푃푛,푀[푀(푋)] − 휇(푃) +���� ≤ 훽, +E +푋←푃푛,푀[|푀(푋) − 휇(푃)|] ≤ 훼, +∫ ∞ +0 +min +� +훿, +P +푋←푃푛,푀[|푀(푋) − 휇(푃)| > 푥] +� +d푥 ≤ 훼 · 휏. +If 16훽 ≤ 훾 ≤ 1/5, then +훼 ≥ +1 +32푛 · sinh(휀) · 훾1/(휆−1) + 16푛 · 휏 · 훾−1 . +Note that, for small values of 휀, sinh(휀) ≈ 휀, but, for large 휀, sinh(휀) ≈ 1 +2푒휀. Since this is the “usual” +dependence on 휀 in many such bounds under the constraint of DP, sinh allows us to capture behaviour +in both regimes with a single function. +The first two accuracy conditions are not hard to interpret: they control the bias and the mean +absolute error. The parameter 훽 bounds the bias of the algorithm, while 훼 bounds the mean absolute +deviation. By Jensen’s inequality, +���� +E +푋←푃푛,푀[푀(푋)] − 휇(푃) +���� ≤ +E +푋←푃푛,푀[|푀(푋) − 휇(푃)|] ≤ +� +E +푋←푃푛,푀 +� +(푀(푋) − 휇(푃))2� +. +14 + +Thus, we can assume that 훽 ≤ 훼. Furthermore, if an estimator 푀(푋) has a bound on the mean squared +error of 훼2, it consequently also has a mean absolute error of at most 훼. Thus, our somewhat uncon- +ventional assumption controlling the mean absolute error only broadens the class of estimators against +which our lower bound holds: for interpretability, one could instead replace this with 훼2 being the mean +squared error of 푀(푋). The standard decomposition gives that the mean squared error 훼2 is simply the +variance of the estimator plus the square of the bias 훽. +The third property and the parameter 휏 is somewhat hard to interpret. We note that this condition +is implied by a bound on the MSE of the estimator via the following lemma. It applies in more general +circumstances, as well, when we may have bounds on higher or lower moments of the estimator’s error. +Lemma 3.2 (Setting 휏 = 훿1−1/휅 in Theorem 3.1). Let 훼, 훿 ≥ 0 and 휅 > 1. Let 푌 be a random variable +satisfying E[|푌 |휅] ≤ 훼휅. Then +∫ ∞ +0 +min{훿, P[|푌 | > 푥]} d푥 ≤ 훼 · 훿1−1/휅. +Proof. We assume, without loss of generality, that 훿 < 1. Suppose, for now, there exists 푐 > 0 such that +P[|푌 | > 푐] = 훿. If 푥 ≥ 푐, then min{훿, P[|푌 | > 푥]} = P[|푌 | > 푥] = P[|푌 | · I[|푌 | > 푐] > 푥]. Likewise, if +푥 ≤ 푐, then min{훿, P[|푌 | > 푥]} = 훿 = P[|푌 | · I[|푌 | > 푐] > 푥]. Thus, +∫ ∞ +0 +min{훿, P[|푌 | > 푥]}d푥 = +∫ ∞ +0 +P[|푌 | · I[|푌 | > 푐] > 푥] += E[|푌 | · I[|푌 | > 푐]] +≤ E[|푌 |휅] +1 +휅 · E +� +I[|푌 | > 푐] +휅 +휅−1 +� 휅−1 +휅 +(Hölder’s Inequality) += E[|푌 |휅] +1 +휅 · P[|푌 | > 푐]1−1/휅 +≤ 훼 · 훿1−1/휅. +If the distribution of 푌 is continuous, then such a quantity 푐 is guaranteed to exist. In general, there +exists 푐 ≥ 0 such that P[|푌 | > 푐] ≤ 훿 ≤ P[|푌 | ≥ 푐]. We can define a random 퐼 : R → {0, 1} such that +I[|푌 | > 푐] ≤ 퐼 (|푌 |) ≤ I[|푌 | ≥ 푐] with probability 1 and E[퐼 (|푌 |)] = 훿. The above proof carries through +in general if we replace I[|푌 | > 푐] with 퐼 (|푌 |). +□ +In particular, if we have a mean squared error bound for the estimator E +� +(푀(푋) − 휇(푃))2� +≤ 훼2, +then the third condition of Theorem 3.1 holds with 휏 = +√ +훿. Larger values of 휅 entail sharper tail bounds +on the estimator, allowing us to set 휏 larger (and thus implying stronger lower bounds), with 휏 → 훿 as +휅 → ∞. +In general, Theorem 3.1’s lower bound on the error 훼 is maximized by setting +훾 = clip[16훽,1/5] +�� (휆 − 1)휏 +2 sinh(휀) +�1−1/휆� +. +(4) +Combining this parameter setting for 훾, along with the bound of 휏 = 훿1−1/휅 given by Lemma 3.2, and +focusing on the most natural case of휅 = 2 (i.e., we assume only that the estimator has bounded variance), +gives the following result. +15 + +Corollary 3.3 (Combining Theorem 3.1, Lemma 3.2 (with 휅 = 2), and Equation 4.). Let 푀 : R푛 → R +be (휀,훿)-DP and satisfy the following bias and accuracy properties. For any distribution 푃 on R with +휇(푃) ≔ +E +푋←푃[푋] ∈ [0, 1] and +E +푋←푃 +� +|푋 − 휇(푃)|휆� +≤ 1, we have +���� +E +푋←푃푛,푀[푀(푋)] − 휇(푃) +���� ≤ 훽 +and +E +푋←푃푛,푀 +� +(푀(푋) − 휇(푃))2� +≤ 훼2. +If 훽 ≤ +1 +80 and 훿 ≤ +� +2·sinh(휀) +51+ +1 +휆−1 ·(휆−1) +�2 +, then +훼 ≥ +1 +32 · 푛 · sinh(휀) · +휆 +휆−1 · max +� +(16훽) +1 +휆−1, +� +(휆−1) +√ +훿 +2·sinh(휀) +�1/휆� . +We illustrate the representative case where the underlying distribution has bounded variance by +further fixing 휆 = 2. Combining the resulting lower bound with the non-private rate (Proposition 2.8) +gives the following result. +Theorem 3.4 (Setting 휆 = 2 in Corollary 3.3 to get Theorem 1.1). Let 푀 : R푛 → R be (휀,훿)-DP and +satisfy the following bias and accuracy properties. For any distribution 푃 on R with 휇(푃) ≔ +E +푋←푃[푋] ∈ +[0, 1] and +E +푋←푃 +� +(푋 − 휇(푃))2� +≤ 1, we have +���� +E +푋←푃푛,푀[푀(푋)] − 휇(푃) +���� ≤ 훽 +and +E +푋←푃푛,푀 +� +(푀(푋) − 휇(푃))2� +≤ 훼2. +If 훽 ≤ 1/80 and 훿 ≤ � 2 +25 sinh휀�2, then +훼 ≥ max + + +1 +� +6(푛 + 2) +, +1 +64 · 푛 · sinh(휀) · max +� +16 · 훽, +� +√ +훿 +2·sinh(휀) +� + + +≥ Ω +� +1 +√푛 + min +� +1 +푛휀훽, +1 +푛 +� +휀 +√ +훿 +�� +. +Now, we prove our general result. Our proof follows what is known as the fingerprinting approach. +Fingerprinting codes were first studied by Boneh and Shaw [BS98] in the context of cryptographic +traitor-tracing schemes; Tardos [Tar08] gave an optimal construction of fingerprinting codes. Finger- +printing codes were used to prove negative results for differentially private algorithms by Bun, Ull- +man, and Vadhan [BUV14]. Subsequently, many works have expanded this methodology [SU15, BSU17, +SU17a, SU17b, KMS22a]. +We will use the following lemma in the proof. +Lemma 3.5 (Fingerprinting Derivative Lemma [SU17b, Lemma 9]). Let 푓 : {0, 1}푛 → R be an arbitrary +function. Define 푔 : [0, 1] → R by 푔(푝) = E[푓 (푋)], where 푋1, . . . ,푋푛 be independent samples from +Bernoulli(푝). Then, for all 푝 ∈ [0, 1], we have 푔′(푝) · 푝(1 −푝) = E +� +푓 (푋) · �푛 +푖 (푋푖 − 푝) +� +, where 푋1, . . . , 푋푛 +be independent samples from Bernoulli(푝). +16 + +Proof of Theorem 3.1. For 푝 ∈ [0, 1] and 푣 > 0, define D푣,푝 = 푣 · Bernoulli(푝) – i.e., a sample from D푣,푝 +is 0 with probability 1 − 푝 and 푣 with probability 푝. Then 휇(D푣,푝) = +E +푋←D푣,푝 [푋] = 푣푝 and +E +푋←D푣,푝 +� +|푋 − 휇(D푣,푝)|휆� += (1 − 푝)(푣푝)휆 + 푝(푣(1 − 푝))휆 ≤ 2푝푣휆. +If we ensure 푣 ≤ (2푝)−1/휆 ≤ 1/푝, then the 휆-th absolute central moment is below 1, and the mean is in +the interval [0, 1], so the bias and accuracy guarantees of 푀 apply. +For 푣 > 0, define 푔푣 : [0, 1] → R by +푔푣(푝) ≔ +E +푋←D푛푣,푝,푀[푀(푋)]. +By Lemma 3.5, for all 푣 > 0 and 푝 ∈ [0, 1], we have +E +푋←D푛 +푣,푝,푀 +� +푀(푋) · +푛 +� +푖 +�1 +푣 푋푖 − 푝 +�� += 푝(1 − 푝)푔′ +푣(푝). +(5) +Fix 0 < 푎 < 푏 ≤ 1/2 and 0 < 푣 ≤ (2푏)−1/휆 (to be determined later). Now, let 푃 ∈ [푎,푏] be a +random variable with density ∝ +1 +푃 (1−푃) – i.e., ∀푡 ∈ [푎,푏], P[푃 ≤ 푡] = +∫ 푡 +푎 +1 +푥 (1−푥) d푥 +∫ 푏 +푎 +1 +푥 (1−푥) d푥 . Conditioned on 푃, let +푋1, . . . , 푋푛 ∈ R be independent samples from D푣,푃. Now, +E +푃,푋,푀 +� +푀(푋) · +푛 +� +푖 +�1 +푣 푋푖 − 푃 +�� += E +푃 +� +푃 (1 − 푃)푔′ +푣(푃) +� +(Equation 5) += +∫ 푏 +푎 푔′ +푣(푝)d푝 +∫ 푏 +푎 +1 +푥 (1−푥) d푥 += +푔푣(푏) − 푔푣(푎) +log(푏/(1 − 푏)) − log(푎/(1 − 푎)) . +By our bias assumption, |푔푣(푏) − 푣푏| ≤ 훽 and |푔푣(푎) − 푣푎| ≤ 훽. Thus, +E +푃,푋,푀 +� +푀(푋) · +푛 +� +푖 +�1 +푣 푋푖 − 푃 +�� +≥ 푣 · (푏 − 푎) − 2훽 +log +� +푏·(1−푎) +푎·(1−푏) +� +. +Since E[푋푖] = 푣푃 for all 푖, we can center 푀(푋) and rearrange slightly: +푛 +� +푖 +E +푃,푋,푀 +� +(푀(푋) − 푣푃) · +�1 +푣 푋푖 − 푃 +�� +≥ 푣 · (푏 − 푎) − 2훽 +log +� +푏·(1−푎) +푎·(1−푏) +� +. +Next, we will use differential privacy to prove an upper bound on this quantity. Fix an arbitrary +푖 ∈ [푛] and fix 푃 = 푝 ∈ [푎,푏]. Our goal is to upper bound +E +푋←D푛푣,푝,푀 +� +(푀(푋) − 푣푝) · � 1 +푣푋푖 − 푝�� +. +Since 푀 satisfies (휀,훿)-DP, the distribution of the pair (푀(푋), 푋푖) is (휀,훿)-indistinguishable from +that of +� +푀(푋−푖, ˜푋푖),푋푖 +� +, where (푋−푖, ˜푋푖) denotes the dataset 푋 with 푋푖 replaced by ˜푋푖; here ˜푋푖 ← D푣,푝 +is a fresh sample from the distribution. Now ˜푋푖 and 푋푖 are interchangeable, this means the distribution +17 + +of +� +푀(푋−푖, ˜푋푖),푋푖 +� +is identical to that of +� +푀(푋), ˜푋푖 +� +. By transitivity, the distribution of (푀(푋), 푋푖) is +(휀,훿)-indistinguishable from that of +� +푀(푋), ˜푋푖 +� +. In particular, +(푀(푋) − 푣푝) · � 1 +푣푋푖 − 푝� ∼휀,훿 +� +푀(푋−푖, ˜푋푖) − 푣푝 +� +· � 1 +푣푋푖 − 푝� ∼ (푀(푋) − 푣푝) · +� +1 +푣 ˜푋푖 − 푝 +� +. +We also have +��(푀(푋) − 푣푝) · � 1 +푣푋푖 − 푝��� ≤ |푀(푋) − 푣푝| with probability 1. +Thus +E +� +(푀(푋) − 푣푝) · +�1 +푣 푋푖 − 푝 +�� += E +� +max{(푀(푋) − 푣푝) · +�1 +푣 푋푖 − 푝 +� +, 0} +� +− E +� +max{−(푀(푋) − 푣푝) · +�1 +푣 푋푖 − 푝 +� +, 0} +� += +∞ +∫ +0 +P +� +(푀(푋) − 푣푝) · +�1 +푣 푋푖 − 푝 +� +> 푥 +� +d푥 +− +∞ +∫ +0 +P +� +(푀(푋) − 푣푝) · +�1 +푣 푋푖 − 푝 +� +< −푥 +� +d푥. +We have +P +� +(푀(푋) − 푣푝) · +�1 +푣 푋푖 − 푝 +� +> 푥 +� +≤ P[|푀(푋) − 푣푝| > 푥] +and simultaneously, +P +� +(푀(푋) − 푣푝) · +�1 +푣 푋푖 − 푝 +� +> 푥 +� +≤ 푒휀 · P +� +(푀(푋) − 푣푝) · +�1 +푣 +˜푋푖 − 푝 +� +> 푥 +� ++ 훿 += 푒휀 · 푝 · P[(푀(푋) − 푣푝) · (1 − 푝) > 푥] ++ 푒휀 · (1 − 푝) · P[(푀(푋) − 푣푝) · (0 − 푝) > 푥] + 훿. +Define 훿(푥) ≔ min{훿, P[|푀(푋) − 푣푝| > 푥]}. Then +∞ +∫ +0 +P +� +(푀(푋) − 푣푝) · +�1 +푣 푋푖 − 푝 +� +> 푥 +� +d푥 ≤ +∞ +∫ +0 +푒휀 · 푝 · P[(푀(푋) − 푣푝) · (1 − 푝) > 푥]d푥 ++ +∞ +∫ +0 +푒휀 · (1 − 푝) · P[(푀(푋) − 푣푝) · (0 − 푝) > 푥]d푥 ++ +∞ +∫ +0 +훿(푥)d푥 += E[푒휀 · 푝 · max{(푀(푋) − 푣푝) · (1 − 푝), 0}] ++ E[푒휀 · (1 − 푝) · max{(푀(푋) − 푣푝) · (0 − 푝), 0}] ++ +∞ +∫ +0 +훿(푥)d푥 +18 + +≤ 푒휀 · 푝(1 − 푝) · E[|푀(푋) − 푣푝|] + 훼 · 휏. +In the above, the final inequality holds because +E[max{푀(푋) − 푣푝, 0}] + E[max{−푀(푋) + 푣푝, 0}] = E[|푀(푋) − 푣푝|], +and due to the third utility assumption in our theorem statement. Similarly, +∞ +∫ +0 +P +� +(푀(푋) − 푣푝) · +�1 +푣 푋푖 − 푝 +� +< −푥 +� +d푥 ≥ +∞ +∫ +0 +max +� +0, +푒−휀 � +P +� +(푀(푋) − 푣푝) · +� +1 +푣 ˜푋푖 − 푝 +� +< −푥 +� +− 훿 +� +� +d푥 += 푒−휀 · +∞ +∫ +0 +P +� +(푀(푋) − 푣푝) · +�1 +푣 +˜푋푖 − 푝 +� +< −푥 +� +d푥 ++ 푒−휀 · +∞ +∫ +0 +max +� +−훿, +−P +� +(푀(푋) − 푣푝) · +� +1 +푣 ˜푋푖 − 푝 +� +< −푥 +� +� +d푥 +≥ 푒−휀 · +∞ +∫ +0 +P +� +(푀(푋) − 푣푝) · +�1 +푣 +˜푋푖 − 푝 +� +< −푥 +� +− 훿(푥)d푥 += 푒−휀 · +∞ +∫ +0 +푝 · P[(푀(푋) − 푣푝) · (1 − 푝) < −푥]d푥 ++ 푒−휀 · +∞ +∫ +0 +(1 − 푝) · P[(푀(푋) − 푣푝) · (0 − 푝) < −푥]d푥 +− 푒−휀 · +∞ +∫ +0 +훿(푥)d푥 +≥ 푒−휀 · 푝(1 − 푝) · E[|푀(푋) − 푣푝|] − 푒−휀 · 훼 · 휏. +Putting these two pieces together, we have: +E +푋1,···,푋푛←D푣,푝 +� +(푀(푋) − 푣푝) · +�1 +푣 푋푖 − 푝 +�� +≤ +∞ +∫ +0 +P +� +(푀(푋) − 푣푝) · +�1 +푣 푋푖 − 푝 +� +> 푥 +� +d푥 +− +∞ +∫ +0 +P +� +(푀(푋) − 푣푝) · +�1 +푣 푋푖 − 푝 +� +< −푥 +� +d푥 +≤ (푒휀 − 푒−휀) · 푝(1 − 푝) · E[|푀(푋) − 푣푝|] + (1 + 푒−휀) · 훼 · 휏 +≤ (푒휀 − 푒−휀) · 푏(1 − 푏) · 훼 + 2훼휏, +as 푝 ≤ 푏 ≤ 1/2. Then, combining this with our lower bound, we have +푣 · (푏 − 푎) − 2훽 +log +� +푏·(1−푎) +푎·(1−푏) +� +≤ +푛 +� +푖 +E +푃,푋,푀 +� +(푀(푋) − 푣푃) · +�1 +푣 푋푖 − 푃 +�� +19 + +≤ 푛 · ((푒휀 − 푒−휀) · 푏(1 − 푏) · 훼 + 2훼휏) +≤ 훼 · 푛 · 2 · (sinh(휀) · 푏 + 휏), +which rearranges to +훼 ≥ +푣 · (푏 − 푎) − 2훽 +2푛 · (sinh(휀) · 푏 + 휏) · log +� +푏·(1−푎) +푎·(1−푏) +� . +It only remains to set the parameters subject to the constraints 0 < 푎 < 푏 ≤ 1/2 and 0 < 푣 ≤ (2푏)−1/휆. +First, we set 푏 = 2푎, and 푣 = (2푏)−1/휆 = (4푎)−1/휆 and assume (8훽) +휆 +휆−1 ≤ 푎 ≤ 1/5, which simplifies the +above expression to +훼 ≥ +푎1−1/휆 · 4−1/휆 − 2훽 +2푛 · (sinh(휀) · 2푎 + 휏) · log +� +2·(1−푎) +1−2푎 +� ≥ +푎1−1/휆 − 8훽 +8푛 · (sinh(휀) · 2푎 + 휏) . +We reparameterize 푎 = 훾 +휆 +휆−1 for some 16훽 ≤ 훾 ≤ 1/5 to obtain +훼 ≥ +훾 − 8훽 +8푛 · (sinh(휀) · 2 · 훾 +휆 +휆−1 + 휏) +≥ +훾/2 +8푛 · (sinh(휀) · 2 · 훾 +휆 +휆−1 + 휏) += +1 +32푛 sinh(휀)훾1/(휆−1) + 16푛휏훾−1 . +This completes our proof. +□ +3.2 +Negative Result via Amplification +In this subsection, we show that known MSE lower bounds (without bias constraints) [KSU20] com- +bined with privacy amplification via shuffling [EFM+19, CSU+19, BBGN19, FMT22, FMT23] can also be +used to derive qualitatively similar lower bounds on MSE for private estimators with low bias as those +yielded by fingerprinting in the previous subsection. Our reduction provides an alternative perspective +on the bias-variance-privacy tradeoff, and could prove useful in future work as it is more “generic” than +the fingerprinting approach. +Specifically, we will use the following lower bound on the MSE of a private estimator in a black-box +manner. +Theorem 3.6 ([KSU20, Theorem 3.8]). Let 푀 : R푛 → R be (휀,훿)-DP. Then there exists a distribution 푃 +on R with 휇(푃) ≔ E푋←푃 [푋] ∈ [−1, +1] and E푋←푃 [(푋 − 휇(푃))2] ≤ 1, such that +E +푋←푃푛 +� +(푀(푋) − 휇(푃))2� +≥ Ω +� +1 +푛(휀 + 훿) +� +. +The other ingredient in our proof is the following extension of the privacy amplification by subsam- +pling result of Feldman, McMillan, and Talwar [FMT22]. Specifically we extend from the setting of local +differential privacy (where each algorithm has one input) to the setting where a dataset is randomly +partitioned into blocks of fixed size 푛 > 1, and these blocks are processed by a sequence of private +mechanisms. To be specific, we randomly partition the dataset as follows. We first arrange the dataset +as a matrix with 푚 columns and 푛 rows. Then, for each row 푖 ∈ [푛], we perform a uniformly random +permutation of the 푚 elements in that row. +Theorem 3.7 (Extension of Privacy Amplification by Shuffling [FMT22] to Larger Inputs). Let 푚,푛 ∈ +Z+, let X be the data universe, and let Y1, . . . , Y푚 be image spaces. Suppose for each 푖 ∈ [푚], we have a +20 + +randomized function 퐿푖 : Y1 × · · · × Y푖−1 × X푛 → Y푖 such that 퐿푖(푦, 푥) is (휀0,훿0)-DP in the parameter +푥 ∈ X푛 for every fixed 푦 ∈ Y1 × · · · × Y푖−1. Consider 퐿푚 ⊗ · · · ⊗ 퐿1 : (X푛)푚 → Y1 × · · · × Y푚 defined by +(퐿푚 ⊗ · · · ⊗ 퐿1)(푥1, . . . , 푥푚) ≔ (푦1, . . . ,푦푚) +where we recursively define 푦푖 ≔ 퐿푖 (푦1, . . . ,푦푖−1,푥푖). In addition, consider the random shuffle operator +Π : (X푛)푚 → (X푛)푚 given by +Π((푥1 +1, . . . ,푥푛 +1 ), . . . , (푥1 +푚, . . . ,푥푛 +푚)) ≔ ((푥1 +휋1(1), . . . ,푥푛 +휋푛 (1)), . . . , (푥1 +휋1(푚), . . . ,푥푛 +휋푛 (푚))) +where 휋1, . . . , 휋푛 are permutations on [푚] chosen independently and uniformly at random. Then, for any +훿1 ∈ [2푒− +푚 +16푒휀0 , 1], the function 퐿푚⊗· · ·⊗퐿1◦Π : (X푛)푚 → Y1×· · ·×Y푚 is (휀1,훿1 + (푒휀1 + 1)(푒−휀0/2 + 1)푚훿0)- +DP, where 휀1 = 푂(휀0 +� +log(1/훿1)/푚) (for 휀0 = 푂(1)) as in Equation 6. +Before we prove Theorem 3.7, we will first show how it can be used to prove a slightly weaker +version of Theorem 1.1 by reduction to Theorem 3.6. +Theorem 3.8 (Bias-Variance-Privacy Tradeoff via Shuffling). Let 푛 ∈ N and 휀,훿 ∈ (0, 1). Let 푀 : R푛 → +R be (휀,훿)-DP and satisfy the following bias and accuracy properties. For any distribution 푃 over R with +휇(푃) ≔ +E +푋←푃[푋] ∈ [±1] and +E +푋←푃 +� +(푋 − 휇(푃))2� +≤ 1, we have +���� +E +푋←푃푛,푀[푀(푋)] − 휇(푃) +���� ≤ 훽 +and +E +푋←푃푛,푀 +� +(푀(푋) − 휇(푃))2� +≤ 훼2. +If 훽2 ≤ �Ω� 1 +푛휀 +� and 훿 ≤ 푂(푛3휀4훽6), then +훼2 ≥ Ω +� +1 +푛2휀2훽2 log(1/푛휀2훽2) +� +. +Proof. Let 푚 ∈ N (we delay our choice of 푚 until later). Consider 퐴푚 : (R푛)푚 → R defined by +∀푥1, . . . ,푥푚 ∈ R푛, +퐴푚((푥1, . . . ,푥푚)) = 1 +푚 +푚 +� +푖=1 +푀(푥푖). +Fix some distribution 푃 with mean and variance bounded by 1. This gives us the following guarantee +about the mean of 퐴푚. +ˇ휇 ≔ +E +(푋1,...,푋푚)←(푃푛)푚,퐴푚 +[퐴푚((푋1, . . . ,푋푚))] = +E +(푋1,...,푋푚)←(푃푛)푚,푀 +� +1 +푚 +푚 +� +푖=1 +푀(푋푖) +� += 1 +푚 +푚 +� +푖=1 +E +푋푖←푃푛,푀[푀(푋푖)] += +E +푋←푃푛,푀[푀(푋)] +Thus, the bias of 퐴푚 is at most 훽, as we see from the following. +| ˇ휇 − 휇(푃)| = +���� +E +(푋1,...,푋푚)←(푃푛)푚,퐴푚 +[퐴푚((푋1, . . . ,푋푚))] − 휇(푃) +���� = +���� +E +푋←푃푛,푀[푀(푋)] − 휇(푃) +���� ≤ 훽 +21 + +Similarly, the MSE of 퐴푚 is +E +(푋1,···,푋푚)←(푃푛)푚,퐴푚 +� +(퐴푚(푋1, · · · ,푋푚) − 휇(푃))2� += ( ˇ휇 − 휇(푃))2 ++ +E +(푋1,···,푋푚)←(푃푛)푚,퐴푚 +� +(퐴푚(푋1, · · · ,푋푚) − ˇ휇)2� += ( ˇ휇 − 휇(푃))2 ++ +E +(푋1,···,푋푚)←(푃푛)푚,푀 + +� +1 +푚 +푚 +� +푖=1 +푀(푋푖) − ˇ휇 +�2 += ( ˇ휇 − 휇(푃))2 + 1 +푚 · +E +푋←푃푛,푀 +� +(푀(푋) − ˇ휇)2� +≤ 훽2 + 훼2 +푚 . +Let Π : (X푛)푚 → (X푛)푚 be the shuffle operator described in Theorem 3.7. Since any set of samples +drawn i.i.d. from a distribution is invariant under shuffling, (퐴푚 ◦ Π)(푋) has the same distribution as +퐴푚(푋) when 푋 ← (푃푛)푚. In particular, 퐴푚 ◦Π has the same bias and MSE as 퐴푚 on inputs from (푃푛)푚. +Privacy amplification by shuffling (Theorem 3.7) and postprocessing (Lemma 2.2), imply that 퐴푚 ◦ Π is +� +휀′ ≔ 푂(휀 +� +log(1/훿1)/푚),훿 ′ ≔ 훿1 + 푂(훿푚) +� +-DP for all 훿1 ∈ [2푒− 푚 +16푒휀 , 1]. +Now, we apply Theorem 3.6 [KSU20, Theorem 3.8] to 퐴푚: There exists a distribution 푃 with mean +and variance bounded by 1, such that +E +(푋1,···,푋푚)←(푃푛)푚,퐴푚 +� +(퐴푚(푋1, · · · , 푋푚) − 휇(푃))2� +≥ Ω +� +1 +푛푚(휀′ + 훿 ′) +� +. +Combining all these inequalities gives +훽2 + 훼2 +푚 ≥ +E +(푋1,···,푋푚)←(푃푛)푚 +� +(퐴푚(푋1, · · · , 푋푚) − 휇(푃))2� +≥ Ω +� +1 +푛푚(휀′ + 훿 ′) +� +≥ Ω +� +1 +푛푚(휀 +� +log(1/훿1)/푚 + 훿1 + 훿푚) +� +. +This rearranges to +훼2 ≥ Ω +� +1 +푛휀 +� +log(1/훿1)/푚 + 푛훿1 + 푛푚훿 +� +− 푚훽2. +It only remains to set 푚 ∈ N and 훿1 ∈ [2푒− 푚 +16푒휀 , 1] to maximize this lower bound. +Now, we assume that 훿1 ≤ 푂(휀/√푚) and 훿 ≤ 푂(휀/푚3/2). Then the first term in the denominator +dominates and we have +훼2 ≥ Ω +� +1 +푛휀 +� +log(1/훿1)/푚 +� +− 푚훽2. +Then setting 푚 = Θ +� +1 +푛2휀2훽4 log(1/훿1) +� +optimizes the expression giving +훼2 ≥ Ω +� +1 +푛2휀2훽2 log(1/훿1) +� +. +22 + +We set 훿1 = 푛휀2훽2 ≤ 푂(휀/√푚). This satisfies 훿1 ∈ [2푒− 푚 +16푒휀 , 1] as long as 훽2 ≤ 1/푛휀2 and 푚 = +Θ +� +1 +푛2휀2훽4 log(1/푛휀2훽2) +� +≥ 푂(푒휀 log(1/푛휀2훽2)). The latter constraint rearranges to 훽2 log(1/푛휀2훽2) ≤ Ω +� +푒−휀 +푛휀 +� +. +To conclude, we note that the assumption 훿 ≤ 푂(휀/푚3/2) is implied by 훿 ≤ 푂(푛3휀4훽6). +□ +We conclude this section by proving the extension of privacy amplification by shuffling (Theo- +rem 3.7). This proof is a direct reduction to the following result of Feldman, McMillan, and Talwar +[FMT22]. +Theorem 3.9 (Local Privacy Amplification by Shuffling [FMT22, Theorem 3.8]). Let 푚 ∈ Z+, let X be +the data universe, and let Y1, . . . , Y푚 be image spaces. Suppose for each 푖 ∈ [푚], we have a randomized +function 푅푖 : Y1 × · · · × Y푖−1 × X → Y푖 such that 푅푖 (푦,푎) is (휀0,훿0)-DP in the parameter 푎 ∈ X for every +fixed 푦 ∈ Y1 × · · · × Y푖−1. Consider 푅푚 ⊗ · · · ⊗ 푅1 : X푚 → Y1 × · · · × Y푚 defined by +(푅푚 ⊗ · · · ⊗ 푅1)(푥1, . . . , 푥푚) ≔ (푦1, . . . ,푦푚) +where we recursively define 푦푖 ≔ 푅푖 (푦1, . . . ,푦푖−1,푥푖). In addition, consider the random shuffle operator +푆 : X푚 → X푚 given by +푆(푥1, . . . ,푥푚) ≔ (푥휋 (1), . . . , 푥휋 (푚)) +where 휋 is a uniformly random permutation on [푚]. Then, for any 훿1 ∈ [2푒− +푚 +16푒휀0 , 1], the function 푅푚 ⊗ +· · · ⊗ 푅1 ◦ 푆 : X푚 → Y1 × · · · Y푚 is (휀1,훿1 + (푒휀1 + 1)(1 + 푒−휀0/2)푚훿0)-DP, where +휀1 = log +� +1 + 8푒휀0 − 1 +푒휀0 + 1 +�� +푒휀0 log(4/훿1) +푚 ++ 푒휀0 +푚 +�� += 푂 +� +휀0 · +� +log(1/훿1) +푚 +� +. +(6) +Proof of Theorem 3.7. Let푥 = ((푥1 +1, . . . , 푥푛 +1 ), . . . , (푥1 +푚, . . . ,푥푛 +푚)),푥 ′ = ((푥 ′1 +1, . . . , 푥 ′푛 +1), . . . , (푥 ′1 +푚, . . . , 푥 ′푛 +푚)) ∈ +(X푛)푚 be neighboring datasets and assume, without loss of generality, that they differ in only the first +entry of the first block. That is, 푥 푗 +푖 = 푥 ′푗 +푖 for all (푖, 푗) ≠ (1, 1). +Now, decompose the operator Π = Π1 ◦ Π−1 as follows. +Π1(푥)푗 +푖 ≔ +� +Π(푥)푗 +푖 +if 푗 = 1 +푥 푗 +푖 +otherwise +and Π−1(푥)푗 +푖 ≔ +� +푥 푗 +푖 +if 푗 = 1 +Π(푥)푗 +푖 +otherwise +In other words, Π1 applies the permutation 휋1 to the first row and leaves the remaining푛−1 rows fixed, +whereas Π−1 applies the permutations 휋2, . . . , 휋푛 to every row except the first. +We claim that +(퐿푚 ⊗ · · · ⊗ 퐿1 ◦ Π1)(푥) ∼휀′,훿′ (퐿푚 ⊗ · · · ⊗ 퐿1 ◦ Π1)(푥 ′) +(7) +with 휀′ = 푂(휀0 +� +log(1/훿)/푚) and 훿 ′ = 훿1 + 푂(훿0푚), as in the conclusion of Theorem 3.9. +To that end, consider the randomized function +푅푖 (푦,푎) ≔ 퐿푖(푦, (푎,푥2 +푖 , . . . , 푥푛 +푖 )) +for 푖 ∈ [푚]. Since (푎,푥2 +푖 , . . . , 푥푛 +푖 ) and (푎′,푥2 +푖 , . . . ,푥푛 +푖 ) are neighboring datasets for any 푎, 푎′ ∈ X, 푅푖 +must be (휀0,훿0)-DP in the parameter 푎 and hence 푅푚 ⊗ · · · ⊗ 푅1 ◦ 푆 is (휀′,훿 ′)-DP by Theorem 3.9. In +particular, since ˆ푥 ≔ (푥1 +1, . . . ,푥1 +푚) and ˆ푥 ′ ≔ (푥 ′1 +1, . . . , 푥 ′1 +푚) are neighbors, (푅푚 ⊗ · · · ⊗ 푅1 ◦ 푆)( ˆ푥) must +be (휀′, 훿 ′)-indistinguishable from (푅푚 ⊗ · · · ⊗ 푅1 ◦ 푆)( ˆ푥 ′). +23 + +Therefore, to prove our claim, it suffices to show that (퐿푚 ⊗· · ·⊗퐿1◦Π1)(푥) is identically distributed +to (푅푚 ⊗ · · · ⊗ 푅1 ◦ 푆)( ˆ푥), and likewise for (퐿푚 ⊗ · · · ⊗ 퐿1 ◦ Π1)(푥 ′) and (푅푚 ⊗ · · · ⊗ 푅1 ◦ 푆)( ˆ푥 ′). Indeed, +퐿푖 (푦, Π1(푥)푖) = 퐿푖 (푦, (푥1 +휋1(푖),푥2 +푖 , . . . , 푥푛 +푖 )) = 푅푖 (푦, 푥1 +휋1(푖)) +for all 푖. So, it follows by induction that +(퐿푚 ⊗ · · · ⊗ 퐿1 ◦ Π1)(푥) = (푅푚 ⊗ · · · ⊗ 푅1)(푥1 +휋1(1), . . . ,푥1 +휋1 (푚)) ∼ (푅푚 ⊗ · · · ⊗ 푅1 ◦ 푆)( ˆ푥). +Analogously, we get +(퐿푚 ⊗ · · · ⊗ 퐿1 ◦ Π1)(푥 ′) ∼ (푅푚 ⊗ · · · ⊗ 푅1 ◦ 푆)( ˆ푥 ′), +as desired. +We can now leverage the decomposition Π = Π1 ◦ Π−1 to prove the theorem. Fixing Π−1, Π−1(푥) +and Π−1(푥 ′) are neighboring datasets differing only on the first element of the first block. So, by the +claim that we proved above (Equivalence 7), which used only the fact that 푥 and 푥 ′ differ at 푥1 +1 ≠ 푥 ′1 +1, +we have that +(퐿푚 ⊗ · · · ⊗ 퐿1 ◦ Π1)(Π−1(푥)) ∼휀′,훿′ (퐿푚 ⊗ · · · ⊗ 퐿1 ◦ Π1)(Π−1(푥 ′)). +But Π−1 depends only on 휋2, . . . , 휋푛 and is, thus, is independent of 퐿푚 ⊗ · · · ⊗ 퐿1 ◦ Π1. Therefore, it +follows that +P[(퐿푚 ⊗ · · · ⊗ 퐿1 ◦ Π)(푥) ∈ 퐸] = E +Π−1[P[(퐿푚 ⊗ · · · ⊗ 퐿1 ◦ Π1)(Π−1(푥)) ∈ 퐸 | Π−1]] +≤ E +Π−1 +� +훿 ′ + 푒휀′P[(퐿푚 ⊗ · · · ⊗ 퐿1 ◦ Π1)(Π−1(푥 ′)) ∈ 퐸 | Π−1] +� += 훿 ′ + 푒휀′P[(퐿푚 ⊗ · · · ⊗ 퐿1 ◦ Π)(푥 ′) ∈ 퐸] +for any measurable 퐸. +□ +4 +Low-Bias Estimators for General Distributions +In this section, we describe and analyze algorithms for private estimation with low or no bias. In +Section 4.1, we provide some technical lemmata, which will be useful in our analysis. In Section 4.2, we +give three algorithms: an (휀, 0)-DP algorithm based on the clipped mean (Proposition 4.3), a (0,훿)-DP +algorithm based on a variant of the “name-and-shame” algorithm (Proposition 4.4), and an (휀,훿)-DP +algorithm obtained by combining the two (Proposition 4.5). By taking the best of the three resulting +bounds, we get Theorem 1.2. +4.1 +Lemmata +We will require the following technical lemmata. +Lemma 4.1. ∀휆 > 1 ∀푐 > 0 ∀푡 ∈ R, +max{0, 푡 − 푐} ≤ (휆−1)휆−1 +휆휆 ·푐휆−1 · |푡|휆. +Proof. If 푡 = 0, the claim holds as an equality. Now, we assume that 푡 ∈ R \ {0}. Define 푓 : R \ {0} → R +by 푓 (푡) = |푡|휆. Then 푓 ′(푡) = 휆 · |푡|휆−1 · sign(푡) and 푓 ′′(푡) = 휆(휆 − 1) · |푡|휆−2 ≥ 0 for all 푡 ∈ R \ {0}. Since +푓 is convex, +∀푎 ≥ 0 ∀푡 ∈ R \ {0}, 푓 (푡) ≥ 푓 (푎) + 푓 ′(푎) · (푡 − 푎) = 푎휆 + 휆 · 푎휆−1 · (푡 − 푎) = 휆 · 푎휆−1 · +� +푡 − 휆 − 1 +휆 +· 푎 +� +. +Taking the maximum over 푎 = 0 and 푎 = +푐휆 +휆−1 and rearranging yields the result. +□ +24 + +The following lemma decomposes the mean squared error of the clipped mean into the sum of the +sampling error and the (squared) population bias introduced (which is further bounded). +Lemma 4.2. Fix 휆 > 1 and 푎 < 푏. Let 푃 be a distribution with mean 휇(푃) ∈ (푎,푏). Let 휇[푎,푏](푃) ≔ +E +푋←푃 +� +clip[푎,푏](푋) +� +∈ [푎,푏]. Let 푋1, . . . ,푋푛 be independent samples from 푃. Then +E + +� +1 +푛 +푛 +� +푖 +clip[푎,푏](푋푖) − 휇(푃) +�2 +≤ +E +푋←푃 +� +(푋 − 휇(푃))2� +푛 ++ (휇[푎,푏](푃) − 휇(푃))2 +and +��휇[푎,푏](푃) − 휇(푃) +�� ≤ (휆 − 1)휆−1 +휆휆 +· +E +푋←푃 +� +|푋 − 휇(푃)|휆� +(min{휇(푃) − 푎,푏 − 휇(푃)})휆−1 ≤ 1 +휆 · +E +푋←푃 +� +|푋 − 휇(푃)|휆� +(min{휇(푃) − 푎,푏 − 휇(푃)})휆−1 . +Proof. We have +E + +� +1 +푛 +푛 +� +푖 +clip[푎,푏](푋푖) − 휇(푃) +�2 += E + +� +1 +푛 +푛 +� +푖 +clip[푎,푏](푋푖) − 휇[푎,푏](푃) +�2 ++ (휇[푎,푏](푃) − 휇(푃))2 += 1 +푛2 +푛 +� +푖 +E +�� +clip[푎,푏](푋푖) − 휇[푎,푏](푃) +�2� ++ (휇[푎,푏](푃) − 휇(푃))2 +≤ 1 +푛2 +푛 +� +푖 +E +�� +clip[푎,푏](푋푖) − 휇(푃) +�2� ++ (휇[푎,푏](푃) − 휇(푃))2 +≤ 1 +푛2 +푛 +� +푖 +E +� +(푋푖 − 휇(푃))2� ++ (휇[푎,푏](푃) − 휇(푃))2 += +E +푋←푃 +� +(푋 − 휇(푃))2� +푛 ++ (휇[푎,푏](푃) − 휇(푃))2. +The first inequality follows from the fact that +E +푋←푃 +� +(푋 − 휇(푃))2� += inf푢∈R E +푋←푃 +� +(푋 − 푢)2� +. The second +inequality follows from the fact that 휇(푃) ∈ [푎,푏] and, hence, (clip[푎,푏](푥) − 휇(푃))2 ≤ (푥 − 휇(푃))2 for +all 푥 ∈ R. +It remains to bound 휇[푎,푏](푃) − 휇(푃). We have +휇[푎,푏](푃) − 휇(푃) = +E +푋←푃 +� +clip[푎,푏](푋) − 푋 +� += +E +푋←푃[I[푋 > 푏](푏 − 푋) + I[푋 < 푎](푎 − 푋)] += +E +푋←푃[max{푎 − 푋, 0}] − +E +푋←푃[max{푋 − 푏, 0}]. +By Lemma 4.1, +0 ≤ +E +푋←푃[max{푋 − 푏, 0}] = +E +푋←푃[max{(푋 − 휇(푃)) − (푏 − 휇(푃)), 0}] +≤ +E +푋←푃 +� +(휆 − 1)휆−1 +휆휆 · (푏 − 휇(푃))휆−1 · |푋 − 휇(푃)|휆 +� +. +25 + +Similarly, +0 ≤ +E +푋←푃[max{푎 − 푋, 0}] ≤ +(휆 − 1)휆−1 +휆휆 · (휇(푃) − 푎)휆−1 · +E +푋←푃 +� +|푋 − 휇(푃)|휆� +. +Thus, +(휆 − 1)휆−1 +휆휆 · (푏 − 휇(푃))휆−1 · +E +푋←푃 +� +|푋 − 휇(푃)|휆� +≤ 휇[푎,푏](푃) − 휇(푃) ≤ +(휆 − 1)휆−1 +휆휆 · (휇(푃) − 푎)휆−1 · +E +푋←푃 +� +|푋 − 휇(푃)|휆� +. +Finally, note that (휆−1)휆−1 +휆휆 += �1 − 1 +휆 +�휆−1 · 1 +휆 ≤ exp(−1+1/휆) +휆 +≤ 1 +휆. +□ +4.2 +Algorithms +We first have a positive result based on clipping and adding noise, which satisfies pure DP. The clipped +and noised mean is folklore in differential privacy. Analyzing such a procedure with bounded moments +has been done in a few works [DJW13, BD14, KSU20]. These works generally set algorithm parameters +to achieve a prescribed bias, towards the goal of minimizing the overall error. As our goal is to explicitly +quantify the bias, we leave it as a free variable. +Proposition 4.3 (휀-DP Algorithm). For all 휀, 훽 > 0, 푎 < 푏, 휆 ≥ 2, and 푛 ∈ N, there exists an 휀-DP +algorithm 푀 : R푛 → R satisfying the following bias and accuracy properties. For any distribution 푃 on R +with 휇(푃) ≔ +E +푋←푃[푋] ∈ [푎,푏] and +E +푋←푃 +� +|푋 − 휇(푃)|휆� +≤ 1, we have +���� +E +푋←푃푛,푀[푀(푋)] − 휇(푃) +���� ≤ 훽, +E +푋←푃푛,푀 +� +(푀(푋) − 휇(푃))2� +≤ 1 +푛 + 훽2 + +2 +휀2푛2 +� +푏 − 푎 + +2 +훽1/(휆−1) +�2 +. +Proof. Fix ˆ푎 = 푎 − 훽−1/(휆−1) and ˆ푏 = 푏 + 훽−1/(휆−1). Our algorithm 푀 : R푛 → R is defined as +푀(푥) = +� +1 +푛 +푛 +� +푖 +clip[ ˆ푎, ˆ푏](푥푖) +� ++ Lap +� ˆ푏 − ˆ푎 +휀푛 +� +. +The properties of the Laplace distribution ensure that 푀 satisfies 휀-DP, as the sensitivity of �휇(푥) ≔ +1 +푛 +�푛 +푖 clip[ ˆ푎, ˆ푏](푥푖) is +ˆ푏− ˆ푎 +푛 +(Lemma 2.6). +It only remains to analyze the bias and accuracy. Fix an arbitrary distribution 푃 which satisfies the +conditions in the proposition statement. By Lemma 4.2, the bias satisfies +E +푋←푃푛,푀[푀(푋)] − 휇(푃) = +E +푋←푃푛[�휇(푋)] − 휇(푃) += +E +푋←푃 +� +clip[ ˆ푎, ˆ푏] (푋) − 푋 +� +≤ (휆 − 1)휆−1 +휆휆 +· +E +푋←푃 +� +|푋 − 휇(푃)|휆� +(min{휇(푃) − ˆ푎, ˆ푏 − 휇(푃)})휆−1 +≤ 1 · +1 +(min{푎 − ˆ푎, ˆ푏 − 푏})휆−1 += 훽, +26 + +and the mean squared error satisfies +E +푋1,···,푋푛←푃,푀 +� +(푀(푋) − 휇(푃))2� += +E +푋 ←푃푛 +휉←Lap( 푏−푎 +휀푛 ) +� +(�휇(푋) + 휉 − 휇(푃))2� += +E +푋←푃푛 + +� +1 +푛 +푛 +� +푖 +clip[ ˆ푎, ˆ푏] (푋푖) − 휇(푃) +�2 ++ +E +휉←Lap +� ˆ푏− ˆ푎 +휀푛 +� +� +휉2� +≤ +E +푋←푃 +� +(푋 − 휇(푃))2� +푛 ++ 훽2 + 2 +� ˆ푏 − ˆ푎 +휀푛 +�2 +≤ 1 +푛 + 훽2 + +2 +휀2푛2 +� +푏 − 푎 + +2 +훽1/(휆−1) +�2 +. +Our proof is complete. +□ +Next, we give an algorithm based on the folklore “name-and-shame” procedure, which is (0, 훿)-DP. +The name-and-shame procedure is generally phrased as randomly selecting a point from a dataset and +outputting it, sans any further privacy protection. It is most commonly used as an illustration of which +values of 훿 may or may not be meaningful when it comes to informal uses of the word “privacy”, and +not as a serious algorithm. However, we note that such a procedure gives an exactly unbiased estimate +of the mean, which the previous (휀, 0)-DP was unable to do. We thus use it to design an unbiased +algorithm for mean estimation, albeit at a high price in the dependence on 훿, which we recall is usually +chosen to be very small. +Proposition 4.4 ((0,훿)-DP Algorithm). For all 훿 ∈ (0, 1] and 푛 ∈ N, there exists a (0,훿)-DP algorithm +푀 : R푛 → R satisfying the following bias and accuracy properties. For any distribution 푃 on R, we have +E +푋←푃푛,푀[푀(푋)] = 휇(푃), +(i.e., 푀 is unbiased) +E +푋←푃푛,푀 +� +(푀(푋) − 휇(푃))2� += +E +푋←푃 +� +(푋 − 휇(푃))2� ++ (1 − 훿) · 휇(푃)2 +훿 · 푛 +. +Proof. Define a randomized algorithm 퐴 : R → R as follows. +퐴(푥) = +� +0 +with probability 1 − 훿 +푥 +훿 +with probability 훿 +Define 푀 : R푛 → R by +푀(푥) = 1 +푛 +푛 +� +푖 +퐴(푥푖), +where each instantiation of 퐴 is independent. Since 퐴 satisfies local (0,훿)-DP, 푀 satisfies (0, 훿)-DP. +Since 퐴 is unbiased (i.e., ∀푥 ∈ R E +퐴[퐴(푥)] = 푥), so is 푀. Finally, we calculate the mean squared error: +E +푋←푃푛,푀 +� +(푀(푋) − 휇(푃))2� += 1 +푛 +E +푋←푃,퐴 +� +(퐴(푋) − 휇(푃))2� += +E +푋←푃,퐴 +� +퐴(푋)2� +− 휇(푃)2 +푛 +27 + += +E +푋←푃 +� +0 + 훿 · (푋/훿)2� +− 휇(푃)2 +푛 += +E +푋←푃 +� +푋 2� +− 훿 · 휇(푃)2 +훿 · 푛 += +E +푋←푃 +� +(푋 − 휇(푃))2� ++ (1 − 훿) · 휇(푃)2 +훿 · 푛 +. +We have the required result. +□ +We can combine both of these methods to obtain a new algorithm for (휀,훿)-DP mean estimation. +Essentially, it decomposes a sample into the non-tail and tail components, releasing the former via the +(휀, 0)-DP clip-and-noise method, and the latter via the (0,훿)-DP name-and-shame approach. Note that +we must consider a higher moment in our assumption about the unknown distribution 푃. +Proposition 4.5 ((휀,훿)-DP Algorithm). For all 휀 > 0, 훿 ∈ (0, 1], 휓 > 0, 휆 > 2, 푎 < 푏, and 푛 ∈ N, there +exists an (휀,훿)-DP algorithm 푀 : R푛 → R satisfying the following bias and accuracy properties. For any +distribution 푃 on R with 휇(푃) ≔ +E +푋←푃[푋] ∈ [푎,푏] and +E +푋←푃 +� +(푋 − 휇(푃))2� +≤ 1 and +E +푋←푃 +� +|푋 − 휇(푃)|휆� +≤ +휓휆, we have +E +푋←푃푛,푀[푀(푋)] = 휇(푃), +(i.e., 푀 is unbiased) +E +푋←푃푛,푀 +� +(푀(푋) − 휇(푃))2� +≤ 2 +푛 + 4(푏 − 푎)2 +푛2휀2 ++ 24휓2 +푛2휀2 · +� 푛휀2 +4휆훿 +�2/휆 +. +Proof. Fix 푐 > 0 and let ˆ푎 = 푎 − 푐 and ˆ푏 = 푏 + 푐. Define a randomized algorithm 퐴 : R → R as follows. +퐴(푥) = +� +0 +with probability 1 − 훿 +푥 +훿 +with probability 훿 +Define 푀 : R푛 → R by +푀(푥) = +� +1 +푛 +푛 +� +푖 +clip[ ˆ푎, ˆ푏](푥푖) +� ++ Lap +� ˆ푏 − ˆ푎 +푛휀 +� +���������������������������������������������������������������������������������� +(휀, 0)-DP ++ +� +1 +푛 +푛 +� +푖 +퐴 +� +푥푖 − clip[ ˆ푎, ˆ푏](푥푖) +�� +������������������������������������������������������������������ +(0, 훿)-DP +, +where the Laplace noise and all of the instantiations of 퐴 are independent. That is, 푀 decomposes each +푥푖 into a clipped component and a tail component. We release the clipped component using Laplace +noise addition (as in Proposition 4.3), and the tail component by exploiting 훿 (as in Proposition 4.5). +Since both parts are unbiased, 푀 is unbiased – i.e. ∀푥 ∈ R푛 E +푀[푀(푥)] = 1 +푛 +�푛 +푖 푥푖. +By composition and postprocessing, 푀 satisfies (휀,훿)-DP. +Now, we bound the mean squared error. Define 휇[ ˆ푎, ˆ푏](푃) ≔ +E +푋←푃 +� +clip[ ˆ푎, ˆ푏] (푋) +� +. We have: +E +푋←푃푛,푀 +� +(푀(푋) − 휇(푃))2� += +E +푋 ←푃푛 +휉←Lap +� ˆ푏− ˆ푎 +푛휀 +� +,퐴 + +� +1 +푛 +�푛 +푖 clip[ ˆ푎, ˆ푏] (푋푖) + 휉 ++ 1 +푛 +�푛 +푖 퐴 +� +푋푖 − clip[ ˆ푎, ˆ푏](푋푖) +� +− 휇(푃) +�2 +28 + += +E +푋←푃푛,퐴 + +� +1 +푛 +�푛 +푖 clip[ ˆ푎, ˆ푏](푋푖) − 휇[ ˆ푎, ˆ푏](푃) ++ 1 +푛 +�푛 +푖 퐴 +� +푋푖 − clip[ ˆ푎, ˆ푏](푋푖) +� +− (휇(푃) − 휇[ ˆ푎, ˆ푏](푃)) +�2 ++ +E +휉←Lap +� ˆ푏− ˆ푎 +푛휀 +� +� +휉2� +≤ 2 +푛 · +E +푋←푃 +�� +clip[ ˆ푎, ˆ푏](푋) − 휇[ ˆ푎, ˆ푏](푃) +�2� ++ 2 +� ˆ푏 − ˆ푎 +푛휀 +�2 ++ 2 +푛 · +E +푋←푃,퐴 +�� +퐴 +� +푋 − clip[ ˆ푎, ˆ푏](푋) +� +− +� +휇(푃) − 휇[ ˆ푎, ˆ푏](푃) +��2� +. +(8) +The final inequality uses the fact that for independent mean-zero random variables 푈 and 푉 , we have +E +� +(푈 + 푉 )2� += E +� +푈 2� ++ E +� +푉 2� +. For the terms that are not independent, we apply the inequality +E +� +(푈 + 푉 )2� +≤ 2E +� +푈 2� ++ 2E +� +푉 2� +. +Since 휇(푃) ≔ +E +푋←푃[푋] ∈ [푎,푏] ⊂ [ ˆ푎, ˆ푏], we have +E +푋←푃 +�� +clip[ ˆ푎, ˆ푏](푋) − 휇[ ˆ푎, ˆ푏](푃) +�2� +≤ +E +푋←푃 +�� +clip[ ˆ푎, ˆ푏](푋) − 휇(푃) +�2� +≤ +E +푋←푃 +� +(푋 − 휇(푃))2� +≤ 1. +Finally we bound the last term: +E +푋←푃,퐴 +�� +퐴 +� +푋 − clip[푎,푏](푋) +� +− +� +휇(푃) − 휇[ ˆ푎, ˆ푏](푃) +��2� +≤ +E +푋←푃,퐴 +�� +퐴 +� +푋 − clip[ ˆ푎, ˆ푏](푋) +��2� += (1 − 훿) · 0 + 훿 · +E +푋←푃 +��1 +훿 +� +푋 − clip[ ˆ푎, ˆ푏](푋) +��2� += 1 +훿 · +E +푋←푃 +�� +푋 − clip[ ˆ푎, ˆ푏] (푋) +�2� += 1 +훿 · +E +푋←푃 +�� +(푋 − 휇(푃))− +clip[ ˆ푎−휇(푃), ˆ푏−휇(푃)] (푋 − 휇(푃)) +�2� +≤ 1 +훿 · +E +푋←푃 +�� +(푋 − 휇(푃)) − clip[−푐,푐](푋 − 휇(푃)) +�2� += 1 +훿 · +E +푋←푃 +� +(max{0, |푋 − 휇(푃)| − 푐})2� +, +where the final inequality holds because 푐 = ˆ푏 −푏 ≤ ˆ푏 − 휇(푃) and 푐 = 푎 − ˆ푎 ≤ 휇(푃) − 푎. By Lemma 4.1, +max{0, |푋 − 휇(푃)| − 푐} ≤ +1 +휆/2 · |푋 − 휇(푃)|휆/2 +푐휆/2−1 +. +Thus, +E +푋←푃 +� +(max{0, |푋 − 휇(푃)| − 푐})2� +≤ +� +1 +(휆/2) · 푐휆/2−1 +�2 +· +E +푋←푃 +� +|푋 − 휇(푃)|휆� +≤ 4 +휆2 · 휓휆 +푐휆−2 . +Now, we set parameters and assemble the bound from Inequality 8: +E +푋←푃푛,푀 +� +(푀(푋) − 휇(푃))2� +≤ 2 +푛 + 2 +�푏 − 푎 + 2푐 +푛휀 +�2 ++ +� 2 +푛 · 1 +훿 · 4 +휆2 · 휓휆 +푐휆−2 +� +29 + +≤ 2 +푛 + 4(푏 − 푎)2 +푛2휀2 ++ 16 +푛2휀2 · 푐2 + 8휓휆 +푛훿휆2 · (푐2)1−휆/2 += 2 +푛 + 4(푏 − 푎)2 +푛2휀2 ++ 16휓2 +푛2휀2 · +� 푛휀2 +4휆훿 +�2/휆 +· +� +휆 +휆 − 2 +�1−2/휆 +≤ 2 +푛 + 4(푏 − 푎)2 +푛2휀2 ++ 24휓2 +푛2휀2 · +� 푛휀2 +4휆훿 +�2/휆 +, +where the final equality follows from setting 푐 = +� +푛휀2휓휆 (휆−2) +4휆2훿 +�1/휆 +to minimize the expression, and the +final inequality follows from the fact that +� +휆 +휆−2 +�1−2/휆 += (1 − 2/휆)−1+2/휆 ≤ exp(exp(−1)) < 3 +2. +□ +Combining Propositions 4.3, 4.4, and 4.5 yields Theorem 1.2. +5 +Unbiased Estimators for Symmetric Distributions +We now present our unbiased private mean estimation algorithm for symmetric distributions over R +that are weakly concentrated, i.e., those that have a bounded second moment. +Definition 5.1 (Symmetric Distribution). We say that a distribution 푃 on R is symmetric if there exists +some 휇(푃) ∈ R, such that if 푋 ← 푃, then 푋 − 휇(푃) and 휇(푃) − 푋 have the same distribution. That is, 푃 +is symmetric iff +∃휇(푃) ∈ R, ∀푥 ∈ R, +P +푋←푃[푋 − 휇(푃) ≤ 푥] = +P +푋←푃[휇(푃) − 푋 ≤ 푥]. +The value 휇(푃) is called the center of the distribution 푃. +Note that the center of the distribution is unique and coincides with the mean and the median +(whenever these two quantities are well-defined). +Our algorithm is based on the approach of Karwa and Vadhan [KV18], but with some modifications +to ensure unbiasedness. First, we obtain a coarse estimate of the mean, and then we use this coarse +estimate to perform clipping to obtain a precise estimate via noise addition. The key observation is that, +if the coarse estimate we use for clipping is unbiased and symmetric (and also independent from the +data used in the second step), then the clipping does not introduce bias. We obtain the coarse estimate +via a DP histogram, where each bucket in the histogram is an interval on the real line. To ensure that +this is unbiased and symmetric, we simply need to apply a random offset to the bucket intervals. +5.1 +Coarse Unbiased Estimation +Our coarse estimator is similar to that of Karwa and Vadhan [KV18], and uses stability-basedhistograms +(Lemma 2.7) to ensure privacy. The key modification we make to ensure unbiasedness is adding a +random offset to the histogram bins. +We define roundZ : R → Z to be the function that rounds real numbers to the nearest integer. I.e., +for any 푥 ∈ R, we have 푥 ∈ [roundZ(푥) − 1/2, roundZ(푥) + 1/2). +It is easy to verify that our coarse estimator is private. Similar to previous work (see, e.g., [Vad17]), +privacy follows from the privacy of the stable histogram algorithm, plus postprocessing via argmax. +Proposition 5.2 (Privacy of DPUCoarse). Algorithm 1 (DPUCoarse휀,훿) satisfies (휀,훿)-DP. +30 + +Algorithm 1: Unbiased DP Coarse Estimator DPUCoarse휀,훿 (푥) +Input: Dataset 푥 = (푥1, . . . ,푥푛) ∈ R푛. +Output: Estimate �휇 ∈ R ∪ {⊥}. +Let 푇 be uniform on the interval [−1/2, +1/2]. +Let 퐾 = {roundZ(푥푖 −푇) : 푖 ∈ [푛]} ⊂ Z. +For each 푘 ∈ 퐾, sample 휉푘 ← Lap(2/휀) independently. +If max푘∈퐾 |{푖 ∈ [푛] : roundZ(푥푖 −푇) = 푘}| + 휉푘 ≤ 2 + 2 log(1/훿) +휀 +, output ⊥. +Otherwise, output 푇 + arg max푘∈퐾 |{푖 ∈ [푛] : roundZ(푥푖 −푇) = 푘}| + 휉푘. +Proof. We will prove that DPUCoarse휀,훿 satisfies (휀/2,훿/2푒휀/2)-DP with respect to addition or removal +of one element of the dataset. By group privacy (Lemma 2.4), this implies (휀,훿)-DP for replacement of +an element. +Consider a fixed pair of datasets 푥 and 푥 ′ = 푥−푖∗, where 푥 ′ is 푥 with 푥푖∗ removed for some 푖∗ ∈ [푛]. +For the privacy analysis, we also consider the offset푇 to be fixed – i.e.,푇 is not needed to ensure privacy. +By postprocessing, we can consider an algorithm that outputs more information. Specifically, we +can assume that for each 푘 ∈ Z, the algorithm outputs +휈푘 (푥) ≔ +� +max +� +|{푖 ∈ [푛] : roundZ(푥푖 −푇) = 푘}| + 휉푘 − 2 − 2 log(1/훿) +휀 +, 0 +� +if 푘 ∈ 퐾 +0 +if 푘 ∈ Z \ 퐾 +� +. +Note that only finitely many of these 휈푘 (푥) values will be nonzero, so the algorithm can output a +compressed version of this infinite vector of values. We can obtain the true output of DPUCoarse휀,훿 +by taking the argmax of this vector or outputting ⊥ if this vector is all zeros. The advantage of this +perspective is that each 휈푘 (푥) is independent, as it depends only on the noise 휉푘 (the input 푥 and offset +푇 are fixed). +The output distributions on the neighboring inputs are the same except for one 휈푘 (푥) ≁ 휈푘 (푥 ′), +namely푘 = roundZ(푥푖∗ −푇). Thus, we must simply show that this value satisfies (휀/2,훿/2푒휀/2)-DP. That +is, we must show 휈푘 (푥) ∼휀/2,훿/2푒휀/2 휈푘 (푥 ′), where휈푘 (푥) and휈푘 (푥 ′) denote the relevant random variables +on the two different inputs. There are two cases to consider: |{푖 ∈ [푛] : roundZ(푥푖 − 푇) = 푘}| = 1 and +|{푖 ∈ [푛] : roundZ(푥푖 − 푇) = 푘}| ≥ 2. (Note that |{푖 ∈ [푛] : roundZ(푥푖 − 푇) = 푘}| = 0 is ruled out +because 푘 = roundZ(푥푖∗ −푇).) +Suppose |{푖 ∈ [푛] : roundZ(푥푖 − 푇) = 푘}| = 1. Then 휈푘 (푥 ′) = 0 deterministically. Therefore, it +suffices to prove that P[휈푘 (푥) = 0] ≥ 1 − 훿/2푒휀/2. We have +P[휈푘 (푥) ≠ 0] = P +� +|{푖 ∈ [푛] : roundZ(푥푖 −푇) = 푘}| + 휉푘 − 2 − 2 log(1/훿) +휀 +> 0 +� += P +� +1 + 휉푘 − 2 − 2 log(1/훿) +휀 +> 0 +� += P +� +휉푘 > 1 + 2 log(1/훿) +휀 +� += 1 +2 exp +� +−휀 +2 · +� +1 + 2 log(1/훿) +휀 +�� += +훿 +2푒휀/2, +31 + +where the penultimate equality follows from the fact that 휉푘 ← Lap(2/휀) (Lemma 2.6). +Now suppose |{푖 ∈ [푛] : roundZ(푥푖 − 푇) = 푘}| ≥ 2. Then 휈푘 (푥) and 휈푘 (푥 ′) are postprocessings of +|{푖 ∈ [푛] : roundZ(푥푖 − 푇) = 푘}| + 휉푘 and, respectively, |{푖 ∈ [푛] : roundZ(푥푖 − 푇) = 푘}| − 1 + 휉푘. Thus, +by the properties of Laplace noise, we have 푒−휀/2P[휈푘 (푥 ′) ∈ 푆] ≤ P[휈푘 (푥) ∈ 푆] ≤ 푒휀/2P[휈푘 (푥 ′) ∈ 푆] for +all 푆, as required. +□ +Now we turn to the utility analysis, which consists of two parts. First, conditioned on not outputting +⊥, the estimate is symmetric and unbiased (Proposition 5.3). Second, we show that the probability of +outputting ⊥ is low for appropriately concentrated distributions, and that the MSE is bounded, as well. +(Proposition 5.6). +Proposition 5.3 (Conditional Symmetry of DPUCoarse). Let 푃 be a symmetric distribution with center +휇(푃). Let 푋1, · · · ,푋푛 ∈ R be independent samples from 푃. Let �휇 = DPUCoarse휀,훿 (푋1, · · · , 푋푛). Let 푄 +be the distribution of �휇 conditioned on �휇 ≠ ⊥. Then 푄 is symmetric with the same center as 푃 – i.e., +휇(푃) = 휇(푄). +Unlike the static histogram bucket approach of prior work, the introduction of the uniformly ran- +dom offset 푇 ∈ [±1/2] in Algorithm 1 endows DPUCoarse휀,훿 with equivalence under translation. +Recall that we use “∼” to denote distributional equivalence. +Lemma 5.4. For any 푥 = (푥1, . . . , 푥푛) ∈ R푛 and 푐 ∈ R, we have +DPUCoarse휀,훿 (푥 + 푐) ∼ DPUCoarse휀,훿 (푥) + 푐 +where ⊥ + 푐 ≔ ⊥ and 푥 + 푐 ≔ (푥1 + 푐, . . . ,푥푛 + 푐). +Proof. It will be easier to proceed by rewriting Algorithm 1 in a non-algorithmic form. To that end, we +define the following notations. +• For 푟 ∈ R, set �푝푟 (푥) ≔ 1 +푛 |푖 ∈ [푛] : 푥푖 ∈ [푟 ± 1/2)| and sample 푇 ← U[±1/2], �푝푟 (푥) ← �푝푟 (푥) + +Lap(0, 2/(휀푛)) such that 푇 and {�푝푟 (푥)}푟 ∈R are all mutually independent. +• For 푡 ∈ R and 푆 ⊆ R, we define 푆 + 푡 ≔ {푠 + 푡 : 푠 ∈ 푆}, and denote by 푆 +푇 the distribution over +the set of sets {푆 + 푡 : 푡 ∈ [±1/2]} induced by the randomness of 푇. +• Set 푅(푥) ≔ {푟 ∈ Z + 푇 : �푝푟 (푥) > 0} and put 푅∗(푥) ≔ arg max푟 ∈푅(푥) �푝푟 (푥), provided there is an +푟 ∈ 푅(푥) for which �푝푟 (푥) > 2 log(2/훿) +휀푛 ++ 2 +푛 ≕ 휂, otherwise 푅∗(푥) ≔ ⊥. +Essentially, we have reparameterizedthe terms of DPUCoarse휀,훿 (푥) so that 푅(푥) = 퐾+푇 and �푝푘+푇 (푥) = +1 +푛 |{푖 ∈ [푛] : roundZ(푥푖 −푇) = 푘}| hold, so it follows that 푅∗(푥) ∼ DPUCoarse휀,훿 (푥). +Now, notice that +�푝푟 (푥 + 푐) = 1 +푛 |{푖 ∈ [푛] : 푥푖 + 푐 ∈ [푟 ± 1/2)}| = 1 +푛 |{푖 ∈ [푛] : 푥푖 ∈ [푟 − 푐 ± 1/2)}| = �푝푟−푐 (푥), +so in particular, we have that �푝푟 (푥 + 푐) is identically distributed to �푝푟−푐 (푥) for any 푟 ∈ R. Moreover, +Z +푇 is identically distributed to Z +푇 − 푐, so it follows that +푅(푥 + 푐) = {푟 ∈ Z + 푇 : �푝푟 (푥 + 푐) > 0} += {푟 ∈ Z + 푇 : �푝푟−푐 (푥) > 0} += {푟 ∈ Z + 푇 − 푐 : �푝푟 (푥) > 0} + 푐 +32 + +∼ {푟 ∈ Z + 푇 : �푝푟 (푥) > 0} + 푐 += 푅(푥) + 푐. +As 푇 and the Laplace noise were all sampled in a mutually independent manner, these distributional +equivalences hold jointly, i.e., ((�푝푟 (푥 + 푐))푟 ∈R, 푅(푥 + 푐)) ∼ ((�푝푟−푐(푥))푟 ∈R, 푅(푥) + 푐). Hence, +푅∗(푥 + 푐) = +� +arg max푟 ∈푅(푥+푐) �푝푟 (푥 + 푐) +if ∃푟 ∈ 푅(푥 + 푐), �푝푟 (푥 + 푐) > 휂 +⊥ +otherwise. +∼ +� +arg max푟 ∈푅(푥)+푐 �푝푟−푐 (푥) +if ∃푟 ∈ 푅(푥) + 푐, �푝푟−푐 (푥) > 휂 +⊥ +otherwise. += +� +arg max푟′∈푅(푥) �푝푟′(푥) + 푐 +if ∃푟 ′ ∈ 푅(푥), �푝푟′(푥) > 휂 +⊥ +otherwise. +(푟 = 푟 ′ + 푐) += 푅∗(푥) + 푐. +(⊥ = ⊥ + 푐) +The equivalence of 푅∗(푥) and DPUCoarse휀,훿 (푥) gives us the desired result. +□ +Lemma 5.5. For any 푥 = (푥1, . . . , 푥푛) ∈ R푛, we have +DPUCoarse휀,훿 (−푥) ∼ −DPUCoarse휀,훿 (푥) +where −⊥ ≔ ⊥ and −푥 ≔ (−푥1, . . . , −푥푛). +Proof. Recall the notation from the proof of Lemma 5.4. Then, we have that +�푝푟 (−푥) = 1 +푛 |{푖 ∈ [푛] : −푥푖 ∈ [푟 ± 1/2)}| = 1 +푛 |{푖 ∈ [푛] : 푥푖 ∈ [−푟 ± 1/2)}| = �푝−푟 (푥), +so it follows that for any 푟 ∈ R, �푝푟 (−푥) is identically distributed to �푝−푟 (푥). Moreover, +푅(−푥) = {푟 ∈ Z + 푇 : �푝푟 (−푥) > 0} += {푟 ∈ Z + 푇 : �푝−푟 (푥) > 0} += −{푟 ∈ −(Z +푇) : �푝푟 (푥) > 0} +∼ −{푟 ∈ Z + 푇 : �푝푟 (푥) > 0} += −푅(푥). +As 푇 and all of the Laplace noise is sampled independently, these distributional equivalences hold si- +multaneously, namely ((�푝푟 (−푥))푟 ∈R, 푅(−푥)) ∼ ((�푝−푟 (푥))푟 ∈R, −푅(푥)). Combining these with −⊥ = ⊥, +we obtain 푅∗(−푥) ∼ −푅∗(푥) by the same argument as the one we used to prove Lemma 5.4. +□ +Proof of Proposition 5.3. Due to Lemma 5.4, we may assume without loss of generality that 푃 has center +0. Then, since 푃 is symmetric, 푋 ← 푃푛 is identically distributed to −푋. So, for any 푎 ≥ 0, +P +푋←푃푛 +� +DPUCoarse휀,훿 (푋) ∈ [푎, ∞) +� += +E +푋←푃푛 +� +P +� +DPUCoarse휀,훿 (푋) ∈ [푎, ∞)|푋 +�� += +E +푋←푃푛 +� +P +� +DPUCoarse휀,훿 (−푋) ∈ (−∞, −푎]|푋 +�� +(Lemma 5.5) += +P +푋←푃푛 +� +DPUCoarse휀,훿 (−푋) ∈ (−∞, −푎] +� += +P +푋←푃푛 +� +DPUCoarse휀,훿 (푋) ∈ (−∞, −푎] +� +. +(푋 ∼ −푋) +33 + +In particular, +푄([푎, ∞)) = P[�휇 ∈ [푎, ∞)] +P[�휇 ≠ ⊥] += P[�휇 ∈ (−∞, −푎]] +P[�휇 ≠ ⊥] += 푄((−∞,푎]) +for all 푎 ≥ 0, so 푄 must also be symmetric with center 0. +□ +Proposition 5.6 (Accuracy of DPUCoarse). Let 푃 be a distribution over R with mean 휇(푃) and variance +E +푋←푃 +� +(푋 − 휇(푃))2� +< 1/64 and +E +푋←푃 +� +|푋 − 휇(푃)|휆� +≤ 휓휆 for some 휆 ≥ 2 and 휓 > 0. Let 푋 = (푋1, . . . , 푋푛) +be independent samples from 푃, and �휇 ← DPUCoarse휀,훿 (푋). If 푛 ≥ 7 + 7 +휀 log(1/훿), then +P[�휇 ≠ ⊥ ∧ |�휇 − 휇(푃)| ≤ 1] ≥ 1 − exp(−푛/128) − 푛 +2 exp(−푛휀/16) +and +E +� +I[�휇 ≠ ⊥] · |�휇 − 휇(푃)|휆� +≤ 1 +2 + 푛 · 2휆−1 · 휓휆. +In particular, for any 훾 > 0, to ensure P[�휇 ≠ ⊥ ∧ |�휇 − 휇(푃)| ≤ 1] ≥ 1 − 훾, it suffices to have +푛 ≥ max +� +7 + 7 +휀 log(1/훿), 128 log(2/훾), 16 +휀 log(푛/훾) +� += 푂(log(푛/훾훿)/휀). +Proof. By Lemma 5.4, we assume, without loss of generality, that 휇(푃) = 0. +Let 푋 ← 푃푛 be the input to DPUCoarse휀,훿 and let �휇 ∈ R ∪ {⊥} be the output. Let 푇 ∈ [± 1 +2], 퐾 ⊂ Z, +and 휉푘 ← Lap(2/휀) be as in the algorithm (and define 휉푘 = 0 for 푘 ∉ 퐾). For 푘 ∈ Z, define +퐶푘 ≔ |{푖 ∈ [푛] : roundZ(푋푖 −푇) = 푘}|. +Recall, from Algorithm 1, that 푘 ∈ 퐾 +⇐⇒ +퐶푘 ≥ 1 and that �휇 = ⊥ +⇐⇒ +max푘∈퐾 퐶푘 + 휉푘 ≤ +2 + 2 +휀 log(1/훿) and, otherwise, �휇 = 푇 + arg max푘∈퐾 퐶푘 + 휉푘. +We begin by showing �휇 ∈ [±1] with high probabiliy. +Define +푘+ = roundZ +�1 +2 −푇 +� +and +푘− = roundZ +� +−1 +2 −푇 +� +. +Note that 푘+ = 푘− + 1 and 푘+,푘− ∈ (−푇 − 1, −푇 + 1]. Thus, if arg max푘∈퐾 퐶푘 + 휉푘 ∈ {푘+,푘−} (and +max푘∈퐾 퐶푘 + 휉푘 > 2 + 2 +휀 log(1/훿)), then �휇 ∈ {푘+ + 푇,푘− + 푇 } ⊂ (−1, +1], as required. In other words, it +suffices for us to show that, with high probability, 퐶푘+ + 휉푘+ or 퐶푘− + 휉푘− are large and all other 퐶푘 + 휉푘 +values are small. +For any 푥 ∈ �− 1 +2, +1 +2 +�, we have roundZ(푥 −푇) ∈ {푘−,푘+}. Thus +퐶푘− + 퐶푘+ ≥ +푛 +� +푖 +I +� +푋푖 ∈ +� +−1 +2, +1 +2 +�� +. +By our assumption that E +푋←푃 +� +(푋 − 휇(푃))2� +< 1/64 and Chebyshev’s inequality, P +푋←푃 +� +푋 ∈ �− 1 +2, +1 +2 +�� +≥ +15 +16. By Hoeffding’s inequality, for all 푠 ≥ 0, we have +P +푋←푃푛 +��푛 +푖 I +� +푋푖 ∈ �− 1 +2, +1 +2 +�� +≥ 15 +16푛 − 푠 +� +≥ 1 − +exp(−2푠2/푛). In particular, +P +푋←푃푛 +� 푛 +� +푖 +I +� +푋푖 ∈ +� +−1 +2, +1 +2 +�� +≥ 7 +8푛 +� +≥ 1 − exp(−푛/128). +34 + +This means P +� +퐶푘− + 퐶푘+ ≥ 7 +8푛 +� +≥ 1 − exp(−푛/128). Define 푘∗ ≔ arg max푘∈{푘+,푘− } 퐶푘 (breaking ties +arbitrarily). If 퐶푘− + 퐶푘+ ≥ 7 +8푛, then 퐶푘∗ ≥ +7 +16푛, while 퐶푘 ≤ 1 +8푛 for all 푘 ∉ {푘+,푘−}. Thus +P +� +퐶푘∗ ≥ 7 +16푛 ∧ +max +푘∈퐾\{푘+,푘− }퐶푘 ≤ 1 +8푛 +� +≥ 1 − exp(−푛/128). +The next step is to analyze the noise. For all 푘 ∈ 퐾 and 푟 ≥ 0, P[휉푘 ≥ 푟] = P[휉푘 ≤ −푟] = 1 +2푒−푟휀/2. +Note that |퐾| ≤ 푛. Setting 푟 = 푛/8 and taking a union bound over 푘 ∈ 퐾, we have +P +� +휉푘∗ ≥ −푛 +8 ∧ +max +푘∈퐾\{푘+,푘− } 휉푘 ≤ 푛 +8 +� +≥ 1 − 푛 +2 exp(−푛휀/16). +Combining the high probability bounds on the noise bound and the data, we have +P +� +퐶푘∗ + 휉푘∗ ≥ 5 +16푛 ∧ +max +푘∈퐾\{푘+,푘− }퐶푘 + 휉푘 ≤ 1 +4푛 +� +≥ 1 − exp(−푛/128) − 푛 +2 exp(−푛휀/16). +Since +5 +16푛 ≥ 2 + 2 +휀 log(1/훿), the event 퐶푘∗ + 휉푘∗ ≥ +5 +16푛 ∧ max푘∈퐾\{푘+,푘− } 퐶푘 + 휉푘 ≤ +1 +4푛 implies �휇 ∈ +{푇 + 푘+,푇 + 푘−} ⊂ [−1, +1], as required. +Finally, we bound E +� +I[�휇 ≠ ⊥] · |�휇 − 휇(푃)|휆� +. Observe that �휇 = 푇 +푘 for some 푇 ∈ [−1/2, +1/2] and +푘 = roundZ(푋푖 − 푇) for some 푖 ∈ [푛]. Thus, |�휇 − 푋푖| ≤ 1/2 for some 푖 ∈ [푛] and, hence, |�휇 − 휇(푃)| ≤ +1/2 + max푖∈[푛] |푋푖 − 휇(푃)|. It follows that +E +� +I[�휇 ≠ ⊥] · |�휇 − 휇(푃)|휆� +≤ E +��1 +2 + max +푖∈[푛] |푋푖 − 휇(푃)| +�휆� +(A) +≤ E +�1 +2 + 2휆−1 · max +푖∈[푛] |푋푖 − 휇(푃)|휆 +� +(B) +≤ 1 +2 + 2휆−1 · +� +푖∈[푛] +E +� +|푋푖 − 휇(푃)|휆� +≤ 1 +2 + 푛 · 2휆−1 · 휓휆, +where Inequality A follows from the fact that ∀푝 ≥ 1 ∀푥,푦 ≥ 0, +(푥 + 푦)푝 ≤ (푥푝 + 푦푝) · 2푝−1, and +Inequality B holds because the maximum among a set of non-negative real numbers should be at most +the sum of those numbers. This completes our proof. +□ +5.2 +Final Algorithm +Now, we present our main algorithm (Algorithm 2) for unbiased mean estimation of symmetric distribu- +tions under (approximate) DP. The idea is straightforward: invoke our coarse estimator (Algorithm 1) +to get a symmetric, unbiased, mildly accurate estimate of the mean privately; then apply the standard +clip-average-noise technique on our dataset. The second step will not create any new bias because the +clipping is performed around a symmetric, unbiased estimate that is independent of the data we are +clipping and averaging, and the added noise has mean 0. There is an additional hiccup though: the +coarse estimator may fail to produce an estimate. In this case, we fall back to a different algorithm that +exploits (0,훿)-DP and does not require a coarse estimate, as in Proposition 4.4. +The following privacy and utility gurantee is the more general version of Theorem 1.3. +35 + +Algorithm 2: Unbiased DP Estimator DPUMean휀,훿,푐,휎,푛1,푛2(푥) +Input: Dataset 푥 = (푥1, . . . ,푥푛1, 푥푛1+1, . . . ,푥푛1+푛2) ∈ R푛1+푛2. Privacy parameters 휀,훿 > 0. +Clipping parameter 푐 > 0. Scale parameter 휎 > 0. Dataset splitting parameters +푛1,푛2 ∈ N. +Output: Estimate �휇 ∈ R. +// Get a coarse unbiased symmetric estimate of the mean privately. +�휇 ← 휎 · DPUCoarse휀,훿 +� +푥1 +휎 , . . . , +푥푛1 +휎 +� +. +If �휇 = ⊥ +// When the coarse estimator fails. +Let 휉1, 휉2, · · · , 휉푛2 ∈ {0, 1} be independent samples from Bernoulli(훿). +Let �휇 = +1 +푛2훿 +�푛2 +푖=1 푥푛1+푖 · 휉푖. +Else +// When the coarse estimator outputs �휇 ∈ R. +Let �휇 = +� +1 +푛2 +푛2� +푖=1 +clip[�휇−푐,�휇+푐](푥푛1+푖) +� ++ Lap +� +2푐 +푛2휀 +� +. +Return (�휇, �휇). +Theorem 5.7 (Unbiased DP Estimator). Fix 휀,훿 ∈ (0, 1), 푛2 ∈ N, 휓 ≥ 1, and 휆 ≥ 2. Set 훾 = 훿2, 휎 = 10, +푐 = 휎 +휓 · (푛2휀)1/휆, 푛1 = 푂(log(푛1/훾훿)/휀) (as in Proposition 5.6), and 푛 = 푛1 + 푛2. +Algorithm 2 (DPUMean휀,훿,푐,휎,푛1,푛2) satisfies (휀,훿)-DP and the following bias and accuracy properties. +Let 푃 be a symmetric distribution over R with center 휇(푃), variance E +푋←푃 +� +(푋 − 휇(푃))2� +≤ 1, and E +푋←푃 +� +|푋 − 휇(푃)|휆� +≤ +휓휆. +Let 푋 = (푋1, · · · ,푋푛) ← 푃푛 and (�휇, �휇) ← DPUMean휀,훿,푐,휎,푛1,푛2(푋). Then +E[�휇] = 휇(푃), +E +� +(�휇 − 휇(푃))2� +≤ 1 +푛2 ++ 푂 +� +휓2 +(푛2휀)2−2/휆 + 훿 · 휇(푃)2 +푛2 ++ 훿2−4/휆 · (푛1 + 푛2휀)2/휆 · 휓2 +� +. +Furthermore, +P[�휇 ≠ ⊥] ≥ 1 − 훾 = 1 − 훿2, +E[�휇 | �휇 ≠ ⊥] = E[�휇 | �휇 ≠ ⊥] = 휇(푃), +E +� +(�휇 − 휇(푃))2 | �휇 ≠ ⊥ +� +≤ 1 +푛2 ++ 푂 +� +휓2 +(푛2휀)2−2/휆 + 훿2−4/휆 · (푛1 + 푛2휀)2/(휆−1) · 휓2 +� +. +In particular, we can apply Theorem 5.7 to Gaussians (or subgaussians). Using the bound on Gaus- +sian moments +E +푋←N(0,1) +� +|푋 |휆� += 푂( +� +log휆)휆 and setting 휆 = Θ(log푛) yields the following. Note that +we also restrict |휇| ≤ 훿−1/2 to remove the 훿 · 휇2/푛 term from the bound. +Corollary 5.8 (Unbiased Gaussian Mean Estimation). Let휀 ∈ (0, 1),훿 ∈ (0, 1/푛), and푛 ≥ 푂(log(1/훿)/휀). +Let 푀 = DPUMean휀,훿,푐,휎,푛1,푛2 be as in Algorithm 2 with appropriate settings of parameters. Then, for all +휇 ∈ [−훿−1/2, +훿1/2], +E +푋←N(휇,1)푛,푀[푀(푋)] = 휇 +and +E +푋←N(휇,1)푛,푀 +� +(푀(푋) − 휇)2� +≤ 푂 +� 1 +푛 + log log푛 +푛2휀2 +� +. +36 + +To prove Theorem 5.7, we use the following lemma that characterizes the symmetry of a clipped +random variable from a symmetric distribution under special circumstances. +Lemma 5.9. Let 푃 and 푄 be symmetric distributions with the same center 휇(푃) = 휇(푄). Let 푐 > 0. Define +a distribution 푅 to be clip[푌−푐,푌+푐](푋) where 푋 ← 푃 and 푌 ← 푄 are independent. Then 푅 is symmetric +with the same center 휇(푅) = 휇(푃) = 휇(푄). +Proof. Assume, without loss of generality, that 휇(푃) = 휇(푄) = 0. Let푋 ← 푃 and푌 ← 푄 be independent. +Let 푍 = clip[푌−푐,푌+푐](푋). +We claim that +∀푥,푦 ∈ R clip[(−푦)−푐,(−푦)+푐] (−푥) = −clip[푦−푐,푦+푐](푥). +This can be verified by analyzing the following cases: (1) 푥 < 푦 −푐; (2) 푥 ∈ [푦 −푐,푦 +푐]; and (3) 푥 > 푦 +푐. +Since 푃 and 푄 are symmetric, clip[(−푌)−푐,(−푌)+푐] (−푋) has the same distribution as 푍. By the claim, +this is simply −푍. Ergo, the distribution of 푍 is symmetric and centered at 0. +□ +Proof of Theorem 5.7. The privacy of Algorithm 2 follows from parallel composition, as we split the +dataset in two, and apply (휀,훿)-DP algorithms to each half. Computing �휇 is (휀,훿)-DP by Proposition 5.2. +If �휇 = ⊥, then we compute �휇 in a (0,훿)-DP manner by sampling a 훿 fraction of the data points. If �휇 ≠ ⊥, +then we compute �휇 in a (휀, 0)-DP manner using clipping and Laplace noise addition (Lemma 2.6). +Note that �휇 is independent from 푋푛1+1, . . . ,푋푛1+푛2, which are the data points used to compute �휇. If +�휇 = ⊥, then we compute �휇 in an unbiased manner: +E[�휇 | �휇 = ⊥] = E +� +1 +푛2훿 +푛1+푛2 +� +푖=푛1+1 +푋푖휉푖 +� += +1 +푛2훿 +푛1+푛2 +� +푖=푛1+1 +E[푋푖]E[휉푖] = +1 +푛2훿 +푛1+푛2 +� +푖=푛1+1 +휇(푃)훿 = 휇(푃). +Now, condition on �휇 ≠ ⊥. By Proposition 5.3, �휇 has a symmetric distribution with center 휇(푃). By +Lemma 5.9, E +� +clip[�휇−푐,�휇+푐] (푋푖) | �휇 ≠ ⊥ +� += 휇(푃), which implies that E[�휇 | �휇 ≠ ⊥] = 휇(푃) because the +Laplace noise has expected value 0. Combining these two cases implies E[�휇] = 휇(푃). +Finally, we analyze the variance: +E +� +(�휇 − 휇(푃))2� += P[�휇 = ⊥] · E +� +(�휇 − 휇(푃))2 | �휇 = ⊥ +� ++ P[�휇 ≠ ⊥] · E +� +(�휇 − 휇(푃))2 | �휇 ≠ ⊥ +� +. +We bound the two terms for �휇 = ⊥ and �휇 ≠ ⊥ separately. For the first term, Proposition 5.6 gives us +P[�휇 = ⊥] ≤ 훾. Then we have the following. +P[�휇 = ⊥] · E +� +(�휇 − 휇(푃))2 | �휇 = ⊥ +� += P[�휇 = ⊥] · E + +� +1 +푛2훿 +푛1+푛2 +� +푖=푛1+1 +푋푖휉푖 − 휇(푃) +�2 += P[�휇 = ⊥] · +1 +푛2 +2훿2 +푛1+푛2 +� +푖=푛1+1 +E +� +(푋푖휉푖 − 휇(푃))2� +≤ P[�휇 = ⊥] · +1 +푛2 +2훿2 +푛1+푛2 +� +푖=푛1+1 +E +� +(푋푖휉푖)2� += P[�휇 = ⊥] · +휇(푃)2 + +E +푋←푃 +� +(푋 − 휇(푃))2� +푛2훿 +≤ 훾 · 휇(푃)2 + 1 +푛2훿 +. +37 + +Now, we bound the second term: P[�휇 ≠ ⊥] · E +� +(�휇 − 휇(푃))2 | �휇 ≠ ⊥ +� += E +� +I[�휇 ≠ ⊥] · (�휇 − 휇(푃))2� +. We +split this into two cases: 퐴 ≔ [�휇 ∈ [휇(푃) − 휎, 휇(푃) + 휎]] and 퐵 ≔ [�휇 ∈ R \ [휇(푃) − 휎, 휇(푃) + 휎]]. Note +that the event 퐴∧�휇 ≠ ⊥ is equivalent to 퐴 because 퐴 cannot happen if �휇 = ⊥, because ⊥ ∉ R. Similarly, +퐵 ∧ �휇 ≠ ⊥ is equivalent to 퐵. Note that �휇 ≠ ⊥ =⇒ 퐴 ∨ 퐵. Thus, we have +P[�휇 ≠ ⊥] · E +� +(�휇 − 휇(푃))2 | �휇 ≠ ⊥ +� += P[�휇 ≠ ⊥] · P[퐴 | �휇 ≠ ⊥] · E +� +(�휇 − 휇(푃))2 | �휇 ≠ ⊥ ∧ 퐴 +� ++ P[�휇 ≠ ⊥] · P[퐵 | �휇 ≠ ⊥] · E +� +(�휇 − 휇(푃))2 | �휇 ≠ ⊥ ∧ 퐵 +� += P[�휇 ≠ ⊥ ∧ 퐴] · E +� +(�휇 − 휇(푃))2 | �휇 ≠ ⊥ ∧ 퐴 +� ++ P[�휇 ≠ ⊥ ∧ 퐵] · E +� +(�휇 − 휇(푃))2 | �휇 ≠ ⊥ ∧ 퐵 +� += P[�휇 ≠ ⊥ ∧ 퐴] · E +� +(�휇 − 휇(푃))2 | �휇 ≠ ⊥ ∧ 퐴 +� ++ E +� +I[�휇 ≠ ⊥] · I[퐵] · (�휇 − 휇(푃))2� +≤ E +� +(�휇 − 휇(푃))2 | �휇 ≠ ⊥ ∧ 퐴 +� ++ E +� +I[�휇 ≠ ⊥] · I[퐵] · (�휇 − 휇(푃))2� += E +� +(�휇 − 휇(푃))2 | 퐴 +� ++ E +� +I[퐵] · (�휇 − 휇(푃))2� +. +(9) +If 퐴 holds (i.e., �휇 ∈ [휇(푃) − 휎, 휇(푃) + 휎]), then 휇(푃) ∈ [�휇 − 휎, �휇 + 휎], so we can bound the first term of +the last line in Inequality 9 as follows. +E +� +(�휇 − 휇(푃))2 | 퐴 +� += E + +� +1 +푛2 +푛1+푛2 +� +푖=푛1+1 +clip[�휇−푐,�휇+푐](푋푖) + Lap +� 2푐 +푛2휀 +� +− 휇(푃) +�2 +| 퐴 + +(A)= E + +� +1 +푛2 +푛1+푛2 +� +푖=푛1+1 +clip[�휇−푐,�휇+푐] (푋푖) − 휇(푃) +�2 +| 퐴 + ++ 2 +� 2푐 +푛2휀 +�2 +(B) +≤ E + +E +푋←푃 +� +(푋 − 휇(푃))2� +푛2 +| 퐴 + ++ E + +�� +� +E +푋←푃 +� +|푋 − 휇(푃)|휆� +휆 · (min{휇(푃) − (�휇 − 푐), (�휇 + 푐) − 휇(푃)})휆−1 +�� +� +2 +| 퐴 + ++ 8푐2 +푛2 +2휀2 +≤ +E +푋←푃 +� +(푋 − 휇(푃))2� +푛2 ++ �� +� +1 +휆 · +E +푋←푃 +� +|푋 − 휇(푃)|휆� +(푐 − 휎)휆−1 +�� +� +2 ++ 8푐2 +푛2 +2휀2 +≤ 1 +푛2 ++ +휓2휆 +휆2 · (푐 − 휎)2(휆−1) + 8푐2 +푛2 +2휀2 +(C)= 1 +푛2 ++ +휓2휆 +휆2 · 휓2(휆−1) · (푛2휀)2−2/휆 + 8푐2 +푛2 +2휀2 += 1 +푛2 ++ 8푐2 +휓2 · (푛2휀)2/휆 · 휆−2 +푛2 +2휀2 +≤ 1 +푛2 ++ 8푐2 +휓2 · (푛2휀)2/휆 +푛2 +2휀2 +. +In the above: Equality A follows from the fact that the Laplace noise is independent from everything +else; Inequality B follows from Lemma 4.2 and linearity of expectations; and Equality C follows from +38 + +the setting of 푐 = 휎 +휓 · (푛2휀)1/휆. +Next, we bound the second term in the last line of Inequality 9. We use the fact that +∀푥 ∈ R |clip[�휇−푐,�휇+푐](푥) − 휇(푃)| ≤ |�휇 − 휇(푃)| + 푐. +We have +E +� +I[퐵] · (�휇 − 휇(푃))2� += E + +I[퐵] · +� +1 +푛2 +푛1+푛2 +� +푖=푛1+1 +clip[�휇−푐,�휇+푐] (푋푖) + Lap +� 2푐 +푛2휀 +� +− 휇(푃) +�2 +(D)= E + +I[퐵] · +� +1 +푛2 +푛1+푛2 +� +푖=푛1+1 +clip[�휇−푐,�휇+푐](푋푖) − 휇(푃) +�2 ++ 2 +� 2푐 +푛2휀 +�2 +· P[퐵] +≤ E +� +I[퐵] · (|�휇 − 휇(푃)| + 푐)2� ++ 8푐2 +푛2 +2휀2 · P[퐵] +(E) +≤ E +� +I[퐵] +휆 +휆−2 +� 휆−2 +휆 · E +� +I[�휇 ≠ ⊥] · (|�휇 − 휇(푃)| + 푐)휆�2/휆 ++ 8푐2 +푛2 +2휀2 · P[퐵] +(F) +≤ E[I[퐵]] +휆−2 +휆 · E +� +I[�휇 ≠ ⊥] · +� +|�휇 − 휇(푃)|휆 + 푐휆� +· 2휆−1�2/휆 ++ 8푐2 +푛2 +2휀2 · P[퐵] +(G) +≤ 훾 +휆−2 +휆 · +�1 +2 + 푛1 · 2휆−1 ·휓휆 + 푐휆 +�2/휆 +· 22−2/휆 + 8푐2 +푛2 +2휀2 · 훾 +(H) +≤ 훾 +휆−2 +휆 · +� +2−2/휆 + 푛2/휆 +1 +· 22−2/휆 · 휓2 + 푐2� +· 22−2/휆 + 8푐2 +푛2 +2휀2 · 훾 +≤ 훾 +휆−2 +휆 · +� +1 + 푛2/휆 +1 +· 4 ·휓2 + 푐2� +· 4 + 8푐2 +푛2 +2휀2 · 훾. +In the above: Equality D follows from the independence of the Laplace noise; Inequality E follows from +Hölder’s inequality; Inequality F holds because ∀푝 ≥ 1 ∀푥,푦 ≥ 0 (푥 +푦)푝 ≤ (푥푝 +푦푝) · 2푝−1; Inequality +G follows from Proposition 5.6; and Inequality H holds because ∀푝 ∈ (0, 1] ∀푥,푦 ≥ 0 (푥 +푦)푝 ≤ 푥푝 +푦푝. +Finally, we can combine all the pieces, and use our parameter settings 훾 = 훿2 ≤ 1 and 푐2 = (10 +휓 · +(푛2휀)1/휆)2 ≤ 2휓2(푛2휀)2/휆 + 200, to get the following. +E +� +(�휇 − 휇(푃))2� +≤ +� +훾 · 휇(푃)2 + 1 +푛2훿 +� ++ +� 1 +푛2 ++ 8푐2 + 휓2 · (푛2휀)2/휆 +푛2 +2휀2 +� ++ +� +훾 +휆−2 +휆 · +� +1 + 푛2/휆 +1 +· 4 · 휓2 + 푐2� +· 4 + 8푐2 +푛2 +2휀2 · 훾 +� +≤ 1 +푛2 ++ 16푐2 + 휓2 · (푛2휀)2/휆 +푛2 +2휀2 ++ 훿 · 1 + 휇(푃)2 +푛2 ++ 훿2−4/휆 · 4 +� +4푛2/휆 +1 +· 휓2 + 푐2 + 1 +� +≤ 1 +푛2 ++ 33휓2(푛2휀)2/휆 + 3200 +푛2 +2휀2 ++ 훿 · 1 + 휇(푃)2 +푛2 ++ 훿2−4/휆 · +� +16휓2푛2/휆 +1 ++ 8휓2(푛2휀)2/휆 + 804 +� += 1 +푛2 ++ 푂 +� +휓2 +(푛2휀)2−2/휆 + 훿 · 휇(푃)2 +푛2 ++ 훿2−4/휆 · (푛1 + 푛2휀)2/휆 ·휓2 +� +. +Our proof is now complete. +□ +39 + +6 +An Impossibility Result for Exponential Families Under Pure DP +In Section 5, we showed that it is possible to perform unbiased mean estimation for symmetric distri- +butions. However, this result only provides approximate DP (i.e., (휀,훿)-DP with 훿 > 0). We now show +that this is inherent. +We show that unbiased estimation is impossible under pure DP (i.e., (휀, 0)-DP) when the data comes +from an exponential family. Exponential families include a wide range of distributions, including Gaus- +sians, exponential distributions, Laplace distributions with fixed mean, and Gamma distributions. +We begin by formally defining exponential families: +Definition 6.1 (Exponential Family). Let 푈 ⊆ R푑 be open, ℎ: 푈 → [0, ∞), and 푇 : 푈 → R푘. Given +휂 ∈ R푘, we say that the probability measure 푃휂 belongs to the exponential family E(푇,ℎ) if its density +can be written as: +푓푇,ℎ,휂(푥) = ℎ(푥) exp�휂⊤푇 (푥) − 푍 (휂)�, ∀푥 ∈ 푈, +where: +푍 (휂) = log�� +� +∫ +푆 +ℎ(푥) exp�휂⊤푇 (푥)� 푑푥�� +� +. +The functions ℎ and 푇 are referred to as the carrier measure and the sufficient statistics of the family, +respectively. Additionally, 푍 is known as the log-partition function and 휂 is the natural parameter vector. +Finally, we denote the range of natural parameters by H ⊆ R푘, which is the set of values of 휂 for which +the log-partition function is well-defined (푍 (휂) < ∞). +The density of a Gaussian N (휇, 1) at 푥 is exp(−(푥 − 휇)2/2) = exp(−푥2/2) · exp(휇 · 푥 − 휇2/2). Thus +we see that Gaussians are an exponential family with natural parameter 휇, carrier measure ℎ(푥) = +exp(−푥2/2), sufficient statistics 푇 (푥) = 푥, and log-partition function 푍 (휇) = 휇2/2. +Now we can state our main result: +Theorem 6.2 (Impossibility of Pure DP Unbiased Estimation for Exponential Families ). Let 푈 ⊆ R be +an interval of infinite length, let {푃휂 : 휂 ∈ 푈 } be an exponential family, and let 퐼 ⊆ 푈 be any interval of +positive length. Then, for any 휀 ≥ 0 and 푛 ≥ 0, there exists no (휀, 0)-DP algorithm 푀 : R푛 → R satisfying +E푋←푃푛휂 ,푀 [푀(푋)] = 휂 for all 휂 ∈ 퐼. +For instance, this theorem shows that there is no pure DP estimator that is unbiased for Gaussians +N (휇, 1), even if the algorithm is only required to be unbiased when, say, 휇 ∈ [2, 4]. Similarly, there +can be no pure-DP estimator that gives an unbiased estimate for the mean of Exponential(휆) for all +휆 ∈ (0, 10). +We remark that in Theorem 6.2, the interval where the algorithm is guaranteed to be unbiased 퐼 +having positive length is essential. It is easy to construct a pathological estimator that is unbiased at +a single point 휂0 ∈ 푈 but not anywhere else, e.g. by setting 푀(푥) = 휂0 for all 푥 ∈ R푛. The interval +on which the parameter 휂 is well-defined 푈 having infinite length is also essential to our analysis. We +emphasize that this is a property of the distribution and not of the algorithm; that is, the algorithm +does not need to “know” about 푈 . Note that the family of distributions {Bernoulli(푝) : 푝 ∈ [0, 1]} is an +exponential family3 and it is possible to estimate the mean 푝 under pure DP. In this case 푈 = [0, 1] has +finite length, so we see that the assumption that 푈 has infinite length is also necessary. +The proof of this result relies on tools and language from complex analysis and measure theory, a +review of which can be found in Appendices A and B, respectively. We first show that, for an estimator +3Definition 6.1 is stated in terms of densities, but it can be extended to discrete distributions. +40 + +휙 : R푛 → R and an exponential family {푃휂 : 휂}, the expected value of the estimator E푋←푃푛휂 [휙(푋)] is +an analytic function in 휂. We then apply the identity theorem for analytic functions to argue that if 휙 is +locally unbiased, i.e., unbiased when 휂 lies in some small set, then 휙 must also be globally-unbiased, i.e., +unbiased for all choices of 휂. On the other hand, we will argue that global unbiasedness over an infinite +interval is impossible for pure DP estimators as a consequence of the strong group privacy properties +of pure DP. +6.1 +Locally Unbiased Estimators Are Globally Unbiased +The following result shows that the the expectation function of any estimator for the parameters of +distributions from an exponential family is analytic. +Theorem 6.3 (Analyticity under Exponential Families). Let {푃휂 : 휂 ∈ 푈 } be an exponential family on R푛 +in canonical form (recall Definition 6.1) and let 휙 : X푛 → R be any well-defined estimator for {푃휂 : 휂 ∈ 푈 }, +i.e., E푋←푃휂 [|휙(푋)|] is finite for all휂 ∈ 푈 . Then푔 : 푈 → R defined by푔(휂) ≔ E푋←푃휂 [휙(푋)] is an analytic +function. +A function being analytic means that its Taylor series provides an exact representation of the func- +tion. Thus, if an analytic function is linear in some nontrivial interval, we can compute the Taylor +series at an interior point of that interval to deduce that the funciton is linear globally, which yields the +following result. +Corollary 6.4 (Local Unbiasedness Implies Global Unbiasedness). Let {푃휂 : 휂 ∈ 푈 } be an exponential +family on R푛 and let 휙 : X푛 → R be any well-defined estimator for {푃휂 : 휂 ∈ 푈 }. Let 퐼 ⊂ 푈 be an interval +of nonzero length. If E푋←푃휂 [휙(푋)] = 휂 for all 휂 ∈ 퐼, then E푋←푃휂 [휙(푋)] = 휂 for all 휂 ∈ 푈 . +The main idea behind the proof of Theorem 6.3 is that analyticity is preserved under integration +under certain circumstances, which we show next. Although a proof for the real plane is possible, it +will be technically convenient to pass to the complex plane where we can wield Morera’s theorem +(Theorem A.2). +Before delving into the proof of Theorem 6.3, we show that the product distribution where each +marginal has the same distribution from an exponential family is also an exponential family. +Proposition 6.5. Let E(푇,ℎ) be an exponential family with support 푈 ⊆ R, with the range of natural +parameters H ⊆ R. Then for any 푛 ∈ N, the family of distributions {푃푛 : 푃 ∈ E(푇,ℎ)} is an exponential +family over 푈 푛( denoted by E(푇푛,ℎ푛), where 푇푛 : 푈 푛 → R and ℎ푛 : 푈 푛 → [0, ∞)) with the range of +natural parameters H푛 = H, such that for every 푃 ∈ E(푇,ℎ), if the natural parameter of 푃 is some 휂 ∈ H, +then the natural parameter of 푃푛 is also 휂, and for all 푥 = (푥1, . . . ,푥푛) ∈ 푈 푛, +푇푛(푥) = +푛 +� +푖=1 +푇 (푥푖) +and ℎ푛(푥) = +푛 +� +푖=1 +ℎ(푥푖). +Proof. Let 휂 ∈ H, and 푃휂 ∈ E(푇,ℎ) with natural parameter 휂 and density function 푓푇,ℎ,휂 as described in +Definition 6.1. Suppose 푓 : 푈 푛 → R is the density function of 푃푛 +휂 . Then for any 푥 = (푥1, . . . , 푥푛) ∈ 푈 푛, +we have the following. +푓 (푥) = +푛 +� +푖=1 +푓푇,ℎ,휂(푥푖) +41 + += +푛 +� +푖=1 +ℎ(푥푖) exp(휂 · 푇 (푥푖) − 푍 (휂)) += +� 푛 +� +푖=1 +ℎ(푥푖) +� +exp +� +휂 +푛 +� +푖=1 +푇 (푥푖) − 푛푍 (휂) +� +This gives us: 푇푛(푥) = +푛� +푖=1 +푇 (푥푖); ℎ푛(푥) = +푛� +푖=1 +ℎ(푥푖); and the natural parameter of 푃푛 +휂 being 휂푛 = 휂 ∈ H. +One can easily verify that the log-partition function (푍푛(휂푛)) of 푃푛 +휂 equals 푛푍 (휂). +□ +Lemma 6.6. Let Ω be a 휎-finite measure space with measure 휈, let 푉 ⊆ C be open, and let 푓 : Ω ×푉 → C. +Assume that 푓 (휔,휂) is analytic in 휂 for every fixed 휔 ∈ Ω and that, for every compact 퐾 ⊆ 푉 , there is +a 휈-integrable function (see Section B) 퐺 : Ω → [0, ∞) for which |푓 (휔,휂)| ≤ 퐺(휔) for all 휂 ∈ 퐾. Then +푔(휂) ≔ +∫ +Ω +푓 (휔,휂) 푑휈(휔) is analytic, as well. +Proof. Our plan is to apply Morera’s theorem (Theorem A.2). To that end, we must first show that 푔 +is continuous, so let (휂푛)푛∈N be any sequence with 휂푛 → 휂 as 푛 → ∞. By our assumption, there is a +휈-integrable퐺 : Ω → [0, ∞) such that |푓 (휔,휂푛)| ≤ 퐺(휔) for all 푛 ∈ N and 휔 ∈ Ω. So, by the dominated +convergence theorem (Theorem B.1), 푔(휂푛) → 푔(휂) as 푛 → ∞. +Now, let 훾 : [0, 1] → C be any closed contour lying in a simply connected (see Section A) subset of +푉 , and let 훾 ′ denote its first derivative. Then, |훾 ′| must be bounded by some 퐶 > 0, so +1 +∫ +0 +∫ +Ω +|푓 (휔,훾(푡))훾 ′(푡)|푑휈(휔) 푑푡 ≤ +1 +∫ +0 +∫ +Ω +퐺(휔)퐶 푑휈(휔) 푑푡 = 퐶 +∫ +Ω +퐺 푑휈 < ∞ +and thus Fubini’s theorem (Theorem B.2) implies that +∮ +훾 +푔(휂) 푑휂 = +1 +∫ +0 +∫ +Ω +푓 (휔,훾(푡))훾 ′(푡) 푑휈(휔) 푑푡 += +∫ +Ω +1 +∫ +0 +푓 (휔,훾(푡))훾 ′(푡) 푑푡 푑휈(휔) += +∫ +Ω +∮ +훾 +푓 (휔,휂) 푑휂 푑휈(휔) += +∫ +Ω +0푑휈(휔) +(Theorem A.1) += 0. +As 훾 was arbitrary, 푔 must be analytic by Morera’s theorem. +□ +Proof of Theorem 6.3. Our main goal is to show that 푔(휂) ≔ E푋←푃휂 [휙(푋)] is analytic. To that end, let +ℎ,푇, and 푍 be the carrier measure, the sufficient statistic, and the log-partition function of {푃휂 : 휂 ∈ 푈 }, +respectively. +We first show that 푒푍 (휂) is analytic by way of Lemma 6.6. Indeed, 푟 (푥,휂) ≔ ℎ(푥)푒휂푇 (푥) is entire +(see Section A) in 휂 ∈ C for each fixed 푥 ∈ R푛. Let 퐾 ⊆ C be an arbitrary compact set, and let 푚 and 푀 +42 + +be the minimum and the maximum real coordinates among the points within 퐾, respectively. Then for +any 푥 ∈ R푛 and 휂 ∈ 퐾, +푇 (푥) < 0 =⇒ |푟 (푥,휂)| = ℎ(푥)푒Re(휂)푇 (푥) ≤ ℎ(푥)푒푚푇 (푥) +and +푇 (푥) ≥ 0 =⇒ |푟 (푥,휂)| ≤ ℎ(푥)푒푀푇 (푥), +so we have +|푟 (푥,휂)| ≤ ℎ(푥)푒푚푇 (푥) + ℎ(푥)푒푀푇 (푥). +But +∫ +R푛 +ℎ(푥)푒푚푇 (푥) +ℎ(푥)푒푀푇 (푥) 푑푥 = 푒푍 (푚) +푒푍 (푀) < ∞, so, since 퐾 was arbitrary, 푒푍 (휂) = +∫ +R푛 +푟 (푥,휂) 푑푥 +must be entire by Lemma 6.6. +As a consequence, ℎ(푥)푒휂푇 (푥)−푍 (휂) is analytic in 휂 for every fixed 푥 ∈ R푛, so we can apply nearly +the same argument to ℎ(푥)푒휂푇 (푥)−푍 (휂) in order to conclude that +푔(휂) = +E +푋←푃휂[휙(푋)] = +∫ +Ω +휙(푥)ℎ(푥)푒휂푇 (푥)−푍 (휂) 푑푥 +is analytic, as well. +□ +6.2 +Pure DP Estimators Are Uniformly Bounded +We now exploit the strong group privacy property of pure DP (Lemma 2.4) to show that a pure DP +estimator that is bounded locally is uniformly bounded globally. +Proposition 6.7 (Pure DP Estimators Are Uniformly Bounded). Let 퐴 : X푛 → R be a randomized +algorithm. If 퐴 is (휀, 0)-DP, then for all 푥, 푥∗ ∈ X푛, +����E +퐴[퐴(푥)] +���� ≤ 푒휀푛 · E +퐴[|퐴(푥∗)|]. +We emphasize that the above result holds for any 푥, 푥∗ (not just neighbouring datasets or ones +drawn from an appropriate distribution). Thus the bound on +����E +퐴[퐴(푥)] +���� is uniform – i.e., it does not +depend on 푥. +Proof. Recall that E[푌] = +∞ +∫ +0 +P[푌 ≥ 푡] 푑푡 for any non-negative random variable 푌. For any 푥 ∈ X푛, we +have +����E +퐴[퐴(푥)] +���� +(a) +≤ E +퐴[|퐴(푥)|] = +∞ +∫ +0 +P +퐴[|퐴(푥)| ≥ 푡] 푑푡 +(b) +≤ 푒휀푛 · +∞ +∫ +0 +P +퐴[|퐴(푥∗)| ≥ 푡] 푑푡 = 푒휀푛 · E +퐴[|퐴(푥∗)|], +where inequalities (a) and (b) follow from Jensen’s inequality and group privacy (Lemma 2.4), respec- +tively. +□ +We remark that Proposition 6.7 does not hold for approximate DP unless we make further assump- +tions about 퐴(푥∗). +Our impossibility result for exponential families now follows by stringing together the tools we +have collected so far. +43 + +Proof of Theorem 6.2. Suppose, for the sake of contradiction, there exist 휀 ≥ 0, 푛 ≥ 0, and an 휀-DP +algorithm 푀 : R푛 → R for which E푋←푃푛휂 ,푀 [푀(푋)] = 휂 when 휂 ∈ 퐼. By Proposition 6.5, {푃푛 +휂 : 휂} is an +exponential family, such that for every 푃 ∈ {푃휂 : 휂}, the natural parameter of 푃푛 is the same as that of +푃. Therefore, by Theorem 6.3, E푋←푃푛휂 ,푀 [푀(푋)] is an analytic function in 휂. But E푋←푃푛휂 ,푀 [푀(푋)] = 휂 +for 휂 ∈ 퐼, so it follows from the identity theorem (Theorem A.4) that E푋←푃푛휂 ,푀 [푀(푋)] = 휂 for all +휂 ∈ 푈 . In particular, sine 푈 is unbounded, E푋←푃푛휂 ,푀 [푀(푋)] must be an unbounded function of 휂, +which contradicts Proposition 6.7. +□ +Acknowledgements +GK was supported by an NSERC Discovery Grant, an unrestricted gift from Google, an unrestricted gift +from Apple, and a University of Waterloo startup grant. AM was supported by an NSERC Discovery +Grant and a David R. Cheriton Graduate Scholarship. MR was supported by a Vector Scholarship in AI +and an NSERC CGS-M. 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JMLR, Inc., 2020. +[ZKKW20] Huanyu Zhang, Gautam Kamath, Janardhan Kulkarni, and Zhiwei Steven Wu. Privately +learning Markov random fields. In Proceedings of the 37th International Conference on Ma- +chine Learning, ICML ’20, pages 11129–11140. JMLR, Inc., 2020. +[ZVHF21] +Keyu Zhu, Pascal Van Hentenryck, and Ferdinando Fioretto. Bias and variance of post- +processing in differential privacy. In Proceedings of the Thirty-Fifth AAAI Conference on +Artificial Intelligence, volume 35 of AAAI ’21, pages 11177–11184, 2021. +A +Background on Complex Analysis +The primary objects of interest in complex analysis are the holomorphic functions in the complex plane, +namely those functions 푓 : 푈 → C that are differentiable at every point 푧 ∈ 푈 . Many familiar functions, +such as the polynomials, are in fact holomorphic or may be extended to a holomorphic function. Note +that when 푈 = C, i.e., 푓 is differentiable on the whole complex plane, we say that 푓 is an entire function. +A basic result of complex analysis asserts that a function 푓 : 푈 → C is holomorphic exactly when +it is analytic, i.e., its Taylor series expansion around any point 푧0 ∈ 푈 converges to 푓 in some neighbor- +hood of 푧0. For this reason, holomorphic functions are typically referred to as analytic functions. We +consider analyticity in our work as there exist useful mathematical tools to check when functions are +analytic, and even more useful tools for constraining functions that we have established to be analytic. +For our purposes, we define a closed contour in a region 퐷 ⊆ C to be a continuously differentiable +map 훾 : [0, 1] → 퐷 with 훾(0) = 훾(1). Informally, we say that a region in the plane is simply connected +if it contains no holes. For instance, the disk {푧 ∈ C : |푧| ≤ 3} is simply connected, whereas the “donut” +{푧 ∈ C : |푧| ∈ [1, 3]} is not. +50 + +A thorough review of the language of complex analysis with the precise definitions of the above +(which are not necessary for the understanding of our application) is outside the scope of this work, so +we recommend the textbook by Ahlfors [Ahl53] for a more comprehensive background. +A useful property of analytic functions is that their closed contour integrals vanish in simply con- +nected regions. The following theorem characterises this more formally. +Theorem A.1 (Cauchy’s Theorem). Let 푈 be an open, simply connected subset of C and let 푓 : 푈 → C +be analytic. Then, for any closed contour 훾 in 푈 , we have +∮ +훾 +푓 (푧) 푑푧 = 0. +The converse is true, as well, and is a convenient technique for establishing analyticity. +Theorem A.2 (Morera’s Theorem). Let 푈 ⊆ C be open and let 푓 : 푈 → C be continuous. Suppose that, +for all simply connected 퐷 ⊆ 푈 and any closed contour 훾 in 퐷, we have +∮ +훾 +푓 (푧) 푑푧 = 0. Then 푓 is analytic. +Next, for functions 푓1, 푓2 : 푈 → C and any 퐿 ⊆ 푈 , we write 푓1|퐿 ≡ 푓2|퐿, if for all 푥 ∈ 퐿, 푓1(푥) = 푓2(푥). +Additionally, we write 푓1 ≡ 푓2, if 푓1|푈 ≡ 푓2|푈 . Finally, we define the limit points of a set. +Definition A.3 (Limit Point of a Set). Given a topological space X and 푆 ⊆ X, we say that 푥 ∈ X is a +limit point of 푆, if for every neighbourhood 퐵 ⊆ X of 푥 (with respect to the topology of X), there exists +a point 푦 ∈ 퐵, such that 푦 ∈ 푆 and 푦 ≠ 푥. +In other words, a limit point 푥 of 푆 can be “approximated by points in 푆.” The main property of +analytic functions that we exploit is the fact that any two analytic functions that agree locally must, in +fact, agree globally, as we show next. +Theorem A.4 (Identity Theorem). Let 푈 ⊆ C be open, and 푓1, 푓2 : 푈 → C be analytic. Suppose there is +a set 퐿 ⊆ 푈 with a limit point in 푈 , such that 푓1|퐿 ≡ 푓2|퐿. Then 푓1 ≡ 푓2. +B +Background on Measure Theory +Recall that a measure space is the combination of a set X with a collection Σ of subsets of X, which +are closed under complement and countable unions, as well as a function 휇 : Σ → [0, ∞] satisfying +휇(∅) = 0 and 휇��∞ +푖=1 퐴푖 +� = �∞ +푖=1 휇(퐴푖) for disjoint 퐴1,퐴2, · · · ∈ Σ. The subsets making up Σ are called +the measurable subsets of X and 휇 is called a measure on X. We say that X is 휎-finite when it can be +decomposed as X = �∞ +푖=1 퐴푖 where 퐴1, 퐴2, · · · ∈ Σ are all of finite measure 휇(퐴푖) < ∞. A function +푓 : X → C is said to be measurable if 푓 −1(푈 ) is a measurable subset of X for any open 푈 ⊆ C. In this +case, we say that 푓 : X → R is 휇-integrable if +∫ +X|푓 | 푑휇, the Lebesgue integral of |푓 | with respect to 휇, +exists and is finite. +Now, in order to apply Morera’s theorem, we will require some standard integral-limit interchange +theorems. The first is the dominated convergence theorem, which asserts that pointwise convergence of +a sequence of functions may be interchanged with integration, provided that the sequence is uniformly +bounded by an integrable function. +Theorem B.1 (Dominated Convergence Theorem). Let X be a measure space. Suppose that (푓푛)푛∈N is +a sequence of measurable functions X → C converging pointwise to some 푓 , i.e., 푓푛(푥) → 푓 (푥) for all +푥 ∈ X as 푛 → ∞. Suppose further that there is some measurable 퐺 : X → [0, ∞) such that +∫ +X +퐺 푑휇 < ∞ +and |푓푛(푥)| ≤ 퐺(푥) for all 푥 ∈ X and 푛 ∈ N. Then 푓 is integrable such that +lim +푛→∞ +∫ +X +푓푛 푑휇 = +∫ +X +푓 푑휇. +51 + +Switching the order of integration is a very useful operation that is permitted under fairly general +measure-theoretic conditions. We describe it as follows. +Theorem B.2 (Fubini’s Theorem). Let X and Y be 휎-finite measure spaces and suppose that 푓 : X×Y → +R is measurable such that +∫ +X +∫ +Y +|푓 (푥,푦)| 푑푦 푑푥 < ∞. +Then +∫ +X +∫ +Y +푓 (푥,푦) 푑푦 푑푥 = +∫ +Y +∫ +X +푓 (푥,푦) 푑푥 푑푦. +C +Impossibility Result for Concentrated DP +We extend the impossibility result for unbiased estimation under pure DP in Section 6 to concentrated +DP. But first we briefly introduce concentrated DP. +Concentrated DP is a variant of DP that has particularly nice composition properties. It was intro- +duced by Dwork and Rothblum [DR16], but we use a slightly different definition due to Bun and Steinke +[BS16] (see also [Ste22]). +Definition C.1 (Concentrated DP). A randomized algorithm 푀 : X푛 → Y satisfies 휌-zCDP if, for all +neighboring inputs 푥, 푥 ′ ∈ X, +∀푡 > 0 D푡+1(푀(푥)∥푀(푥 ′)) ≔ 1 +푡 log�� +� +E +푌←푀(푥) + +�� +� +P +푀[푀(푥) = 푌] +P +푀[푀(푥 ′) = 푌] +�� +� +푡 +�� +� +≤ (푡 + 1)휌. +The quantity D푡+1(·∥·) is the Rényi divergence of order 푡 + 1. The above definition applies when +the distributions of 푀(푥) and 푀(푥 ′) are discrete; in the continuous case, we replace +P +푀 [푀(푥)=푌 ] +P +푀 [푀(푥′)=푌 ] with +the Radon-Nikodym derivative. +Concentrated DP is intermediate between pure DP and approximate DP. Specifically, (휀, 0)-DP im- +plies 1 +2휀2-zCDP and 휌-zCDP implies (휌 + 2 +� +휌 · log(1/훿),훿)-DP for all 훿 > 0. Concentrated DP cap- +tures most common DP algorithms, including Laplace and Gaussian noise addition and the exponential +mechanism. Thus our impossibility result for concentrated DP provides a barrier against a wide range +of techniques. +Concentrated DP also has strong group privacy properties (cf. Lemma 2.4), which forms the basis +of our impossibility result. +Lemma C.2 (Group Privacy for Concentrated DP). Let 퐴 : X푛 → Y be 휌-zCDP. Then for any integer +푘 ∈ {0, . . . ,푛} and pairs of datasets 푥, 푥 ′ ∈ X푛 differing in 푘 entries, +∀푡 > 0 D푡+1(퐴(푥)∥퐴(푥 ′)) ≤ (푡 + 1)푘2휌. +We can also use Rényi divergences to bound expectations. +Lemma C.3 ([BS16, Lemma C.2]). Let 푋 and 푌 be random variables with D2(푋 ∥푌) and E +� +푌 2� +being +finite. Then +|E[푋]| ≤ +� +E[푌 2] · (exp(D2(푋 ∥푌)) − 1). +52 + +Thus, we can give an analog of Proposition 6.7. +Proposition C.4 (Concentrated DP Estimators Are Uniformly Bounded). Let 퐴 : X푛 → R be a ran- +domized algorithm. If 퐴 is 휌-zCDP, then for all 푥, 푥∗ ∈ X푛, +����E +퐴[퐴(푥)] +���� ≤ +� +(푒2푛2휌 − 1) · E +퐴 +� +퐴(푥∗)2� +. +This yields an extension of Theorem 6.2. +Theorem C.5 (Impossibility of Concentrated DP Unbiased Estimation for Exponential Families ). Let +푈 ⊆ R be an interval of infinite length, let {푃휂 : 휂 ∈ 푈 } be an exponential family, and let 퐼 ⊆ 푈 be any +interval of positive length. Then, for any 휌 ≥ 0 and 푛 ≥ 0, there exists no 휌-zCDP algorithm 푀 : R푛 → R +satisfying E푋←푃푛휂 ,푀 [푀(푋)] = 휂 and E푋←푃푛휂 ,푀 [푀(푋)2] < ∞ for all 휂 ∈ 퐼. +53 + diff --git a/YtFQT4oBgHgl3EQfeDaR/content/tmp_files/load_file.txt b/YtFQT4oBgHgl3EQfeDaR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..97eba04bdd8d32979599d457ae4a3e7c0eee7391 --- /dev/null +++ b/YtFQT4oBgHgl3EQfeDaR/content/tmp_files/load_file.txt @@ -0,0 +1,2384 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf,len=2383 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='13334v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='ST] 30 Jan 2023 A Bias-Variance-Privacy Trilemma for Statistical Estimation∗ Gautam Kamath† Argyris Mouzakis‡ Matthew Regehr§ Vikrant Singhal� Thomas Steinke∥ Jonathan Ullman∗∗ Abstract The canonical algorithm for differentially private mean estimation is to first clip the samples to a bounded range and then add noise to their empirical mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Clipping controls the sensitivity and, hence, the variance of the noise that we add for privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' But clipping also introduces statistical bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We prove that this tradeoff is inherent – no algorithm can simultaneously have low bias, low variance, and low privacy loss for arbitrary distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' On the positive side, we show that unbiased mean estimation is possible under approximate differential privacy if we assume that the distribution is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Furthermore, we show that, even if we assume that the data is sampled from a Gaussian, unbiased mean estimation is impossible under pure or concentrated differential privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ∗Authors are ordered alphabetically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' †g@csail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Cheriton School of Computer Science, University of Waterloo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ‡amouzaki@uwaterloo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Cheriton School of Computer Science, University of Waterloo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' §matt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='regehr@uwaterloo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Cheriton School of Computer Science, University of Waterloo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' �vikrant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='singhal@uwaterloo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Cheriton School of Computer Science, University of Waterloo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ∥bias@thomas-steinke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Google Research, Brain Team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ∗∗jullman@ccs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='neu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Khoury College of Computer Sciences, Northeastern University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 1 Contents 1 Introduction 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 Our Results .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 30 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2 Final Algorithm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 35 6 An Impossibility Result for Exponential Families Under Pure DP 40 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 Locally Unbiased Estimators Are Globally Unbiased .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 41 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2 Pure DP Estimators Are Uniformly Bounded .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 43 Acknowledgements 44 References 44 A Background on Complex Analysis 50 B Background on Measure Theory 51 C Impossibility Result for Concentrated DP 52 2 1 Introduction While the goal of statistical inference and machine learning is to learn about a population, rather than the sample, most statistical and learning algorithms reveal lot of information that is specific to their sample, raising concerns about the privacy of the individuals who contribute their data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In response, differential privacy [DMNS06] has emerged as the standard framework for addressing these privacy concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Informally, a differentially private algorithm guarantees that no attacker can infer much more about any one individual in the sample than they could have inferred in a hypothetical world where that person’s data was never collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' There is a rich literature providing differentially private algorithms for various statistical inference and machine learning tasks, and many of these are now deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Adding the constraint of differential privacy to a statistical inference or machine learning task can, and often does, incur an inherent cost (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=', [BUV14, DSS+15, KV18, KLSU19]), and there has been a large body of work aimed at pinning down these costs for a variety of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The costs are typically studied via the two-way tradeoff between privacy and error, where error is measured according to a single loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' However, in many applications, we have multiple desiderata for the estimator, not all of which can be captured by a single loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In this work, we study the statistical bias of differentially private estimators, which adds an extra dimension to the tradeoff betweeen privacy and error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Estimators with little or no bias are desirable because error due to variance can be averaged out by combining multiple estimates, whereas error due to bias can be difficult to eliminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Unfortunately, for most estimation tasks, the existing private estimators with optimal error are significantly biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Our main contribution is to show that statistical bias is an inherent feature of private mean estima- tion, by establishing a trilemma among bias, variance, and privacy for the fundamental task of mean estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Estimating the mean of a distribution is both a ubiquitous task on its own, and a subroutine in algorithms for more sophisticated tasks such as stochastic first-order optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We also identify asymmetry as the primary cause of bias in private mean estimation by constructing unbiased private estimators for symmetric distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Bias in Private Mean Estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We study mean estimation for a univariate distribution – given a set of 푛 independent samples 푋1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' , 푋푛 ∈ R from an unknown distribution 푃, estimate the mean 휇(푃) ≔ E푋←푃 [푋], subject to the constraint that the estimator ˆ휇(푋) be differentially private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Without a privacy constraint, we could solve this problem using the empirical mean ¯푋 = 1 푛 �푛 푖=1 푋푖, which is both unbiased and provides optimal error bounds for all the settings we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Research on private mean estimation has pinned down the optimal mean squared error (MSE) E[( ˆ휇(푋) − 휇)2] for a variety of families of distributions, such as subgaussian distributions [KV18, BS19] and distributions satisfying bounded moment conditions [BD14, KSU20], but these estimators can be very biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' There are a variety of methods for optimal mean estimation, all of which introduce bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' To un- derstand the source of bias, it is useful to review one common approach – the noisy clipped mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Specifically, consider the following algorithm 푀(푋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' First, it clips the samples to some bounded range [푎,푏], defined by clip[푎,푏](푥) ≔ \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f4\uf8f3 푎 푥 < 푎 푥 푎 ≤ 푥 ≤ 푏 푏 푏 < 푥 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Next, it computes the empirical mean of the clipped samples ˆ휇[푎,푏](푋) ≔ 1 푛 �푛 푖=1 clip[푎,푏](푋푖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Finally, it perturbs the clipped mean with random noise whose variance is calibrated to the sensitivity of the 3 clipped mean – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=', the width of the clipping interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Specifically, to ensure 휀-differential privacy (휀-DP), we have 푀(푋) = 1 푛 푛 � 푖=1 clip[푎,푏](푋푖) + Lap �푏 − 푎 휀푛 � , where Lap denotes the Laplace distribution, which has mean 0 and variance 2(푏−푎)2 휀2푛2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Since the Laplace distribution has mean 0, we have E[푀(푋)] = E[clip[푎,푏](푋푖)], so the only step that can introduce bias is the clipping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' If we choose a large enough interval so that the support of the distribution 푃 is contained in [푎,푏], then clipping has no effect, and the estimator is unbiased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' However, in this case, [푎,푏] might have to be very wide, resulting in a large variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' On the other hand, if we reduce the variance by choosing a small interval [푎,푏], then we will have E[clip[푎,푏](푋푖)] ≠ E[푋푖] and the estimator will be biased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Thus, we are faced with a non-trivial bias-variance-privacy tradeoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The exact form of the bias and the variance depends on what assumptions we make about 푃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In particular, if we consider the class of distributions 푃 with bounded variance E푋←푃 [(푋 − 휇(푃))2] ≤ 1, then for any 훽 > 0, one can instantiate the noisy-clipped-mean estimator 푀 with an appropriate interval so that it satisfies 휀-DP, has bias at most 훽, and has MSE E � (푀(푋) − 휇(푃))2� ≤ 푂 �1 푛 + 훽2 + 1 푛2 · 휀2 · 훽2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' (1) We show that no private estimator with bias bounded by 훽 can achieve a smaller MSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 Our Results Our first result is the following optimal lower bound on the MSE of any differentially private estimator for the mean of an arbitrary distribution with bounded variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 (Bias-Variance-Privacy Tradeoff).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푀 : R푛 → R be an (휀,훿)-DP algorithm, for some 휀,훿 satisfying 0 < 훿 ≤ 휀2/200 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Suppose 푀 satisfies the following bounds on its bias 훽 and MSE 훼2: for every distribution 푃 with E 푋←푃[푋] = 휇 ∈ [0, 1]1 and E푋←푃 [(푋 − 휇)2] ≤ 1, ���� E 푋←푃푛,푀[푀(푋) − 휇] ���� ≤ 훽 ≤ 1 100 and E 푋←푃푛,푀 � (푀(푋) − 휇)2� ≤ 훼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then 훼2 ≥ Ω � min � 1 푛2 · 휀2 · 훽2 , 1 푛2 · 휀 · 훿1/2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' (2) To interpret the lower bound in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 and compare it to the upper bound (1), it helps to start by assuming 훿 is small – specifically, 훿 ≤ 훽4휀2 – so that the first term in the minimum dominates and the bound simplifies to 훼2 ≥ Ω(1/푛2휀2훽2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' This is the most interesting case, because (휀,훿)-DP is only a meaningful privacy constraint when 훿 is quite small (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=', [KS14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Observe that the upper bound (1) has two other terms, which are not reflected in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1’s lower bound (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' These terms are also inherent, but for reasons unrelated to the privacy constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' First, we also know that 훼2 ≥ Ω(1/푛), 1Since this theorem proves a lower bound, restricting the mean only strengthens the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In particular, even if our estimator is provided a coarse estimate of the mean, we still face the same bias-variance-privacy tradeoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' It is common to consider coarse and fine private mean estimation separately, see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2 for more discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 4 which is a lower bound on the MSE of any mean estimator, even those that are not private (such as the unperturbed empirical mean).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Second, since E � (푀(푋) − 휇)2� ���������������������������������� MSE = E[푀(푋) − 휇]2 ���������������������������� squared bias + E � (푀(푋) − E[푀(푋)])2� �������������������������������������������������������� variance , we have that 훼(푃)2 ≥ 훽(푃)2 for each distribution 푃, where 훽(푃) and 훼(푃)2 denote, respectively, the bias and MSE of the estimator on that distribution 푃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Thus, if we set2 훽 = sup푃 훽(푃) and combine the three lower bounds, we conclude that 훼2 ≥ Ω �1 푛 + 훽2 + 1 푛2 · 휀2 · 훽2 � , (3) which matches the upper bound (1) up to constant factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' However, there is also a corner case when 훿 ≫ 훽4휀2, where the second term in the minimum of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 dominates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For small enough 훽, this privacy guarantee is still meaningful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Thus, for completeness, we address this corner case by constructing an estimator that nearly matches the lower bound of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 in most parameter regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2 (Tightness of Bias-Variance-PrivacyTradeoff).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For all 휀, 훿, 훽 > 0 and푛 ∈ N, there exists an (휀,훿)-DP algorithm 푀 : R푛 → R satisfying the following bias and accuracy properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For any distribution 푃 on R with E 푋←푃[푋] = 휇 ∈ [0, 1], E 푋←푃 � (푋 − 휇)2� ≤ 1, and E 푋←푃 � (푋 − 휇)4� ≤ 휓4, we have ���� E 푋←푃푛,푀[푀(푋) − 휇] ���� ≤ 훽 and E 푋←푃푛,푀 � (푀(푋) − 휇)2� = 푂 �1 푛 + min � 1 푛2 · 휀2 · 훽2 + 훽2, 휓2 푛3/2 · 휀 · 훿1/2 + 1 푛2 · 휀2, 1 푛 · 훿 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The lower bound in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 applies to private mean estimators that are accurate for the entire class of distributions with bounded variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Can we obtain unbiased private estimators by making stronger assumptions on the distribution?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Our subsequent results show that the answer depends on what assumptions we are willing to make.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Namely, we show that a generalization of the lower bound in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 holds even for the case of distributions with bounded higher moments, but we also show that unbiased private mean estimation is possible for symmetric distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Generalization to Higher Moment Bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' If the distribution 푃 is supported on a bounded interval, then unbiased private mean estimation is possible, as clipping to this support becomes an identity operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' More generally, if 푃 is more tightly concentrated, then we can clip more aggressively and obtain better bias-variance-privacy tradeoffs for the noisy clipped mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We consider the class of distributions that satisfy the stronger assumption E푋←푃 [|푋 − 휇|휆] ≤ 1 for some 휆 > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For bias 훽 we can achieve MSE E � (푀(푋) − 휇(푃))2� ≤ 푂 � 1 푛 + 훽2 + 1 푛2 · 휀2 · 훽2/(휆−1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 2Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 permits us to set 훽 ≫ sup푃 훽(푃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Thus, we cannot conclude 훼2 ≥ 훽2 in the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 5 Note that, although we can achieve a lower MSE for the same bias, this tradeoff is still qualitatively similar to the case of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 in the sense that any estimator with vanishing bias 훽 → 0 must have high MSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We prove an analogue of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 showing that this tradeoff is tight for this class of distributions, for every 휆 > 2, and conclude that bias remains an essential feature of private estimation even under stronger concentration assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Symmetric Distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The reason the noisy clipped mean method leads to bias is because clipping to the interval [푎,푏] might affect the distribution asymmetrically, introducing bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Thus, it is natural to consider whether we can achieve unbiased private estimation when the distribution is symmetric around its mean, which holds for many families of distributions like Gaussians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' If the distribution is symmetric and we could clip to an interval [푎,푏] = [휇 − 푐, 휇 + 푐], then the clipped mean would be unbiased, but this would require us to already know 휇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Nonetheless, we construct a private, unbiased mean estimator for any symmetric distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='3 (Unbiased Private Mean Estimation for Symmetric Distributions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For all 휀, 훿 > 0, 휆 > 2, and 푛 ≥ 푂(log(1/훿)/휀) with 훿 ≤ 1/푛, there exists an (휀,훿)-DP algorithm 푀 : R푛 → R satisfying the following: Let 푃 be a symmetric distribution on R (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=', there exists 휇 ∈ R so that 푋 − 휇 and 휇 − 푋 are identically distributed) satisfying E 푋←푃 � |푋 − 휇|휆� ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' If 푋 ← 푃푛, then E[푀(푋)] = 휇, and E � (푀(푋) − 휇)2� ≤ 푂 � 1 푛 + 1 (푛 · 휀)2−2/휆 + 훿 · 휇2 푛 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Note that the MSE in the theorem has a dependance on 휇, which can be unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' However, if we assume that we know some 푟 such that |휇| ≤ 푟, then we can remove this term by setting 훿 ≤ 1/푂(푛푟2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Furthermore, if the distribution 푃 is Gaussian (or subgaussian), then the central moments satisfy E푋←푃 [|푋 − 휇|휆] ≤ 푂(log휆)휆/2 for all 휆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In particular, for the special case of Gaussians with bounded mean, we can set 휆 = Θ(log푛) and the guarantee of our algorithm simplifies to 휇2 ≤ 1/훿 =⇒ E 푋←N(휇,1)푛,푀 � (푀(푋) − 휇)2� ≤ 푂 � 1 푛 + log log푛 푛2 · 휀2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' This matches what is possible without the unbiasedness constraint [KV18, KLSU19, KSU20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We note that, unlike the noisy clipped mean method, and many other methods for private mean estimation, the estimator of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='3 only satisfies (휀,훿)-DP for 훿 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' This is fundamental to the techniques we use;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' our estimator cannot be made to satisfy (휀, 0)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We show that this is inherent by proving that every unbiased mean estimator even for the restricted classes of distributions like Gaussians cannot satisfy (휀, 0)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='4 (Impossibilty of Unbiased Estimators under Pure DP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푀 : R푛 → R be a randomized algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Assume that 푀 satisfies the following guarantee of unbiased estimation for Gaussian data: there is a nonempty open interval (푎,푏) such that for every 휇 ∈ (푎,푏), E 푋←N(휇,1)푛,푀[푀(푋)] = 휇 and E 푋←N(휇,1)푛,푀[|푀(푋) − 휇|] < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then 푀 does not satisfy (휀, 0)-DP for any 휀 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We also extend this impossibility result beyond Gaussians to exponential families and from pure DP to concentrated DP [DR16, BS16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2 Our Techniques We provide two different methods for proving lower bounds on the MSE of low-bias private estimators, such as Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 and its generalization to higher moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' While this is redundant, we hope that offering multiple perspectives can provide deeper insight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Negative Results via the Fingerprinting Method (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The fingerprinting method [BUV14, DSS+15] (alternatively called “tracing attacks” or “membership-inference attacks”) is the main approach for proving optimal lower bounds on the error of differentially private estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Our proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 is based on a refinement of this method that separately accounts for the bias and mean squared error of the estimator, and thus allows for us to prove tradeoffs between these two parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' To give intuition for the argument, we consider the case where 푀 is an unbiased estimator, in which case the argument is similar to the proof of the Cramér-Rao bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Assume that we have a suitable family of distributions 푃휇 with mean E푋←푃휇 [푋] = 휇 and an unbiased estimator 푀 such that E푋←푃푛휇 [푀(푋)] = 휇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' As in the proof of the Cramér-Rao bound, we take the derivative of the unbiased- ness constraint, which gives 1 = d d휇 � E 푋←푃푛휇 ,푀[푀(푋)] � = 푛 � 푖=1 E 푋←푃푛휇 ,푀 � 푀(푋) · d d휇 log푃휇(푋푖) � , where 푃휇(푥) denotes the probability mass or density function of 푃휇 evaluated at푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The (휀,훿)-differential privacy guarantee of 푀 says that 푀(푋) and푋푖 are close to being independent where휀 and훿 quantify the distance from independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Moreover, a straightforward calculation shows that E푋←푃휇 [ d d휇 log푃휇(푋)] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Thus, for all 푖 ∈ [푛], we have E 푋←푃푛휇 ,푀 � 푀(푋) · d d휇 log푃휇(푋푖) � ≈휀,훿 E 푋←푃푛휇 ,푀[푀(푋)] · E 푋←푃푛휇 � d d휇 log 푃휇(푋푖) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Intuitively, this leads to the contradiction 1 = 푛 � 푖=1 E 푋←푃푛휇 ,푀 � 푀(푋) · d d휇 log 푃휇(푋푖) � ≈휀,훿 푛 � 푖=1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' To make this argument precise, we must exactly quantify the approximation ≈휀,훿, which depends both on the privacy parameters 휀 and 훿, as well as on the variances of 푀(푋) and of d d휇 log 푃휇(푋푖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The variance of 푀(푋) is the quantity that we are trying to bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The variance of d d휇 log 푃휇(푋푖) (which is known as the Fisher information) is something we control by choosing the distribution 푃휇 to be a distribution supported on two points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The above proof sketch applies to the unbiased case (훽 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The general case (훽 > 0) introduces some additional complications to the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In particular, we cannot simply consider a single fixed value of the mean parameter 휇, as we must rule out the pathological algorithm that ignores its input sample and outputs 휇, which has somehow been hardcoded into the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' This pathological algorithm trivially satisfies privacy and is unbiased for the single distribution 푃휇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' To rule out this algorithm, we consider a distribution over the parameter 휇 and average over this distribution, where the distribution’s support is wider than the allowable bias 훽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' While we can no longer assume that 1 = d d휇 � E 푋←푃푛휇 ,푀[푀(푋)] � , we can still argue that the derivative must be ≥ Ω(1) on average over the choice of 휇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 7 Negative Results via Amplification (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 Revisited).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We present an alternativeapproach for proving MSE lower bounds on low-bias private estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' While this approach gives slightly weaker bounds than Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1, it exploits less structure of the problem, and thus may be easier to adapt to other settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' This method is a proof by contradiction: We start by assuming the existence of a private estimator and we show that running such an algorithm on independent datasets and av- eraging the results would violate previous lower bounds on the mean squared error of private mean estimators [KSU20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We start with an (휀,훿)-DP private estimator 푀 : R푛 → R that takes 푛 samples from some dis- tribution and estimates its mean with bias 훽 and variance 휎2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then we construct a new estimator 퐴푚 that takes 푛푚 samples, randomly splits them into 푚 blocks of 푛 samples each, runs 푀 on each block, and averages the outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' This averaging won’t reduce the bias but will reduce the variance by a factor of 푚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Thus, the MSE of 퐴푚 is 훽2 + 휎2/푚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Moreover, privacy amplification by shuffling [EFM+19, CSU+19, BBGN19, FMT22, FMT23] shows that 퐴푚 is (휀′, 훿 ′)-DP for 휀′ = � 푂(휀 ·푚−1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' To com- plete the proof, we can apply any lower bound on the MSE of the private estimator 퐴푚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In particular, if we consider the class of distributions with bounded variance, then we can use the lower bound of Kamath, Singhal, and Ullman [KSU20], which shows that the MSE of 퐴푚 is 훽2 + 휎2/푚 ≥ Ω(1/푛푚휀′) ≥ �Ω(1/푛푚1/2휀).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Setting 푚 appropriately yields a lower bound on 휎 that roughly matches Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' General-Purpose Low-Bias Mean Estimation (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The estimator we construct in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2 combines two well-known techniques: the noisy-clipped-mean that we already discussed and the so-called name-and-shame algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' As discussed above, the noisy-clipped-mean satisfies pure DP, but it leads to a tradeoff between bias and variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Name-and-shame is a pathological algorithm, which with probability 훿푛, outputs one random sam- ple in the dataset without any privacy protection, and otherwise outputs nothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' This algorithm sat- isfies (0,훿)-DP and can be used as the basis for an unbiased estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Specifically, given a sample 푋 ← 푃푛, with probability 훿푛, output 푋퐼/훿푛 for a uniformly random 퐼 ∈ [푛], and otherwise output 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' This estimator is unbiased, but the variance scales with 1/훿푛, which is impractically large for reasonable values of 훿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Intuitively, our estimator breaks the distribution into the tail portion (far from the mean) and the head portion (close to the mean), and uses the noisy-clipped mean on the head and then uses name- and-shame on the tail to correct the bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' That is, suppose we estimate 1 푛 �푛 푖=1 clip[푎,푏](푋푖) with noise addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then we can estimate the tail 1 푛 �푛 푖=1(푋푖 − clip[푎,푏](푋푖))) using name-and-shame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Combining these two estimates yields an unbiased estimator, such that the variance of name-and-shame is greatly reduced because 푋푖 − clip[푎,푏](푋푖) = 0 with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Unbiased Mean Estimation for Symmetric Distributions (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Our unbiased private estimator for symmetric distributions is based on the estimator of Karwa and Vadhan [KV18], with some modifications to ensure unbiasedness, so we begin by reviewing the key ideas in their estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Their estimator has three steps: First, obtain a coarse estimate �휇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Second, use this coarse estimate to compute a clipped mean �휇 ≔ 1 푛 �푛 푖=1 clip[�휇−푐,�휇+푐] (푋푖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Finally, add noise to this fine estimate �휇, so 푀(푋) = �휇 + Lap� 2푐 푛휀 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The coarse estimate only needs to satisfy a minimal accuracy guarantee;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' roughly, we require |�휇 − 휇(푃)| ≤ 푂(휎) with high probability, where 휎2 is the variance of 푃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We can compute such an estimate using a histogram where we split the real line into intervals of length 푂(휎) and pick an interval that contains many samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' With high probability, there is at least one interval containing many samples and the midpoint of any such interval is a good coarse estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We can privately select such an interval 8 with high probability – even though there are infinitely many intervals to choose from – using standard techniques in DP [KKMN09, Vad17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Intuitively, we modify the algorithm of Karwa and Vadhan so that the symmetry of the distribution 푃 is preserved in our estimates �휇 and �휇 of the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The only part of the algorithm that breaks the symmetry is when we split the real line into intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We fix this by adding a random offset to the intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' That is, up to scaling, our intervals are of the form {[ℓ + 푇, ℓ + 1 + 푇) : ℓ ∈ Z}, where 푇 ∈ [−1/2, +1/2) is uniformly random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Note that the distribution of this set is equivariant under translation, so we can proceed with the analysis as if we had translated it to be symmetric around the unknown mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The key observation for the analysis is that, if the coarse estimate �휇 has a symmetric distribution with center 휇, then the clipping does not introduce bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' This is because the clipping is equally likely to introduce positive bias or negative bias and this averages out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' There are two additional technicalities in the algorithm: First, we must ensure that the coarse estimate �휇 is independent by using independent samples for the two stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Second, the coarse estimation procedure may fail to produce an estimate �휇 because no interval contains enough samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In this case, we fall back on a version of the name-and- shame algorithm that is unbiased but has high variance 푂(휇2/훿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Since the probabiltiy of needing this fallback will be 푂(훿2), this case does not contribute much to the overall variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Negative Result for Pure DP Unbiased Mean Estimation (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Suppose, for the sake of contradiction, that 푀 is an 휀-DP unbiased estimator for 휇 ∈ (푎,푏) given samples from N (휇, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Consider the function 푔(휇) ≔ E 푋←N(휇,1)푛,푀[푀(푋)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We first observe that, if 푀 is unbiased for 휇 ∈ (푎,푏), then it must be unbiased for all 휇 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Intuitively, this holds because of the smoothing induced by the Gaussian distribution – if 푀(푋) were biased when the data is drawn from N (휇∗, 1) for some 휇∗, then the same data could occur under any other distribution N (휇, 1) with some tiny but non-zero probability, and thus 푀(푋) would also be biased for 휇 ∈ (푎,푏), as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Formally, we show 푔 is analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Thus, its value on the whole real line can be determined from the interval (푎,푏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Second we observe that, if 푀(푋) has bounded mean absolute deviation E푋←N(휇∗,1)푛 [|푀(푋) − 휇∗|] for some 휇∗, then E[|푀(푥)|] must be uniformly bounded for all inputs 푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Namely, |푔(휇)| ≤ E 푋←N(휇,1)푛[|푀(푋)|] ≤ |휇∗| + E 푋←N(휇,1)푛[|푀(푋) − 휇∗|] ≤ |휇∗| + 푒휀푛 E 푋←N(휇∗,1)푛[|푀(푋) − 휇∗|] for all 휇 and some fixed 휇∗ that does not depend on 휇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' This follows from the strong group privacy property of (휀, 0)-DP algorithms – changing the entire sample 푋 ← N (휇∗, 1)푛 to any other sample 푥 can only change the distribution of 푀(푋) by a pointwise multiplicative factor of 푒휀푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The first observation shows that 푔(휇) = 휇 on the entire real line 휇 ∈ R, and the second observation shows that 푔(휇) is uniformly bounded on the entire real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' This is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' This result extends beyond the Gaussian distribution to exponential families.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Furthermore, it ex- tends from pure (휀, 0)-DP to concentrated DP [DR16, BS16] and other variants of differential privacy that satisfy a strong group privacy property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' However, it does not apply to approximate (휀,훿)-DP be- cause the group privacy guarantee underpinning the second observation breaks when 훿 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='3 Related Work Unbiased estimators have long been a topic of interest in statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For example, topics such as the min- imum variance unbiased estimator (MVUE) and the best linear unbiased estimator (BLUE) are textbook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 9 A number of celebrated results derive properties of estimators with low or no bias, often proving certain estimators are optimal within this class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Some examples include the Gauss-Markov theorem [Gau23, Mar00], the Lehman-Scheffé theorem [LS11, LS12], and the Cramèr-Rao bound [Cra99, Rao92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' These results often focus on unbiased estimators for mathematicalconvenience: it is easier to prove optimality within this restricted class than for general estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Within the context of differential privacy, relatively little work has considered the bias of private estimators separately from their overall mean squared error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' A number of works [DJW13, BD14, KV18, KLSU19, KSU20] bound the bias of the clipped mean, though only to the ends of trying to minimize the overall error of the estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Amin, Kulesza, Munoz, and Vassilvitskii [AKMV19] examine bias- variance tradeoffs of a similar procedure in the context of private empirical risk minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Ka- math, Liu, and Zhang [KLZ22] employ the mean estimation approach of Kamath, Singhal, and Ull- man [KSU20] as an oracle for stochastic first-order optimization, but, due to specifics of their setting, employ a different balance between bias and noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' They raise the question of whether unbiased al- gorithms for mean estimation exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Barrientos, Williams, Snoke, and Bowen [BWSB21b, BWSB21a] empirically measure the bias induced by various mean estimation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Zhu, van Hentenryck, and Fioretto [ZVHF21] study bias due to post-processing of differentially private statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Evans, King, and Thakurta [EK21, EKST22] and Covington, He, Honaker, and Kamath [CHHK21] give methods for unbiased private estimation, though these rely upon strong assumptions or caveat their unbiasedness guarantees (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=', guaranteeing a statistic is unbiased only with high probability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Beyond considerations of bias, private statistical estimation has been a topic of much recent in- terest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Mean estimation is perhaps the most fundamental question in this space, enjoying significant attention (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=', [BD14, KV18, BS19, KLSU19, KSU20, WXDX20, DFM+20, BDKU20, CWZ21, BGS+21, HLY21, LKKO21, LKO22, KLZ22, HKM22, KMV22, TCK+22, DHK23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Most relevant to our work are those which focus on estimation in settings with bounds on only the low-order central moments of the underlying distribution [BD14, KSU20, HKM22], as the bias introduced due to clipping is more sig- nificant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Other related problems involve private covariance or density estimation [BKSW19, AAK21, KMS+22b, AL22, AKT+22, HKMN22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Beyond these settings, other works have examined statistical estimation under privacy constraints for mixtures of Gaussians [KSSU19, AAL21, CCd+23], graphical models [ZKKW20], discrete distributions [DHS15], median estimation [AMB19, TVGZ20, RC21, RJC22, BEMZ22], and more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Several recent works explore connections between private and robust estima- tion [LKKO21, HKM22, GH22, LKO22, KMV22, AKT+22, HKMN22, CCd+23] and between privacy and generalization [HU14, DFH+15, SU15, BNS+16a, RRST16, FS17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Emerging directions of interest include guaranteeing privacy when one person may contribute multiple samples [LSY+20, LSA+21, GRST22], a combination of local and central DP for different users [ADK19], or estimation with access to some public data [BKS22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' See [KU20] for more coverage of recent work on private statistical estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 2 Preliminaries Here, we provide a brief background on certain families of distributions and on DP, along with a few notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We defer the information on the other tools (in complex analysis and measure theory) that are used in this work to Appendices A and B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We denote the set of all distributions over R by Δ(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We denote that 푋 is a random variable drawn from a distribution 푃 by 푋 ← 푃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We denote that two random variables 푋 and 푌 have the same distribution by 푋 ∼ 푌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 10 Let 푃,푄 be distributionsover a set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then for휀,훿 ≥ 0, we say that푃 and푄 are (휀,훿)-indistinguishable (denoted by 푃 ∼휀,훿 푄), if for all measurable 푆 ⊆ X, 푒−휀 · � P 푋←푄[푋 ∈ 푆] − 훿 � ≤ P 푋←푃[푋 ∈ 푆] ≤ 푒휀 · P 푋←푄[푋 ∈ 푆] + 훿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For any distribution 푃 over a set X and 푆 ⊆ X, we say 푃 (푆) ≔ P 푋←푃[푋 ∈ 푆].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For a distribution 푃 over 푈 ⊆ R, we denote the product distribution over 푈 푑, with each marginal being 푃, by 푃푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푀 : R푛 → R be an estimator that takes 푛 samples 푥 = (푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ,푥푛) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' from a distribution 푃 ∈ Δ(R) (with 휇(푃) ≔ E 푋←푃[푋]) as input, and outputs an estimate 푀(푥) of 휇(푃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 휇푀 ≔ E 푋←푃푛,푀[푀(푋)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then we define the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' – Bias of 푀: 훽 ≔ |휇푀 − 휇(푃)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' – Variance of 푀: 휎2 ≔ E 푋←푃푛,푀 � (푀(푋) − 휇푀)2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' – Mean Squared Error of 푀: 훼2 ≔ E 푋←푃푛,푀 � (푀(푋) − 휇(푃))2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The standard bias-variance decomposition of mean squared error gives 훼2 = 훽2 + 휎2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Throughout this manuscript, for any 푥 ∈ R, we use the following notation to denote clipping of 푥 to within an interval [푎,푏] ⊆ R: clip[푎,푏](푥) ≔ min{max{푥, 푎},푏}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 Differential Privacy A dataset 푥 = (푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ,푥푛) ∈ X푛 is a collection of elements from a data universe X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 (Differential Privacy (DP) [DMNS06]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We say that two datasets 푥, 푥 ′ ∈ X푛 are neigh- boring (denoted by, 푥 ∼ 푥 ′) if they differ in at most one entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For 휀,훿 ≥ 0, a randomized algorithm 퐴 : X푛 → Y satisfies (휀,훿)-differential privacy if for every neighboring pair of datasets 푥 ∼ 푥 ′ ∈ X푛, ∀푌 ⊆ Y, P[퐴(푥) ∈ 푌] ≤ 푒휀 · P[퐴(푥 ′) ∈ 푌] + 훿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' This property is called pure DP (or 휀-DP) when 훿 = 0, and approximate DP when 훿 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' This definition is closed under post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2 (Post-Processing [DMNS06]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' If 푀 : X푛 → Y is (휀,훿)-DP and 푃 : Y → Z is any random- ized function, then the algorithm 푃 ◦ 푀 is (휀,훿)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' It is also closed under pre-processing that respects the neighboring relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='3 (Pre-Processing).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' If 푀 : X푛 → Y is (휀,훿)-DP and 푃 : X푛 → X푛 is any randomized function such that 푥 ∼ 푥 ′ implies 푃 (푥) ∼ 푃 (푥 ′) with probability 1, then the algorithm 푀 ◦ 푃 is (휀,훿)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For any 푌 ⊆ Y and 푥 ∼ 푥 ′ ∈ X푛, 푃 (푥) ∼ 푃 (푥 ′) with probability 1, so P[(퐴 ◦ 푃)(푥) ∈ 푌] = E 푃 [P[퐴(푃 (푥)) ∈ 푌]] ≤ E 푃 [푒휀P[퐴(푃 (푥 ′)) ∈ 푌] + 훿] = 푒휀P[(퐴 ◦ 푃)(푥 ′) ∈ 푌] + 훿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We have our result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' □ 11 The following result quantifies the privacy guaranteed by a DP algorithm for a group of individuals within a dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='4 (Group Privacy [DMNS06, DR14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 퐴 : X푛 → Y be (휀,훿)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then for any integer 푘 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ,푛}, subset 푌 ⊆ Y, and pairs of datasets 푥, 푥 ′ ∈ X푛 differing in 푘 entries, P[퐴(푥) ∈ 푌] ≤ 푒푘휀 · P[퐴(푥 ′) ∈ 푌] + 푒푘휀 − 1 푒휀 − 1 · 훿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Common Differentially Private Mechanisms The following defines the sensitivity of a function to changing a single point within a dataset in the worst case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='5 (ℓ1-Sensitivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푓 : X푛 → R be a function, its ℓ1-sensitivity is Δ푓 ,1 ≔ sup 푥∼푥′∈X푛|푓 (푥) − 푓 (푥 ′)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Our estimators will use the well-known Laplace mechanism to ensure DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='6 (Laplace Mechanism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푓 : X푛 → R be a function with ℓ1-sensitivity Δ푓 ,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then the Laplace mechanism 푀(푋) ≔ 푓 (푋) + Lap(Δ푓 ,1/휀) satisfies 휀-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Furthermore, P � |푀(푋) − 푓 (푋)| ≥ Δ푓 ,1 · log(1/훽) 휀 � ≤ 훽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Our estimators will also make use of DP mechanisms for computing noisy histograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In particular, we will use well-known approximate DP algorithms that release a histogram in such a way that the maximum error on every bucket depends only on the privacy parameters, and is independent of the number of buckets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='7 (DP Histograms [KKMN09, BNS16b, Vad17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푥 = (푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ,푥푛) be a multiset of samples from some (possibly infinite) data universe 푈 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let Ω be a partition of 푈 into a (possibly infinite) collection of disjoint subsets (the “bins” of the histogram) and define the vector ℎ(푥) ∈ RΩ as ℎ(푥)휔 = |푥 ∩휔|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' There is an (휀,훿)-DP algorithm 푀 : 푈 푛 → RΩ with the following guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' With probability at least 1 − 훽, max 휔∈Ω |푀(푥)휔 − ℎ(푥)휔 | = 푂(log(1/훿훽)/휀), and the algorithm runs in time poly(푛, log(1/휀훽)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2 Non-Private Error of Mean Estimation Given independent samples 푋1, · · · , 푋푛 ∈ R from an unknown distribution 푃, the empirical mean ˆ휇(푋) ≔ 1 푛 �푛 푖 푋푖 is an unbiased estimator of the distribution mean 휇(푃) ≔ E 푋←푃[푋] and its mean squared error is 훼2 ≔ E 푋←푃푛 � ( ˆ휇(푋) − 휇(푃))2� = E 푋←푃 � (푋 − 휇(푃))2� 푛 = 푂(1/푛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' This mean squared error is asymptotically optimal in a minimax sense and is optimal for the univariate Gaussian case 푃 = N (휇, 휎2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We have the following well-known result which shows that the empirical mean is asymptotically optimal even for the simple case of Bernoulli data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 12 Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푀 : {0, 1}푛 → R be an estimator with the following property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ∀푝 ∈ [0, 1] E 푋←Bernoulli(푝)푛 � (푀(푋) − 푝)2� ≤ 훼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then 훼2 ≥ 1 6(푛+2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The empirical mean attains MSE E 푋←Bernoulli(푝)푛 � ( ˆ휇(푋) − 푝)2� = 푝 (1−푝) 푛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' However, this is not the minimax optimal estimator of the mean of a Bernoulli distribution, rather it is the biased estimator ˇ휇(푋) ≔ 1 푛 + √푛 �√푛 2 + 푛 � 푖 푋푖 � , which has MSE E 푋←Bernoulli(푝)푛 � ( ˇ휇(푋) − 푝)2� = 1 4(√푛 + 1)2 for all 푝 ∈ [0, 1] [HL50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푃 ∈ [0, 1] be uniform and, conditioned on 푃, let 푋 ← Bernoulli(푃)푛 be 푛 independent bits, each with conditional expectation 푃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Note that the marginal distribution of �푛 푖 푋푖 is uniform on {0, 1, · · · ,푛}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Given 푋 = 푥, the conditional distribution of 푃 is 푃|푋=푥 ∼ Beta � 1 + 푛 � 푖 푥푖, 1 + 푛 � 푖 (1 − 푥푖) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In terms of mean squared error, the best estimator of 푃 is simply the mean of this conditional dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' That is, the function 푓 : {0, 1}푛 → R that minimizes E 푃,푋 � (푃 − 푓 (푋))2� is the conditional expectation 푓 (푥) = E[푃 | 푋 = 푥].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Indeed, this is the definition of conditional expectation in the general measure theoretic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Consequently, the best possible mean squared error of an estimator of 푃 given 푋 is the variance of this conditional distribution 푃|푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The distribution Beta(푎,푏) has mean 푎 푎+푏 and variance 푎푏 (푎+푏)2(푎+푏+1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Now we have 훼2 ≥ E 푃←[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='푋←Bernoulli(푃)푛 � (푀(푋) − 푃)2� ≥ E 푋 � E 푃 �� E 푃 [푃 | 푋] − 푃 �2 | 푋 �� = E 푃←[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='푋←Bernoulli(푃)푛 ��1 + �푛 푖 푋푖 2 + 푛 − 푃 �2� = E 푃←[0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='푋←Bernoulli(푃)푛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='� (1 + �푛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='푖 푋푖)(1 + �푛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='푖 (1 − 푋푖)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='(푛 + 2)2(푛 + 3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='푛 + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='푛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='푘=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='(1 + 푘)(1 + 푛 − 푘) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='(푛 + 2)2(푛 + 3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='(푛 + 1)(푛 + 2)2(푛 + 3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='푛 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='푘=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='(1 + 푛) + 푛 · 푘 − 푘2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='(푛 + 1)(푛 + 2)2(푛 + 3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='(1 + 푛) · (푛 + 1) + 푛 · 푛(푛 + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='− 푛(푛 + 1)(2푛 + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='= 6(푛 + 1)2 + 3푛2(푛 + 1) − 푛(푛 + 1)(2푛 + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='6(푛 + 1)(푛 + 2)2(푛 + 3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='= 6(푛 + 1) + 3푛2 − 푛(2푛 + 1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='6(푛 + 2)2(푛 + 3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='5푛 + 6 + 푛2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='6(푛 + 2)2(푛 + 3) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='(푛 + 2)(푛 + 3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='6(푛 + 2)2(푛 + 3) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='6(푛 + 2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' □ If we change the distribution of 푃 ∈ [0, 1] from uniform to Beta(√푛/2, √푛/2) in the above proof, then we obtain the stronger conclusion 훼2 ≥ 1 4(√푛+1)2 , which is exactly optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' However, this requires a more complicated calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 3 Bias-Variance-Privacy Trilemma for General-Purpose Estimators We now prove our main negative result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Informally, we show that if an algorithm is differentially private and has low bias, then it must have high error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' There are, of course, other parameters that arise in the analysis, such as bounds on the tails of the unknown distribution 푃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We provide two different proofs, which give slightly different results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The first proof directly applies the fingerprinting technique for lower bounds on differentially private estimation [BUV14], while the second proof is a “black-box” reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 Negative Result via Fingerprinting We begin by stating our general result and, before giving the proof, we provide some remarks and corollaries to help interpret the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 (Bias-Variance-Privacy Tradeoff).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 휀,훿, 훽, 훼,휏 ≥ 0 and 휆 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푀 : R푛 → R be an (휀,훿)-DP algorithm that satisfies the following bias and accuracy properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For any distribution 푃 on R with 휇(푃) ≔ E 푋←푃[푋] ∈ [0, 1] and E 푋←푃 � |푋 − 휇(푃)|휆� ≤ 1, we have the following: ���� E 푋←푃푛,푀[푀(푋)] − 휇(푃) ���� ≤ 훽, E 푋←푃푛,푀[|푀(푋) − 휇(푃)|] ≤ 훼, ∫ ∞ 0 min � 훿, P 푋←푃푛,푀[|푀(푋) − 휇(푃)| > 푥] � d푥 ≤ 훼 · 휏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' If 16훽 ≤ 훾 ≤ 1/5, then 훼 ≥ 1 32푛 · sinh(휀) · 훾1/(휆−1) + 16푛 · 휏 · 훾−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Note that, for small values of 휀, sinh(휀) ≈ 휀, but, for large 휀, sinh(휀) ≈ 1 2푒휀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Since this is the “usual” dependence on 휀 in many such bounds under the constraint of DP, sinh allows us to capture behaviour in both regimes with a single function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The first two accuracy conditions are not hard to interpret: they control the bias and the mean absolute error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The parameter 훽 bounds the bias of the algorithm, while 훼 bounds the mean absolute deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' By Jensen’s inequality, ���� E 푋←푃푛,푀[푀(푋)] − 휇(푃) ���� ≤ E 푋←푃푛,푀[|푀(푋) − 휇(푃)|] ≤ � E 푋←푃푛,푀 � (푀(푋) − 휇(푃))2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 14 Thus, we can assume that 훽 ≤ 훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Furthermore, if an estimator 푀(푋) has a bound on the mean squared error of 훼2, it consequently also has a mean absolute error of at most 훼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Thus, our somewhat uncon- ventional assumption controlling the mean absolute error only broadens the class of estimators against which our lower bound holds: for interpretability, one could instead replace this with 훼2 being the mean squared error of 푀(푋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The standard decomposition gives that the mean squared error 훼2 is simply the variance of the estimator plus the square of the bias 훽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The third property and the parameter 휏 is somewhat hard to interpret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We note that this condition is implied by a bound on the MSE of the estimator via the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' It applies in more general circumstances, as well, when we may have bounds on higher or lower moments of the estimator’s error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2 (Setting 휏 = 훿1−1/휅 in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 훼, 훿 ≥ 0 and 휅 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푌 be a random variable satisfying E[|푌 |휅] ≤ 훼휅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then ∫ ∞ 0 min{훿, P[|푌 | > 푥]} d푥 ≤ 훼 · 훿1−1/휅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We assume, without loss of generality, that 훿 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Suppose, for now, there exists 푐 > 0 such that P[|푌 | > 푐] = 훿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' If 푥 ≥ 푐, then min{훿, P[|푌 | > 푥]} = P[|푌 | > 푥] = P[|푌 | · I[|푌 | > 푐] > 푥].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Likewise, if 푥 ≤ 푐, then min{훿, P[|푌 | > 푥]} = 훿 = P[|푌 | · I[|푌 | > 푐] > 푥].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Thus, ∫ ∞ 0 min{훿, P[|푌 | > 푥]}d푥 = ∫ ∞ 0 P[|푌 | · I[|푌 | > 푐] > 푥] = E[|푌 | · I[|푌 | > 푐]] ≤ E[|푌 |휅] 1 휅 · E � I[|푌 | > 푐] 휅 휅−1 � 휅−1 휅 (Hölder’s Inequality) = E[|푌 |휅] 1 휅 · P[|푌 | > 푐]1−1/휅 ≤ 훼 · 훿1−1/휅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' If the distribution of 푌 is continuous, then such a quantity 푐 is guaranteed to exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In general, there exists 푐 ≥ 0 such that P[|푌 | > 푐] ≤ 훿 ≤ P[|푌 | ≥ 푐].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We can define a random 퐼 : R → {0, 1} such that I[|푌 | > 푐] ≤ 퐼 (|푌 |) ≤ I[|푌 | ≥ 푐] with probability 1 and E[퐼 (|푌 |)] = 훿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The above proof carries through in general if we replace I[|푌 | > 푐] with 퐼 (|푌 |).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' □ In particular, if we have a mean squared error bound for the estimator E � (푀(푋) − 휇(푃))2� ≤ 훼2, then the third condition of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 holds with 휏 = √ 훿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Larger values of 휅 entail sharper tail bounds on the estimator, allowing us to set 휏 larger (and thus implying stronger lower bounds), with 휏 → 훿 as 휅 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In general, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1’s lower bound on the error 훼 is maximized by setting 훾 = clip[16훽,1/5] �� (휆 − 1)휏 2 sinh(휀) �1−1/휆� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' (4) Combining this parameter setting for 훾, along with the bound of 휏 = 훿1−1/휅 given by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2, and focusing on the most natural case of휅 = 2 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=', we assume only that the estimator has bounded variance), gives the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 15 Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='3 (Combining Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2 (with 휅 = 2), and Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푀 : R푛 → R be (휀,훿)-DP and satisfy the following bias and accuracy properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For any distribution 푃 on R with 휇(푃) ≔ E 푋←푃[푋] ∈ [0, 1] and E 푋←푃 � |푋 − 휇(푃)|휆� ≤ 1, we have ���� E 푋←푃푛,푀[푀(푋)] − 휇(푃) ���� ≤ 훽 and E 푋←푃푛,푀 � (푀(푋) − 휇(푃))2� ≤ 훼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' If 훽 ≤ 1 80 and 훿 ≤ � 2·sinh(휀) 51+ 1 휆−1 ·(휆−1) �2 , then 훼 ≥ 1 32 · 푛 · sinh(휀) · 휆 휆−1 · max � (16훽) 1 휆−1, � (휆−1) √ 훿 2·sinh(휀) �1/휆� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We illustrate the representative case where the underlying distribution has bounded variance by further fixing 휆 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Combining the resulting lower bound with the non-private rate (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='8) gives the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='4 (Setting 휆 = 2 in Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='3 to get Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푀 : R푛 → R be (휀,훿)-DP and satisfy the following bias and accuracy properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For any distribution 푃 on R with 휇(푃) ≔ E 푋←푃[푋] ∈ [0, 1] and E 푋←푃 � (푋 − 휇(푃))2� ≤ 1, we have ���� E 푋←푃푛,푀[푀(푋)] − 휇(푃) ���� ≤ 훽 and E 푋←푃푛,푀 � (푀(푋) − 휇(푃))2� ≤ 훼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' If 훽 ≤ 1/80 and 훿 ≤ � 2 25 sinh휀�2, then 훼 ≥ max \uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f4\uf8f4\uf8f3 1 � 6(푛 + 2) , 1 64 · 푛 · sinh(휀) · max � 16 · 훽, � √ 훿 2·sinh(휀) � \uf8fc\uf8f4\uf8f4\uf8f4\uf8f4\uf8fd \uf8f4\uf8f4\uf8f4\uf8f4\uf8fe ≥ Ω � 1 √푛 + min � 1 푛휀훽, 1 푛 � 휀 √ 훿 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Now, we prove our general result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Our proof follows what is known as the fingerprinting approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Fingerprinting codes were first studied by Boneh and Shaw [BS98] in the context of cryptographic traitor-tracing schemes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Tardos [Tar08] gave an optimal construction of fingerprinting codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Finger- printing codes were used to prove negative results for differentially private algorithms by Bun, Ull- man, and Vadhan [BUV14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Subsequently, many works have expanded this methodology [SU15, BSU17, SU17a, SU17b, KMS22a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We will use the following lemma in the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='5 (Fingerprinting Derivative Lemma [SU17b, Lemma 9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푓 : {0, 1}푛 → R be an arbitrary function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Define 푔 : [0, 1] → R by 푔(푝) = E[푓 (푋)], where 푋1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ,푋푛 be independent samples from Bernoulli(푝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then, for all 푝 ∈ [0, 1], we have 푔′(푝) · 푝(1 −푝) = E � 푓 (푋) · �푛 푖 (푋푖 − 푝) � , where 푋1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' , 푋푛 be independent samples from Bernoulli(푝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 16 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For 푝 ∈ [0, 1] and 푣 > 0, define D푣,푝 = 푣 · Bernoulli(푝) – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=', a sample from D푣,푝 is 0 with probability 1 − 푝 and 푣 with probability 푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then 휇(D푣,푝) = E 푋←D푣,푝 [푋] = 푣푝 and E 푋←D푣,푝 � |푋 − 휇(D푣,푝)|휆� = (1 − 푝)(푣푝)휆 + 푝(푣(1 − 푝))휆 ≤ 2푝푣휆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' If we ensure 푣 ≤ (2푝)−1/휆 ≤ 1/푝, then the 휆-th absolute central moment is below 1, and the mean is in the interval [0, 1], so the bias and accuracy guarantees of 푀 apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For 푣 > 0, define 푔푣 : [0, 1] → R by 푔푣(푝) ≔ E 푋←D푛푣,푝,푀[푀(푋)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='5, for all 푣 > 0 and 푝 ∈ [0, 1], we have E 푋←D푛 푣,푝,푀 � 푀(푋) · 푛 � 푖 �1 푣 푋푖 − 푝 �� = 푝(1 − 푝)푔′ 푣(푝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' (5) Fix 0 < 푎 < 푏 ≤ 1/2 and 0 < 푣 ≤ (2푏)−1/휆 (to be determined later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Now, let 푃 ∈ [푎,푏] be a random variable with density ∝ 1 푃 (1−푃) – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=', ∀푡 ∈ [푎,푏], P[푃 ≤ 푡] = ∫ 푡 푎 1 푥 (1−푥) d푥 ∫ 푏 푎 1 푥 (1−푥) d푥 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Conditioned on 푃, let 푋1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' , 푋푛 ∈ R be independent samples from D푣,푃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Now, E 푃,푋,푀 � 푀(푋) · 푛 � 푖 �1 푣 푋푖 − 푃 �� = E 푃 � 푃 (1 − 푃)푔′ 푣(푃) � (Equation 5) = ∫ 푏 푎 푔′ 푣(푝)d푝 ∫ 푏 푎 1 푥 (1−푥) d푥 = 푔푣(푏) − 푔푣(푎) log(푏/(1 − 푏)) − log(푎/(1 − 푎)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' By our bias assumption, |푔푣(푏) − 푣푏| ≤ 훽 and |푔푣(푎) − 푣푎| ≤ 훽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Thus, E 푃,푋,푀 � 푀(푋) · 푛 � 푖 �1 푣 푋푖 − 푃 �� ≥ 푣 · (푏 − 푎) − 2훽 log � 푏·(1−푎) 푎·(1−푏) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Since E[푋푖] = 푣푃 for all 푖, we can center 푀(푋) and rearrange slightly: 푛 � 푖 E 푃,푋,푀 � (푀(푋) − 푣푃) · �1 푣 푋푖 − 푃 �� ≥ 푣 · (푏 − 푎) − 2훽 log � 푏·(1−푎) 푎·(1−푏) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Next, we will use differential privacy to prove an upper bound on this quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Fix an arbitrary 푖 ∈ [푛] and fix 푃 = 푝 ∈ [푎,푏].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Our goal is to upper bound E 푋←D푛푣,푝,푀 � (푀(푋) − 푣푝) · � 1 푣푋푖 − 푝�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Since 푀 satisfies (휀,훿)-DP, the distribution of the pair (푀(푋), 푋푖) is (휀,훿)-indistinguishable from that of � 푀(푋−푖, ˜푋푖),푋푖 � , where (푋−푖, ˜푋푖) denotes the dataset 푋 with 푋푖 replaced by ˜푋푖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' here ˜푋푖 ← D푣,푝 is a fresh sample from the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Now ˜푋푖 and 푋푖 are interchangeable, this means the distribution 17 of � 푀(푋−푖, ˜푋푖),푋푖 � is identical to that of � 푀(푋), ˜푋푖 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' By transitivity, the distribution of (푀(푋), 푋푖) is (휀,훿)-indistinguishable from that of � 푀(푋), ˜푋푖 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In particular, (푀(푋) − 푣푝) · � 1 푣푋푖 − 푝� ∼휀,훿 � 푀(푋−푖, ˜푋푖) − 푣푝 � � 1 푣푋푖 − 푝� ∼ (푀(푋) − 푣푝) · � 1 푣 ˜푋푖 − 푝 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We also have ��(푀(푋) − 푣푝) · � 1 푣푋푖 − 푝��� ≤ |푀(푋) − 푣푝| with probability 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Thus E � (푀(푋) − 푣푝) · �1 푣 푋푖 − 푝 �� = E � max{(푀(푋) − 푣푝) · �1 푣 푋푖 − 푝 � , 0} � − E � max{−(푀(푋) − 푣푝) · �1 푣 푋푖 − 푝 � , 0} � = ∞ ∫ 0 P � (푀(푋) − 푣푝) · �1 푣 푋푖 − 푝 � > 푥 � d푥 − ∞ ∫ 0 P � (푀(푋) − 푣푝) · �1 푣 푋푖 − 푝 � < −푥 � d푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We have P � (푀(푋) − 푣푝) · �1 푣 푋푖 − 푝 � > 푥 � ≤ P[|푀(푋) − 푣푝| > 푥] and simultaneously, P � (푀(푋) − 푣푝) · �1 푣 푋푖 − 푝 � > 푥 � ≤ 푒휀 · P � (푀(푋) − 푣푝) · �1 푣 ˜푋푖 − 푝 � > 푥 � + 훿 = 푒휀 · 푝 · P[(푀(푋) − 푣푝) · (1 − 푝) > 푥] + 푒휀 · (1 − 푝) · P[(푀(푋) − 푣푝) · (0 − 푝) > 푥] + 훿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Define 훿(푥) ≔ min{훿, P[|푀(푋) − 푣푝| > 푥]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then ∞ ∫ 0 P � (푀(푋) − 푣푝) · �1 푣 푋푖 − 푝 � > 푥 � d푥 ≤ ∞ ∫ 0 푒휀 · 푝 · P[(푀(푋) − 푣푝) · (1 − 푝) > 푥]d푥 + ∞ ∫ 0 푒휀 · (1 − 푝) · P[(푀(푋) − 푣푝) · (0 − 푝) > 푥]d푥 + ∞ ∫ 0 훿(푥)d푥 = E[푒휀 · 푝 · max{(푀(푋) − 푣푝) · (1 − 푝), 0}] + E[푒휀 · (1 − 푝) · max{(푀(푋) − 푣푝) · (0 − 푝), 0}] + ∞ ∫ 0 훿(푥)d푥 18 ≤ 푒휀 · 푝(1 − 푝) · E[|푀(푋) − 푣푝|] + 훼 · 휏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In the above, the final inequality holds because E[max{푀(푋) − 푣푝, 0}] + E[max{−푀(푋) + 푣푝, 0}] = E[|푀(푋) − 푣푝|], and due to the third utility assumption in our theorem statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ∞ ∫ 0 P � (푀(푋) − 푣푝) · �1 푣 푋푖 − 푝 � < −푥 � d푥 ≥ ∞ ∫ 0 max � 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 푒−휀 � P � (푀(푋) − 푣푝) · � 1 푣 ˜푋푖 − 푝 � < −푥 � − 훿 � � d푥 = 푒−휀 · ∞ ∫ 0 P � (푀(푋) − 푣푝) · �1 푣 ˜푋푖 − 푝 � < −푥 � d푥 + 푒−휀 · ∞ ∫ 0 max � −훿,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' −P � (푀(푋) − 푣푝) · � 1 푣 ˜푋푖 − 푝 � < −푥 � � d푥 ≥ 푒−휀 · ∞ ∫ 0 P � (푀(푋) − 푣푝) · �1 푣 ˜푋푖 − 푝 � < −푥 � − 훿(푥)d푥 = 푒−휀 · ∞ ∫ 0 푝 · P[(푀(푋) − 푣푝) · (1 − 푝) < −푥]d푥 + 푒−휀 · ∞ ∫ 0 (1 − 푝) · P[(푀(푋) − 푣푝) · (0 − 푝) < −푥]d푥 − 푒−휀 · ∞ ∫ 0 훿(푥)d푥 ≥ 푒−휀 · 푝(1 − 푝) · E[|푀(푋) − 푣푝|] − 푒−휀 · 훼 · 휏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Putting these two pieces together, we have: E 푋1,···,푋푛←D푣,푝 � (푀(푋) − 푣푝) · �1 푣 푋푖 − 푝 �� ≤ ∞ ∫ 0 P � (푀(푋) − 푣푝) · �1 푣 푋푖 − 푝 � > 푥 � d푥 − ∞ ∫ 0 P � (푀(푋) − 푣푝) · �1 푣 푋푖 − 푝 � < −푥 � d푥 ≤ (푒휀 − 푒−휀) · 푝(1 − 푝) · E[|푀(푋) − 푣푝|] + (1 + 푒−휀) · 훼 · 휏 ≤ (푒휀 − 푒−휀) · 푏(1 − 푏) · 훼 + 2훼휏, as 푝 ≤ 푏 ≤ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then, combining this with our lower bound, we have 푣 · (푏 − 푎) − 2훽 log � 푏·(1−푎) 푎·(1−푏) � ≤ 푛 � 푖 E 푃,푋,푀 � (푀(푋) − 푣푃) · �1 푣 푋푖 − 푃 �� 19 ≤ 푛 · ((푒휀 − 푒−휀) · 푏(1 − 푏) · 훼 + 2훼휏) ≤ 훼 · 푛 · 2 · (sinh(휀) · 푏 + 휏), which rearranges to 훼 ≥ 푣 · (푏 − 푎) − 2훽 2푛 · (sinh(휀) · 푏 + 휏) · log � 푏·(1−푎) 푎·(1−푏) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' It only remains to set the parameters subject to the constraints 0 < 푎 < 푏 ≤ 1/2 and 0 < 푣 ≤ (2푏)−1/휆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' First, we set 푏 = 2푎, and 푣 = (2푏)−1/휆 = (4푎)−1/휆 and assume (8훽) 휆 휆−1 ≤ 푎 ≤ 1/5, which simplifies the above expression to 훼 ≥ 푎1−1/휆 · 4−1/휆 − 2훽 2푛 · (sinh(휀) · 2푎 + 휏) · log � 2·(1−푎) 1−2푎 � ≥ 푎1−1/휆 − 8훽 8푛 · (sinh(휀) · 2푎 + 휏) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We reparameterize 푎 = 훾 휆 휆−1 for some 16훽 ≤ 훾 ≤ 1/5 to obtain 훼 ≥ 훾 − 8훽 8푛 · (sinh(휀) · 2 · 훾 휆 휆−1 + 휏) ≥ 훾/2 8푛 · (sinh(휀) · 2 · 훾 휆 휆−1 + 휏) = 1 32푛 sinh(휀)훾1/(휆−1) + 16푛휏훾−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' This completes our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2 Negative Result via Amplification In this subsection, we show that known MSE lower bounds (without bias constraints) [KSU20] com- bined with privacy amplification via shuffling [EFM+19, CSU+19, BBGN19, FMT22, FMT23] can also be used to derive qualitatively similar lower bounds on MSE for private estimators with low bias as those yielded by fingerprinting in the previous subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Our reduction provides an alternative perspective on the bias-variance-privacy tradeoff, and could prove useful in future work as it is more “generic” than the fingerprinting approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Specifically, we will use the following lower bound on the MSE of a private estimator in a black-box manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='6 ([KSU20, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푀 : R푛 → R be (휀,훿)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then there exists a distribution 푃 on R with 휇(푃) ≔ E푋←푃 [푋] ∈ [−1, +1] and E푋←푃 [(푋 − 휇(푃))2] ≤ 1, such that E 푋←푃푛 � (푀(푋) − 휇(푃))2� ≥ Ω � 1 푛(휀 + 훿) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The other ingredient in our proof is the following extension of the privacy amplification by subsam- pling result of Feldman, McMillan, and Talwar [FMT22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Specifically we extend from the setting of local differential privacy (where each algorithm has one input) to the setting where a dataset is randomly partitioned into blocks of fixed size 푛 > 1, and these blocks are processed by a sequence of private mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' To be specific, we randomly partition the dataset as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We first arrange the dataset as a matrix with 푚 columns and 푛 rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then, for each row 푖 ∈ [푛], we perform a uniformly random permutation of the 푚 elements in that row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='7 (Extension of Privacy Amplification by Shuffling [FMT22] to Larger Inputs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푚,푛 ∈ Z+, let X be the data universe, and let Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' , Y푚 be image spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Suppose for each 푖 ∈ [푚], we have a 20 randomized function 퐿푖 : Y1 × · · · × Y푖−1 × X푛 → Y푖 such that 퐿푖(푦, 푥) is (휀0,훿0)-DP in the parameter 푥 ∈ X푛 for every fixed 푦 ∈ Y1 × · · · × Y푖−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Consider 퐿푚 ⊗ · · · ⊗ 퐿1 : (X푛)푚 → Y1 × · · · × Y푚 defined by (퐿푚 ⊗ · · · ⊗ 퐿1)(푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' , 푥푚) ≔ (푦1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ,푦푚) where we recursively define 푦푖 ≔ 퐿푖 (푦1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ,푦푖−1,푥푖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In addition, consider the random shuffle operator Π : (X푛)푚 → (X푛)푚 given by Π((푥1 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ,푥푛 1 ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' , (푥1 푚, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ,푥푛 푚)) ≔ ((푥1 휋1(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ,푥푛 휋푛 (1)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' , (푥1 휋1(푚), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ,푥푛 휋푛 (푚))) where 휋1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' , 휋푛 are permutations on [푚] chosen independently and uniformly at random.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then, for any 훿1 ∈ [2푒− 푚 16푒휀0 , 1], the function 퐿푚⊗· · ·⊗퐿1◦Π : (X푛)푚 → Y1×· · ·×Y푚 is (휀1,훿1 + (푒휀1 + 1)(푒−휀0/2 + 1)푚훿0)- DP, where 휀1 = 푂(휀0 � log(1/훿1)/푚) (for 휀0 = 푂(1)) as in Equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Before we prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='7, we will first show how it can be used to prove a slightly weaker version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 by reduction to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='8 (Bias-Variance-Privacy Tradeoff via Shuffling).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푛 ∈ N and 휀,훿 ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푀 : R푛 → R be (휀,훿)-DP and satisfy the following bias and accuracy properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For any distribution 푃 over R with 휇(푃) ≔ E 푋←푃[푋] ∈ [±1] and E 푋←푃 � (푋 − 휇(푃))2� ≤ 1, we have ���� E 푋←푃푛,푀[푀(푋)] − 휇(푃) ���� ≤ 훽 and E 푋←푃푛,푀 � (푀(푋) − 휇(푃))2� ≤ 훼2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' If 훽2 ≤ �Ω� 1 푛휀 � and 훿 ≤ 푂(푛3휀4훽6), then 훼2 ≥ Ω � 1 푛2휀2훽2 log(1/푛휀2훽2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푚 ∈ N (we delay our choice of 푚 until later).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Consider 퐴푚 : (R푛)푚 → R defined by ∀푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ,푥푚 ∈ R푛, 퐴푚((푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ,푥푚)) = 1 푚 푚 � 푖=1 푀(푥푖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Fix some distribution 푃 with mean and variance bounded by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' This gives us the following guarantee about the mean of 퐴푚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ˇ휇 ≔ E (푋1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=',푋푚)←(푃푛)푚,퐴푚 [퐴푚((푋1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ,푋푚))] = E (푋1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=',푋푚)←(푃푛)푚,푀 � 1 푚 푚 � 푖=1 푀(푋푖) � = 1 푚 푚 � 푖=1 E 푋푖←푃푛,푀[푀(푋푖)] = E 푋←푃푛,푀[푀(푋)] Thus, the bias of 퐴푚 is at most 훽, as we see from the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' | ˇ휇 − 휇(푃)| = ���� E (푋1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=',푋푚)←(푃푛)푚,퐴푚 [퐴푚((푋1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ,푋푚))] − 휇(푃) ���� = ���� E 푋←푃푛,푀[푀(푋)] − 휇(푃) ���� ≤ 훽 21 Similarly, the MSE of 퐴푚 is E (푋1,···,푋푚)←(푃푛)푚,퐴푚 � (퐴푚(푋1, · · · ,푋푚) − 휇(푃))2� = ( ˇ휇 − 휇(푃))2 + E (푋1,···,푋푚)←(푃푛)푚,퐴푚 � (퐴푚(푋1, · · · ,푋푚) − ˇ휇)2� = ( ˇ휇 − 휇(푃))2 + E (푋1,···,푋푚)←(푃푛)푚,푀 \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 � 1 푚 푚 � 푖=1 푀(푋푖) − ˇ휇 �2\uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb = ( ˇ휇 − 휇(푃))2 + 1 푚 · E 푋←푃푛,푀 � (푀(푋) − ˇ휇)2� ≤ 훽2 + 훼2 푚 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let Π : (X푛)푚 → (X푛)푚 be the shuffle operator described in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Since any set of samples drawn i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' from a distribution is invariant under shuffling, (퐴푚 ◦ Π)(푋) has the same distribution as 퐴푚(푋) when 푋 ← (푃푛)푚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In particular, 퐴푚 ◦Π has the same bias and MSE as 퐴푚 on inputs from (푃푛)푚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Privacy amplification by shuffling (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='7) and postprocessing (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2), imply that 퐴푚 ◦ Π is � 휀′ ≔ 푂(휀 � log(1/훿1)/푚),훿 ′ ≔ 훿1 + 푂(훿푚) � DP for all 훿1 ∈ [2푒− 푚 16푒휀 , 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Now, we apply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='6 [KSU20, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='8] to 퐴푚: There exists a distribution 푃 with mean and variance bounded by 1, such that E (푋1,···,푋푚)←(푃푛)푚,퐴푚 � (퐴푚(푋1, · · · , 푋푚) − 휇(푃))2� ≥ Ω � 1 푛푚(휀′ + 훿 ′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Combining all these inequalities gives 훽2 + 훼2 푚 ≥ E (푋1,···,푋푚)←(푃푛)푚 � (퐴푚(푋1, · · · , 푋푚) − 휇(푃))2� ≥ Ω � 1 푛푚(휀′ + 훿 ′) � ≥ Ω � 1 푛푚(휀 � log(1/훿1)/푚 + 훿1 + 훿푚) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' This rearranges to 훼2 ≥ Ω � 1 푛휀 � log(1/훿1)/푚 + 푛훿1 + 푛푚훿 � − 푚훽2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' It only remains to set 푚 ∈ N and 훿1 ∈ [2푒− 푚 16푒휀 , 1] to maximize this lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Now, we assume that 훿1 ≤ 푂(휀/√푚) and 훿 ≤ 푂(휀/푚3/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then the first term in the denominator dominates and we have 훼2 ≥ Ω � 1 푛휀 � log(1/훿1)/푚 � − 푚훽2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then setting 푚 = Θ � 1 푛2휀2훽4 log(1/훿1) � optimizes the expression giving 훼2 ≥ Ω � 1 푛2휀2훽2 log(1/훿1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 22 We set 훿1 = 푛휀2훽2 ≤ 푂(휀/√푚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' This satisfies 훿1 ∈ [2푒− 푚 16푒휀 , 1] as long as 훽2 ≤ 1/푛휀2 and 푚 = Θ � 1 푛2휀2훽4 log(1/푛휀2훽2) � ≥ 푂(푒휀 log(1/푛휀2훽2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The latter constraint rearranges to 훽2 log(1/푛휀2훽2) ≤ Ω � 푒−휀 푛휀 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' To conclude, we note that the assumption 훿 ≤ 푂(휀/푚3/2) is implied by 훿 ≤ 푂(푛3휀4훽6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' □ We conclude this section by proving the extension of privacy amplification by shuffling (Theo- rem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' This proof is a direct reduction to the following result of Feldman, McMillan, and Talwar [FMT22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='9 (Local Privacy Amplification by Shuffling [FMT22, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푚 ∈ Z+, let X be the data universe, and let Y1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' , Y푚 be image spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Suppose for each 푖 ∈ [푚], we have a randomized function 푅푖 : Y1 × · · · × Y푖−1 × X → Y푖 such that 푅푖 (푦,푎) is (휀0,훿0)-DP in the parameter 푎 ∈ X for every fixed 푦 ∈ Y1 × · · · × Y푖−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Consider 푅푚 ⊗ · · · ⊗ 푅1 : X푚 → Y1 × · · · × Y푚 defined by (푅푚 ⊗ · · · ⊗ 푅1)(푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' , 푥푚) ≔ (푦1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ,푦푚) where we recursively define 푦푖 ≔ 푅푖 (푦1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ,푦푖−1,푥푖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In addition, consider the random shuffle operator 푆 : X푚 → X푚 given by 푆(푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ,푥푚) ≔ (푥휋 (1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' , 푥휋 (푚)) where 휋 is a uniformly random permutation on [푚].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then, for any 훿1 ∈ [2푒− 푚 16푒휀0 , 1], the function 푅푚 ⊗ · · ⊗ 푅1 ◦ 푆 : X푚 → Y1 × · · · Y푚 is (휀1,훿1 + (푒휀1 + 1)(1 + 푒−휀0/2)푚훿0)-DP, where 휀1 = log � 1 + 8푒휀0 − 1 푒휀0 + 1 �� 푒휀0 log(4/훿1) 푚 + 푒휀0 푚 �� = 푂 � 휀0 · � log(1/훿1) 푚 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' (6) Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let푥 = ((푥1 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' , 푥푛 1 ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' , (푥1 푚, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ,푥푛 푚)),푥 ′ = ((푥 ′1 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' , 푥 ′푛 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' , (푥 ′1 푚, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' , 푥 ′푛 푚)) ∈ (X푛)푚 be neighboring datasets and assume, without loss of generality, that they differ in only the first entry of the first block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' That is, 푥 푗 푖 = 푥 ′푗 푖 for all (푖, 푗) ≠ (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Now, decompose the operator Π = Π1 ◦ Π−1 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Π1(푥)푗 푖 ≔ � Π(푥)푗 푖 if 푗 = 1 푥 푗 푖 otherwise and Π−1(푥)푗 푖 ≔ � 푥 푗 푖 if 푗 = 1 Π(푥)푗 푖 otherwise In other words, Π1 applies the permutation 휋1 to the first row and leaves the remaining푛−1 rows fixed, whereas Π−1 applies the permutations 휋2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' , 휋푛 to every row except the first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We claim that (퐿푚 ⊗ · · · ⊗ 퐿1 ◦ Π1)(푥) ∼휀′,훿′ (퐿푚 ⊗ · · · ⊗ 퐿1 ◦ Π1)(푥 ′) (7) with 휀′ = 푂(휀0 � log(1/훿)/푚) and 훿 ′ = 훿1 + 푂(훿0푚), as in the conclusion of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' To that end, consider the randomized function 푅푖 (푦,푎) ≔ 퐿푖(푦, (푎,푥2 푖 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' , 푥푛 푖 )) for 푖 ∈ [푚].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Since (푎,푥2 푖 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' , 푥푛 푖 ) and (푎′,푥2 푖 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ,푥푛 푖 ) are neighboring datasets for any 푎, 푎′ ∈ X, 푅푖 must be (휀0,훿0)-DP in the parameter 푎 and hence 푅푚 ⊗ · · · ⊗ 푅1 ◦ 푆 is (휀′,훿 ′)-DP by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In particular, since ˆ푥 ≔ (푥1 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ,푥1 푚) and ˆ푥 ′ ≔ (푥 ′1 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' , 푥 ′1 푚) are neighbors, (푅푚 ⊗ · · · ⊗ 푅1 ◦ 푆)( ˆ푥) must be (휀′, 훿 ′)-indistinguishable from (푅푚 ⊗ · · · ⊗ 푅1 ◦ 푆)( ˆ푥 ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 23 Therefore, to prove our claim, it suffices to show that (퐿푚 ⊗· · ·⊗퐿1◦Π1)(푥) is identically distributed to (푅푚 ⊗ · · · ⊗ 푅1 ◦ 푆)( ˆ푥), and likewise for (퐿푚 ⊗ · · · ⊗ 퐿1 ◦ Π1)(푥 ′) and (푅푚 ⊗ · · · ⊗ 푅1 ◦ 푆)( ˆ푥 ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Indeed, 퐿푖 (푦, Π1(푥)푖) = 퐿푖 (푦, (푥1 휋1(푖),푥2 푖 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' , 푥푛 푖 )) = 푅푖 (푦, 푥1 휋1(푖)) for all 푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' So, it follows by induction that (퐿푚 ⊗ · · · ⊗ 퐿1 ◦ Π1)(푥) = (푅푚 ⊗ · · · ⊗ 푅1)(푥1 휋1(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ,푥1 휋1 (푚)) ∼ (푅푚 ⊗ · · · ⊗ 푅1 ◦ 푆)( ˆ푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Analogously, we get (퐿푚 ⊗ · · · ⊗ 퐿1 ◦ Π1)(푥 ′) ∼ (푅푚 ⊗ · · · ⊗ 푅1 ◦ 푆)( ˆ푥 ′), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We can now leverage the decomposition Π = Π1 ◦ Π−1 to prove the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Fixing Π−1, Π−1(푥) and Π−1(푥 ′) are neighboring datasets differing only on the first element of the first block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' So, by the claim that we proved above (Equivalence 7), which used only the fact that 푥 and 푥 ′ differ at 푥1 1 ≠ 푥 ′1 1, we have that (퐿푚 ⊗ · · · ⊗ 퐿1 ◦ Π1)(Π−1(푥)) ∼휀′,훿′ (퐿푚 ⊗ · · · ⊗ 퐿1 ◦ Π1)(Π−1(푥 ′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' But Π−1 depends only on 휋2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' , 휋푛 and is, thus, is independent of 퐿푚 ⊗ · · · ⊗ 퐿1 ◦ Π1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Therefore, it follows that P[(퐿푚 ⊗ · · · ⊗ 퐿1 ◦ Π)(푥) ∈ 퐸] = E Π−1[P[(퐿푚 ⊗ · · · ⊗ 퐿1 ◦ Π1)(Π−1(푥)) ∈ 퐸 | Π−1]] ≤ E Π−1 � 훿 ′ + 푒휀′P[(퐿푚 ⊗ · · · ⊗ 퐿1 ◦ Π1)(Π−1(푥 ′)) ∈ 퐸 | Π−1] � = 훿 ′ + 푒휀′P[(퐿푚 ⊗ · · · ⊗ 퐿1 ◦ Π)(푥 ′) ∈ 퐸] for any measurable 퐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' □ 4 Low-Bias Estimators for General Distributions In this section, we describe and analyze algorithms for private estimation with low or no bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1, we provide some technical lemmata, which will be useful in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2, we give three algorithms: an (휀, 0)-DP algorithm based on the clipped mean (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='3), a (0,훿)-DP algorithm based on a variant of the “name-and-shame” algorithm (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='4), and an (휀,훿)-DP algorithm obtained by combining the two (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' By taking the best of the three resulting bounds, we get Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 Lemmata We will require the following technical lemmata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ∀휆 > 1 ∀푐 > 0 ∀푡 ∈ R, max{0, 푡 − 푐} ≤ (휆−1)휆−1 휆휆 ·푐휆−1 · |푡|휆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' If 푡 = 0, the claim holds as an equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Now, we assume that 푡 ∈ R \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Define 푓 : R \\ {0} → R by 푓 (푡) = |푡|휆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then 푓 ′(푡) = 휆 · |푡|휆−1 · sign(푡) and 푓 ′′(푡) = 휆(휆 − 1) · |푡|휆−2 ≥ 0 for all 푡 ∈ R \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Since 푓 is convex, ∀푎 ≥ 0 ∀푡 ∈ R \\ {0}, 푓 (푡) ≥ 푓 (푎) + 푓 ′(푎) · (푡 − 푎) = 푎휆 + 휆 · 푎휆−1 · (푡 − 푎) = 휆 · 푎휆−1 · � 푡 − 휆 − 1 휆 푎 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Taking the maximum over 푎 = 0 and 푎 = 푐휆 휆−1 and rearranging yields the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' □ 24 The following lemma decomposes the mean squared error of the clipped mean into the sum of the sampling error and the (squared) population bias introduced (which is further bounded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Fix 휆 > 1 and 푎 < 푏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푃 be a distribution with mean 휇(푃) ∈ (푎,푏).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 휇[푎,푏](푃) ≔ E 푋←푃 � clip[푎,푏](푋) � ∈ [푎,푏].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푋1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ,푋푛 be independent samples from 푃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then E \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 � 1 푛 푛 � 푖 clip[푎,푏](푋푖) − 휇(푃) �2\uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ≤ E 푋←푃 � (푋 − 휇(푃))2� 푛 + (휇[푎,푏](푃) − 휇(푃))2 and ��휇[푎,푏](푃) − 휇(푃) �� ≤ (휆 − 1)휆−1 휆휆 E 푋←푃 � |푋 − 휇(푃)|휆� (min{휇(푃) − 푎,푏 − 휇(푃)})휆−1 ≤ 1 휆 · E 푋←푃 � |푋 − 휇(푃)|휆� (min{휇(푃) − 푎,푏 − 휇(푃)})휆−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We have E \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 � 1 푛 푛 � 푖 clip[푎,푏](푋푖) − 휇(푃) �2\uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb = E \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 � 1 푛 푛 � 푖 clip[푎,푏](푋푖) − 휇[푎,푏](푃) �2\uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb + (휇[푎,푏](푃) − 휇(푃))2 = 1 푛2 푛 � 푖 E �� clip[푎,푏](푋푖) − 휇[푎,푏](푃) �2� + (휇[푎,푏](푃) − 휇(푃))2 ≤ 1 푛2 푛 � 푖 E �� clip[푎,푏](푋푖) − 휇(푃) �2� + (휇[푎,푏](푃) − 휇(푃))2 ≤ 1 푛2 푛 � 푖 E � (푋푖 − 휇(푃))2� + (휇[푎,푏](푃) − 휇(푃))2 = E 푋←푃 � (푋 − 휇(푃))2� 푛 + (휇[푎,푏](푃) − 휇(푃))2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The first inequality follows from the fact that E 푋←푃 � (푋 − 휇(푃))2� = inf푢∈R E 푋←푃 � (푋 − 푢)2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The second inequality follows from the fact that 휇(푃) ∈ [푎,푏] and, hence, (clip[푎,푏](푥) − 휇(푃))2 ≤ (푥 − 휇(푃))2 for all 푥 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' It remains to bound 휇[푎,푏](푃) − 휇(푃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We have 휇[푎,푏](푃) − 휇(푃) = E 푋←푃 � clip[푎,푏](푋) − 푋 � = E 푋←푃[I[푋 > 푏](푏 − 푋) + I[푋 < 푎](푎 − 푋)] = E 푋←푃[max{푎 − 푋, 0}] − E 푋←푃[max{푋 − 푏, 0}].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1, 0 ≤ E 푋←푃[max{푋 − 푏, 0}] = E 푋←푃[max{(푋 − 휇(푃)) − (푏 − 휇(푃)), 0}] ≤ E 푋←푃 � (휆 − 1)휆−1 휆휆 · (푏 − 휇(푃))휆−1 · |푋 − 휇(푃)|휆 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 25 Similarly, 0 ≤ E 푋←푃[max{푎 − 푋, 0}] ≤ (휆 − 1)휆−1 휆휆 · (휇(푃) − 푎)휆−1 · E 푋←푃 � |푋 − 휇(푃)|휆� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Thus, (휆 − 1)휆−1 휆휆 · (푏 − 휇(푃))휆−1 · E 푋←푃 � |푋 − 휇(푃)|휆� ≤ 휇[푎,푏](푃) − 휇(푃) ≤ (휆 − 1)휆−1 휆휆 · (휇(푃) − 푎)휆−1 · E 푋←푃 � |푋 − 휇(푃)|휆� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Finally, note that (휆−1)휆−1 휆휆 = �1 − 1 휆 �휆−1 · 1 휆 ≤ exp(−1+1/휆) 휆 ≤ 1 휆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2 Algorithms We first have a positive result based on clipping and adding noise, which satisfies pure DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The clipped and noised mean is folklore in differential privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Analyzing such a procedure with bounded moments has been done in a few works [DJW13, BD14, KSU20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' These works generally set algorithm parameters to achieve a prescribed bias, towards the goal of minimizing the overall error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' As our goal is to explicitly quantify the bias, we leave it as a free variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='3 (휀-DP Algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For all 휀, 훽 > 0, 푎 < 푏, 휆 ≥ 2, and 푛 ∈ N, there exists an 휀-DP algorithm 푀 : R푛 → R satisfying the following bias and accuracy properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For any distribution 푃 on R with 휇(푃) ≔ E 푋←푃[푋] ∈ [푎,푏] and E 푋←푃 � |푋 − 휇(푃)|휆� ≤ 1, we have ���� E 푋←푃푛,푀[푀(푋)] − 휇(푃) ���� ≤ 훽, E 푋←푃푛,푀 � (푀(푋) − 휇(푃))2� ≤ 1 푛 + 훽2 + 2 휀2푛2 � 푏 − 푎 + 2 훽1/(휆−1) �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Fix ˆ푎 = 푎 − 훽−1/(휆−1) and ˆ푏 = 푏 + 훽−1/(휆−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Our algorithm 푀 : R푛 → R is defined as 푀(푥) = � 1 푛 푛 � 푖 clip[ ˆ푎, ˆ푏](푥푖) � + Lap � ˆ푏 − ˆ푎 휀푛 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The properties of the Laplace distribution ensure that 푀 satisfies 휀-DP, as the sensitivity of �휇(푥) ≔ 1 푛 �푛 푖 clip[ ˆ푎, ˆ푏](푥푖) is ˆ푏− ˆ푎 푛 (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' It only remains to analyze the bias and accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Fix an arbitrary distribution 푃 which satisfies the conditions in the proposition statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' the bias satisfies E 푋←푃푛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='푀[푀(푋)] − 휇(푃) = E 푋←푃푛[�휇(푋)] − 휇(푃) = E 푋←푃 � clip[ ˆ푎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ˆ푏] (푋) − 푋 � ≤ (휆 − 1)휆−1 휆휆 E 푋←푃 � |푋 − 휇(푃)|휆� (min{휇(푃) − ˆ푎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ˆ푏 − 휇(푃)})휆−1 ≤ 1 · 1 (min{푎 − ˆ푎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ˆ푏 − 푏})휆−1 = 훽,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 26 and the mean squared error satisfies E 푋1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='···,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='푋푛←푃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='푀 � (푀(푋) − 휇(푃))2� = E 푋 ←푃푛 휉←Lap( 푏−푎 휀푛 ) � (�휇(푋) + 휉 − 휇(푃))2� = E 푋←푃푛 \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 � 1 푛 푛 � 푖 clip[ ˆ푎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ˆ푏] (푋푖) − 휇(푃) �2\uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb + E 휉←Lap � ˆ푏− ˆ푎 휀푛 � � 휉2� ≤ E 푋←푃 � (푋 − 휇(푃))2� 푛 + 훽2 + 2 � ˆ푏 − ˆ푎 휀푛 �2 ≤ 1 푛 + 훽2 + 2 휀2푛2 � 푏 − 푎 + 2 훽1/(휆−1) �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Our proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' □ Next, we give an algorithm based on the folklore “name-and-shame” procedure, which is (0, 훿)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The name-and-shame procedure is generally phrased as randomly selecting a point from a dataset and outputting it, sans any further privacy protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' It is most commonly used as an illustration of which values of 훿 may or may not be meaningful when it comes to informal uses of the word “privacy”, and not as a serious algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' However, we note that such a procedure gives an exactly unbiased estimate of the mean, which the previous (휀, 0)-DP was unable to do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We thus use it to design an unbiased algorithm for mean estimation, albeit at a high price in the dependence on 훿, which we recall is usually chosen to be very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='4 ((0,훿)-DP Algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For all 훿 ∈ (0, 1] and 푛 ∈ N, there exists a (0,훿)-DP algorithm 푀 : R푛 → R satisfying the following bias and accuracy properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For any distribution 푃 on R, we have E 푋←푃푛,푀[푀(푋)] = 휇(푃), (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=', 푀 is unbiased) E 푋←푃푛,푀 � (푀(푋) − 휇(푃))2� = E 푋←푃 � (푋 − 휇(푃))2� + (1 − 훿) · 휇(푃)2 훿 · 푛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Define a randomized algorithm 퐴 : R → R as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 퐴(푥) = � 0 with probability 1 − 훿 푥 훿 with probability 훿 Define 푀 : R푛 → R by 푀(푥) = 1 푛 푛 � 푖 퐴(푥푖), where each instantiation of 퐴 is independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Since 퐴 satisfies local (0,훿)-DP, 푀 satisfies (0, 훿)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Since 퐴 is unbiased (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=', ∀푥 ∈ R E 퐴[퐴(푥)] = 푥), so is 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Finally, we calculate the mean squared error: E 푋←푃푛,푀 � (푀(푋) − 휇(푃))2� = 1 푛 E 푋←푃,퐴 � (퐴(푋) − 휇(푃))2� = E 푋←푃,퐴 � 퐴(푋)2� − 휇(푃)2 푛 27 = E 푋←푃 � 0 + 훿 · (푋/훿)2� − 휇(푃)2 푛 = E 푋←푃 � 푋 2� − 훿 · 휇(푃)2 훿 · 푛 = E 푋←푃 � (푋 − 휇(푃))2� + (1 − 훿) · 휇(푃)2 훿 · 푛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We have the required result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' □ We can combine both of these methods to obtain a new algorithm for (휀,훿)-DP mean estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Essentially, it decomposes a sample into the non-tail and tail components, releasing the former via the (휀, 0)-DP clip-and-noise method, and the latter via the (0,훿)-DP name-and-shame approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Note that we must consider a higher moment in our assumption about the unknown distribution 푃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='5 ((휀,훿)-DP Algorithm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For all 휀 > 0, 훿 ∈ (0, 1], 휓 > 0, 휆 > 2, 푎 < 푏, and 푛 ∈ N, there exists an (휀,훿)-DP algorithm 푀 : R푛 → R satisfying the following bias and accuracy properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For any distribution 푃 on R with 휇(푃) ≔ E 푋←푃[푋] ∈ [푎,푏] and E 푋←푃 � (푋 − 휇(푃))2� ≤ 1 and E 푋←푃 � |푋 − 휇(푃)|휆� ≤ 휓휆, we have E 푋←푃푛,푀[푀(푋)] = 휇(푃), (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=', 푀 is unbiased) E 푋←푃푛,푀 � (푀(푋) − 휇(푃))2� ≤ 2 푛 + 4(푏 − 푎)2 푛2휀2 + 24휓2 푛2휀2 · � 푛휀2 4휆훿 �2/휆 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Fix 푐 > 0 and let ˆ푎 = 푎 − 푐 and ˆ푏 = 푏 + 푐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Define a randomized algorithm 퐴 : R → R as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 퐴(푥) = � 0 with probability 1 − 훿 푥 훿 with probability 훿 Define 푀 : R푛 → R by 푀(푥) = � 1 푛 푛 � 푖 clip[ ˆ푎, ˆ푏](푥푖) � + Lap � ˆ푏 − ˆ푎 푛휀 � ���������������������������������������������������������������������������������� (휀, 0)-DP + � 1 푛 푛 � 푖 퐴 � 푥푖 − clip[ ˆ푎, ˆ푏](푥푖) �� ������������������������������������������������������������������ (0, 훿)-DP , where the Laplace noise and all of the instantiations of 퐴 are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' That is, 푀 decomposes each 푥푖 into a clipped component and a tail component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We release the clipped component using Laplace noise addition (as in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='3), and the tail component by exploiting 훿 (as in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Since both parts are unbiased, 푀 is unbiased – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ∀푥 ∈ R푛 E 푀[푀(푥)] = 1 푛 �푛 푖 푥푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' By composition and postprocessing, 푀 satisfies (휀,훿)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Now, we bound the mean squared error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Define 휇[ ˆ푎, ˆ푏](푃) ≔ E 푋←푃 � clip[ ˆ푎, ˆ푏] (푋) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We have: E 푋←푃푛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='푀 � (푀(푋) − 휇(푃))2� = E 푋 ←푃푛 휉←Lap � ˆ푏− ˆ푎 푛휀 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='퐴 \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 � 1 푛 �푛 푖 clip[ ˆ푎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ˆ푏] (푋푖) + 휉 + 1 푛 �푛 푖 퐴 � 푋푖 − clip[ ˆ푎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ˆ푏](푋푖) � − 휇(푃) �2\uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb 28 = E 푋←푃푛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='퐴 \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 � 1 푛 �푛 푖 clip[ ˆ푎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ˆ푏](푋푖) − 휇[ ˆ푎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ˆ푏](푃) + 1 푛 �푛 푖 퐴 � 푋푖 − clip[ ˆ푎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ˆ푏](푋푖) � − (휇(푃) − 휇[ ˆ푎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ˆ푏](푃)) �2\uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb + E 휉←Lap � ˆ푏− ˆ푎 푛휀 � � 휉2� ≤ 2 푛 · E 푋←푃 �� clip[ ˆ푎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ˆ푏](푋) − 휇[ ˆ푎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ˆ푏](푃) �2� + 2 � ˆ푏 − ˆ푎 푛휀 �2 + 2 푛 · E 푋←푃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='퐴 �� 퐴 � 푋 − clip[ ˆ푎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ˆ푏](푋) � − � 휇(푃) − 휇[ ˆ푎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ˆ푏](푃) ��2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' (8) The final inequality uses the fact that for independent mean-zero random variables 푈 and 푉 , we have E � (푈 + 푉 )2� = E � 푈 2� + E � 푉 2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For the terms that are not independent, we apply the inequality E � (푈 + 푉 )2� ≤ 2E � 푈 2� + 2E � 푉 2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Since 휇(푃) ≔ E 푋←푃[푋] ∈ [푎,푏] ⊂ [ ˆ푎, ˆ푏], we have E 푋←푃 �� clip[ ˆ푎, ˆ푏](푋) − 휇[ ˆ푎, ˆ푏](푃) �2� ≤ E 푋←푃 �� clip[ ˆ푎, ˆ푏](푋) − 휇(푃) �2� ≤ E 푋←푃 � (푋 − 휇(푃))2� ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Finally we bound the last term: E 푋←푃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='퐴 �� 퐴 � 푋 − clip[푎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='푏](푋) � − � 휇(푃) − 휇[ ˆ푎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ˆ푏](푃) ��2� ≤ E 푋←푃,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='퐴 �� 퐴 � 푋 − clip[ ˆ푎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ˆ푏](푋) ��2� = (1 − 훿) · 0 + 훿 · E 푋←푃 ��1 훿 � 푋 − clip[ ˆ푎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ˆ푏](푋) ��2� = 1 훿 · E 푋←푃 �� 푋 − clip[ ˆ푎,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ˆ푏] (푋) �2� = 1 훿 · E 푋←푃 �� (푋 − 휇(푃))− clip[ ˆ푎−휇(푃),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ˆ푏−휇(푃)] (푋 − 휇(푃)) �2� ≤ 1 훿 · E 푋←푃 �� (푋 − 휇(푃)) − clip[−푐,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='푐](푋 − 휇(푃)) �2� = 1 훿 · E 푋←푃 � (max{0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' |푋 − 휇(푃)| − 푐})2� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' where the final inequality holds because 푐 = ˆ푏 −푏 ≤ ˆ푏 − 휇(푃) and 푐 = 푎 − ˆ푎 ≤ 휇(푃) − 푎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1, max{0, |푋 − 휇(푃)| − 푐} ≤ 1 휆/2 · |푋 − 휇(푃)|휆/2 푐휆/2−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Thus, E 푋←푃 � (max{0, |푋 − 휇(푃)| − 푐})2� ≤ � 1 (휆/2) · 푐휆/2−1 �2 E 푋←푃 � |푋 − 휇(푃)|휆� ≤ 4 휆2 · 휓휆 푐휆−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' we set parameters and assemble the bound from Inequality 8: E 푋←푃푛,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='푀 � (푀(푋) − 휇(푃))2� ≤ 2 푛 + 2 �푏 − 푎 + 2푐 푛휀 �2 + � 2 푛 · 1 훿 · 4 휆2 · 휓휆 푐휆−2 � 29 ≤ 2 푛 + 4(푏 − 푎)2 푛2휀2 + 16 푛2휀2 · 푐2 + 8휓휆 푛훿휆2 · (푐2)1−휆/2 = 2 푛 + 4(푏 − 푎)2 푛2휀2 + 16휓2 푛2휀2 · � 푛휀2 4휆훿 �2/휆 � 휆 휆 − 2 �1−2/휆 ≤ 2 푛 + 4(푏 − 푎)2 푛2휀2 + 24휓2 푛2휀2 · � 푛휀2 4휆훿 �2/휆 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' where the final equality follows from setting 푐 = � 푛휀2휓휆 (휆−2) 4휆2훿 �1/휆 to minimize the expression,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' and the final inequality follows from the fact that � 휆 휆−2 �1−2/휆 = (1 − 2/휆)−1+2/휆 ≤ exp(exp(−1)) < 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' □ Combining Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='4, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='5 yields Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 5 Unbiased Estimators for Symmetric Distributions We now present our unbiased private mean estimation algorithm for symmetric distributions over R that are weakly concentrated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=', those that have a bounded second moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 (Symmetric Distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We say that a distribution 푃 on R is symmetric if there exists some 휇(푃) ∈ R, such that if 푋 ← 푃, then 푋 − 휇(푃) and 휇(푃) − 푋 have the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' That is, 푃 is symmetric iff ∃휇(푃) ∈ R, ∀푥 ∈ R, P 푋←푃[푋 − 휇(푃) ≤ 푥] = P 푋←푃[휇(푃) − 푋 ≤ 푥].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The value 휇(푃) is called the center of the distribution 푃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Note that the center of the distribution is unique and coincides with the mean and the median (whenever these two quantities are well-defined).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Our algorithm is based on the approach of Karwa and Vadhan [KV18], but with some modifications to ensure unbiasedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' First, we obtain a coarse estimate of the mean, and then we use this coarse estimate to perform clipping to obtain a precise estimate via noise addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The key observation is that, if the coarse estimate we use for clipping is unbiased and symmetric (and also independent from the data used in the second step), then the clipping does not introduce bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We obtain the coarse estimate via a DP histogram, where each bucket in the histogram is an interval on the real line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' To ensure that this is unbiased and symmetric, we simply need to apply a random offset to the bucket intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 Coarse Unbiased Estimation Our coarse estimator is similar to that of Karwa and Vadhan [KV18], and uses stability-basedhistograms (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='7) to ensure privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The key modification we make to ensure unbiasedness is adding a random offset to the histogram bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We define roundZ : R → Z to be the function that rounds real numbers to the nearest integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=', for any 푥 ∈ R, we have 푥 ∈ [roundZ(푥) − 1/2, roundZ(푥) + 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' It is easy to verify that our coarse estimator is private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Similar to previous work (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=', [Vad17]), privacy follows from the privacy of the stable histogram algorithm, plus postprocessing via argmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2 (Privacy of DPUCoarse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Algorithm 1 (DPUCoarse휀,훿) satisfies (휀,훿)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 30 Algorithm 1: Unbiased DP Coarse Estimator DPUCoarse휀,훿 (푥) Input: Dataset 푥 = (푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ,푥푛) ∈ R푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Output: Estimate �휇 ∈ R ∪ {⊥}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푇 be uniform on the interval [−1/2, +1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 퐾 = {roundZ(푥푖 −푇) : 푖 ∈ [푛]} ⊂ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For each 푘 ∈ 퐾, sample 휉푘 ← Lap(2/휀) independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' If max푘∈퐾 |{푖 ∈ [푛] : roundZ(푥푖 −푇) = 푘}| + 휉푘 ≤ 2 + 2 log(1/훿) 휀 , output ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Otherwise, output 푇 + arg max푘∈퐾 |{푖 ∈ [푛] : roundZ(푥푖 −푇) = 푘}| + 휉푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We will prove that DPUCoarse휀,훿 satisfies (휀/2,훿/2푒휀/2)-DP with respect to addition or removal of one element of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' By group privacy (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='4), this implies (휀,훿)-DP for replacement of an element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Consider a fixed pair of datasets 푥 and 푥 ′ = 푥−푖∗, where 푥 ′ is 푥 with 푥푖∗ removed for some 푖∗ ∈ [푛].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For the privacy analysis, we also consider the offset푇 to be fixed – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=',푇 is not needed to ensure privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' By postprocessing, we can consider an algorithm that outputs more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Specifically, we can assume that for each 푘 ∈ Z, the algorithm outputs 휈푘 (푥) ≔ � max � |{푖 ∈ [푛] : roundZ(푥푖 −푇) = 푘}| + 휉푘 − 2 − 2 log(1/훿) 휀 , 0 � if 푘 ∈ 퐾 0 if 푘 ∈ Z \\ 퐾 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Note that only finitely many of these 휈푘 (푥) values will be nonzero, so the algorithm can output a compressed version of this infinite vector of values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We can obtain the true output of DPUCoarse휀,훿 by taking the argmax of this vector or outputting ⊥ if this vector is all zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The advantage of this perspective is that each 휈푘 (푥) is independent, as it depends only on the noise 휉푘 (the input 푥 and offset 푇 are fixed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The output distributions on the neighboring inputs are the same except for one 휈푘 (푥) ≁ 휈푘 (푥 ′), namely푘 = roundZ(푥푖∗ −푇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Thus, we must simply show that this value satisfies (휀/2,훿/2푒휀/2)-DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' That is, we must show 휈푘 (푥) ∼휀/2,훿/2푒휀/2 휈푘 (푥 ′), where휈푘 (푥) and휈푘 (푥 ′) denote the relevant random variables on the two different inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' There are two cases to consider: |{푖 ∈ [푛] : roundZ(푥푖 − 푇) = 푘}| = 1 and |{푖 ∈ [푛] : roundZ(푥푖 − 푇) = 푘}| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' (Note that |{푖 ∈ [푛] : roundZ(푥푖 − 푇) = 푘}| = 0 is ruled out because 푘 = roundZ(푥푖∗ −푇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=') Suppose |{푖 ∈ [푛] : roundZ(푥푖 − 푇) = 푘}| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then 휈푘 (푥 ′) = 0 deterministically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Therefore, it suffices to prove that P[휈푘 (푥) = 0] ≥ 1 − 훿/2푒휀/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We have P[휈푘 (푥) ≠ 0] = P � |{푖 ∈ [푛] : roundZ(푥푖 −푇) = 푘}| + 휉푘 − 2 − 2 log(1/훿) 휀 > 0 � = P � 1 + 휉푘 − 2 − 2 log(1/훿) 휀 > 0 � = P � 휉푘 > 1 + 2 log(1/훿) 휀 � = 1 2 exp � −휀 2 · � 1 + 2 log(1/훿) 휀 �� = 훿 2푒휀/2, 31 where the penultimate equality follows from the fact that 휉푘 ← Lap(2/휀) (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Now suppose |{푖 ∈ [푛] : roundZ(푥푖 − 푇) = 푘}| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then 휈푘 (푥) and 휈푘 (푥 ′) are postprocessings of |{푖 ∈ [푛] : roundZ(푥푖 − 푇) = 푘}| + 휉푘 and, respectively, |{푖 ∈ [푛] : roundZ(푥푖 − 푇) = 푘}| − 1 + 휉푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Thus, by the properties of Laplace noise, we have 푒−휀/2P[휈푘 (푥 ′) ∈ 푆] ≤ P[휈푘 (푥) ∈ 푆] ≤ 푒휀/2P[휈푘 (푥 ′) ∈ 푆] for all 푆, as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' □ Now we turn to the utility analysis, which consists of two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' First, conditioned on not outputting ⊥, the estimate is symmetric and unbiased (Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Second, we show that the probability of outputting ⊥ is low for appropriately concentrated distributions, and that the MSE is bounded, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' (Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='3 (Conditional Symmetry of DPUCoarse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푃 be a symmetric distribution with center 휇(푃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푋1, · · · ,푋푛 ∈ R be independent samples from 푃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let �휇 = DPUCoarse휀,훿 (푋1, · · · , 푋푛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푄 be the distribution of �휇 conditioned on �휇 ≠ ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then 푄 is symmetric with the same center as 푃 – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=', 휇(푃) = 휇(푄).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Unlike the static histogram bucket approach of prior work, the introduction of the uniformly ran- dom offset 푇 ∈ [±1/2] in Algorithm 1 endows DPUCoarse휀,훿 with equivalence under translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Recall that we use “∼” to denote distributional equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For any 푥 = (푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' , 푥푛) ∈ R푛 and 푐 ∈ R, we have DPUCoarse휀,훿 (푥 + 푐) ∼ DPUCoarse휀,훿 (푥) + 푐 where ⊥ + 푐 ≔ ⊥ and 푥 + 푐 ≔ (푥1 + 푐, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ,푥푛 + 푐).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' It will be easier to proceed by rewriting Algorithm 1 in a non-algorithmic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' To that end, we define the following notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For 푟 ∈ R, set �푝푟 (푥) ≔ 1 푛 |푖 ∈ [푛] : 푥푖 ∈ [푟 ± 1/2)| and sample 푇 ← U[±1/2], �푝푟 (푥) ← �푝푟 (푥) + Lap(0, 2/(휀푛)) such that 푇 and {�푝푟 (푥)}푟 ∈R are all mutually independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For 푡 ∈ R and 푆 ⊆ R, we define 푆 + 푡 ≔ {푠 + 푡 : 푠 ∈ 푆}, and denote by 푆 +푇 the distribution over the set of sets {푆 + 푡 : 푡 ∈ [±1/2]} induced by the randomness of 푇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Set 푅(푥) ≔ {푟 ∈ Z + 푇 : �푝푟 (푥) > 0} and put 푅∗(푥) ≔ arg max푟 ∈푅(푥) �푝푟 (푥), provided there is an 푟 ∈ 푅(푥) for which �푝푟 (푥) > 2 log(2/훿) 휀푛 + 2 푛 ≕ 휂, otherwise 푅∗(푥) ≔ ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Essentially, we have reparameterizedthe terms of DPUCoarse휀,훿 (푥) so that 푅(푥) = 퐾+푇 and �푝푘+푇 (푥) = 1 푛 |{푖 ∈ [푛] : roundZ(푥푖 −푇) = 푘}| hold, so it follows that 푅∗(푥) ∼ DPUCoarse휀,훿 (푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Now, notice that �푝푟 (푥 + 푐) = 1 푛 |{푖 ∈ [푛] : 푥푖 + 푐 ∈ [푟 ± 1/2)}| = 1 푛 |{푖 ∈ [푛] : 푥푖 ∈ [푟 − 푐 ± 1/2)}| = �푝푟−푐 (푥), so in particular, we have that �푝푟 (푥 + 푐) is identically distributed to �푝푟−푐 (푥) for any 푟 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Moreover, Z +푇 is identically distributed to Z +푇 − 푐, so it follows that 푅(푥 + 푐) = {푟 ∈ Z + 푇 : �푝푟 (푥 + 푐) > 0} = {푟 ∈ Z + 푇 : �푝푟−푐 (푥) > 0} = {푟 ∈ Z + 푇 − 푐 : �푝푟 (푥) > 0} + 푐 32 ∼ {푟 ∈ Z + 푇 : �푝푟 (푥) > 0} + 푐 = 푅(푥) + 푐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' As 푇 and the Laplace noise were all sampled in a mutually independent manner, these distributional equivalences hold jointly, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=', ((�푝푟 (푥 + 푐))푟 ∈R, 푅(푥 + 푐)) ∼ ((�푝푟−푐(푥))푟 ∈R, 푅(푥) + 푐).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Hence, 푅∗(푥 + 푐) = � arg max푟 ∈푅(푥+푐) �푝푟 (푥 + 푐) if ∃푟 ∈ 푅(푥 + 푐), �푝푟 (푥 + 푐) > 휂 ⊥ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ∼ � arg max푟 ∈푅(푥)+푐 �푝푟−푐 (푥) if ∃푟 ∈ 푅(푥) + 푐, �푝푟−푐 (푥) > 휂 ⊥ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' = � arg max푟′∈푅(푥) �푝푟′(푥) + 푐 if ∃푟 ′ ∈ 푅(푥), �푝푟′(푥) > 휂 ⊥ otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' (푟 = 푟 ′ + 푐) = 푅∗(푥) + 푐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' (⊥ = ⊥ + 푐) The equivalence of 푅∗(푥) and DPUCoarse휀,훿 (푥) gives us the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' □ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For any 푥 = (푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' , 푥푛) ∈ R푛, we have DPUCoarse휀,훿 (−푥) ∼ −DPUCoarse휀,훿 (푥) where −⊥ ≔ ⊥ and −푥 ≔ (−푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' , −푥푛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Recall the notation from the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then, we have that �푝푟 (−푥) = 1 푛 |{푖 ∈ [푛] : −푥푖 ∈ [푟 ± 1/2)}| = 1 푛 |{푖 ∈ [푛] : 푥푖 ∈ [−푟 ± 1/2)}| = �푝−푟 (푥), so it follows that for any 푟 ∈ R, �푝푟 (−푥) is identically distributed to �푝−푟 (푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Moreover, 푅(−푥) = {푟 ∈ Z + 푇 : �푝푟 (−푥) > 0} = {푟 ∈ Z + 푇 : �푝−푟 (푥) > 0} = −{푟 ∈ −(Z +푇) : �푝푟 (푥) > 0} ∼ −{푟 ∈ Z + 푇 : �푝푟 (푥) > 0} = −푅(푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' As 푇 and all of the Laplace noise is sampled independently, these distributional equivalences hold si- multaneously, namely ((�푝푟 (−푥))푟 ∈R, 푅(−푥)) ∼ ((�푝−푟 (푥))푟 ∈R, −푅(푥)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Combining these with −⊥ = ⊥, we obtain 푅∗(−푥) ∼ −푅∗(푥) by the same argument as the one we used to prove Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' □ Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Due to Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='4, we may assume without loss of generality that 푃 has center 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then, since 푃 is symmetric, 푋 ← 푃푛 is identically distributed to −푋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' So, for any 푎 ≥ 0, P 푋←푃푛 � DPUCoarse휀,훿 (푋) ∈ [푎, ∞) � = E 푋←푃푛 � P � DPUCoarse휀,훿 (푋) ∈ [푎, ∞)|푋 �� = E 푋←푃푛 � P � DPUCoarse휀,훿 (−푋) ∈ (−∞, −푎]|푋 �� (Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='5) = P 푋←푃푛 � DPUCoarse휀,훿 (−푋) ∈ (−∞, −푎] � = P 푋←푃푛 � DPUCoarse휀,훿 (푋) ∈ (−∞, −푎] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' (푋 ∼ −푋) 33 In particular, 푄([푎, ∞)) = P[�휇 ∈ [푎, ∞)] P[�휇 ≠ ⊥] = P[�휇 ∈ (−∞, −푎]] P[�휇 ≠ ⊥] = 푄((−∞,푎]) for all 푎 ≥ 0, so 푄 must also be symmetric with center 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' □ Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='6 (Accuracy of DPUCoarse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푃 be a distribution over R with mean 휇(푃) and variance E 푋←푃 � (푋 − 휇(푃))2� < 1/64 and E 푋←푃 � |푋 − 휇(푃)|휆� ≤ 휓휆 for some 휆 ≥ 2 and 휓 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푋 = (푋1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' , 푋푛) be independent samples from 푃, and �휇 ← DPUCoarse휀,훿 (푋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' If 푛 ≥ 7 + 7 휀 log(1/훿), then P[�휇 ≠ ⊥ ∧ |�휇 − 휇(푃)| ≤ 1] ≥ 1 − exp(−푛/128) − 푛 2 exp(−푛휀/16) and E � I[�휇 ≠ ⊥] · |�휇 − 휇(푃)|휆� ≤ 1 2 + 푛 · 2휆−1 · 휓휆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In particular, for any 훾 > 0, to ensure P[�휇 ≠ ⊥ ∧ |�휇 − 휇(푃)| ≤ 1] ≥ 1 − 훾, it suffices to have 푛 ≥ max � 7 + 7 휀 log(1/훿), 128 log(2/훾), 16 휀 log(푛/훾) � = 푂(log(푛/훾훿)/휀).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='4, we assume, without loss of generality, that 휇(푃) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푋 ← 푃푛 be the input to DPUCoarse휀,훿 and let �휇 ∈ R ∪ {⊥} be the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푇 ∈ [± 1 2], 퐾 ⊂ Z, and 휉푘 ← Lap(2/휀) be as in the algorithm (and define 휉푘 = 0 for 푘 ∉ 퐾).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For 푘 ∈ Z, define 퐶푘 ≔ |{푖 ∈ [푛] : roundZ(푋푖 −푇) = 푘}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Recall, from Algorithm 1, that 푘 ∈ 퐾 ⇐⇒ 퐶푘 ≥ 1 and that �휇 = ⊥ ⇐⇒ max푘∈퐾 퐶푘 + 휉푘 ≤ 2 + 2 휀 log(1/훿) and, otherwise, �휇 = 푇 + arg max푘∈퐾 퐶푘 + 휉푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We begin by showing �휇 ∈ [±1] with high probabiliy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Define 푘+ = roundZ �1 2 −푇 � and 푘− = roundZ � −1 2 −푇 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Note that 푘+ = 푘− + 1 and 푘+,푘− ∈ (−푇 − 1, −푇 + 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Thus, if arg max푘∈퐾 퐶푘 + 휉푘 ∈ {푘+,푘−} (and max푘∈퐾 퐶푘 + 휉푘 > 2 + 2 휀 log(1/훿)), then �휇 ∈ {푘+ + 푇,푘− + 푇 } ⊂ (−1, +1], as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In other words, it suffices for us to show that, with high probability, 퐶푘+ + 휉푘+ or 퐶푘− + 휉푘− are large and all other 퐶푘 + 휉푘 values are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For any 푥 ∈ �− 1 2, +1 2 �, we have roundZ(푥 −푇) ∈ {푘−,푘+}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Thus 퐶푘− + 퐶푘+ ≥ 푛 � 푖 I � 푋푖 ∈ � −1 2, +1 2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' By our assumption that E 푋←푃 � (푋 − 휇(푃))2� < 1/64 and Chebyshev’s inequality, P 푋←푃 � 푋 ∈ �− 1 2, +1 2 �� ≥ 15 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' By Hoeffding’s inequality, for all 푠 ≥ 0, we have P 푋←푃푛 ��푛 푖 I � 푋푖 ∈ �− 1 2, +1 2 �� ≥ 15 16푛 − 푠 � ≥ 1 − exp(−2푠2/푛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In particular, P 푋←푃푛 � 푛 � 푖 I � 푋푖 ∈ � −1 2, +1 2 �� ≥ 7 8푛 � ≥ 1 − exp(−푛/128).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 34 This means P � 퐶푘− + 퐶푘+ ≥ 7 8푛 � ≥ 1 − exp(−푛/128).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Define 푘∗ ≔ arg max푘∈{푘+,푘− } 퐶푘 (breaking ties arbitrarily).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' If 퐶푘− + 퐶푘+ ≥ 7 8푛, then 퐶푘∗ ≥ 7 16푛, while 퐶푘 ≤ 1 8푛 for all 푘 ∉ {푘+,푘−}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Thus P � 퐶푘∗ ≥ 7 16푛 ∧ max 푘∈퐾\\{푘+,푘− }퐶푘 ≤ 1 8푛 � ≥ 1 − exp(−푛/128).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The next step is to analyze the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For all 푘 ∈ 퐾 and 푟 ≥ 0, P[휉푘 ≥ 푟] = P[휉푘 ≤ −푟] = 1 2푒−푟휀/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Note that |퐾| ≤ 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Setting 푟 = 푛/8 and taking a union bound over 푘 ∈ 퐾, we have P � 휉푘∗ ≥ −푛 8 ∧ max 푘∈퐾\\{푘+,푘− } 휉푘 ≤ 푛 8 � ≥ 1 − 푛 2 exp(−푛휀/16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Combining the high probability bounds on the noise bound and the data, we have P � 퐶푘∗ + 휉푘∗ ≥ 5 16푛 ∧ max 푘∈퐾\\{푘+,푘− }퐶푘 + 휉푘 ≤ 1 4푛 � ≥ 1 − exp(−푛/128) − 푛 2 exp(−푛휀/16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Since 5 16푛 ≥ 2 + 2 휀 log(1/훿), the event 퐶푘∗ + 휉푘∗ ≥ 5 16푛 ∧ max푘∈퐾\\{푘+,푘− } 퐶푘 + 휉푘 ≤ 1 4푛 implies �휇 ∈ {푇 + 푘+,푇 + 푘−} ⊂ [−1, +1], as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Finally, we bound E � I[�휇 ≠ ⊥] · |�휇 − 휇(푃)|휆� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Observe that �휇 = 푇 +푘 for some 푇 ∈ [−1/2, +1/2] and 푘 = roundZ(푋푖 − 푇) for some 푖 ∈ [푛].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Thus, |�휇 − 푋푖| ≤ 1/2 for some 푖 ∈ [푛] and, hence, |�휇 − 휇(푃)| ≤ 1/2 + max푖∈[푛] |푋푖 − 휇(푃)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' It follows that E � I[�휇 ≠ ⊥] · |�휇 − 휇(푃)|휆� ≤ E ��1 2 + max 푖∈[푛] |푋푖 − 휇(푃)| �휆� (A) ≤ E �1 2 + 2휆−1 · max 푖∈[푛] |푋푖 − 휇(푃)|휆 � (B) ≤ 1 2 + 2휆−1 · � 푖∈[푛] E � |푋푖 − 휇(푃)|휆� ≤ 1 2 + 푛 · 2휆−1 · 휓휆, where Inequality A follows from the fact that ∀푝 ≥ 1 ∀푥,푦 ≥ 0, (푥 + 푦)푝 ≤ (푥푝 + 푦푝) · 2푝−1, and Inequality B holds because the maximum among a set of non-negative real numbers should be at most the sum of those numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' This completes our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2 Final Algorithm Now, we present our main algorithm (Algorithm 2) for unbiased mean estimation of symmetric distribu- tions under (approximate) DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The idea is straightforward: invoke our coarse estimator (Algorithm 1) to get a symmetric, unbiased, mildly accurate estimate of the mean privately;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' then apply the standard clip-average-noise technique on our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The second step will not create any new bias because the clipping is performed around a symmetric, unbiased estimate that is independent of the data we are clipping and averaging, and the added noise has mean 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' There is an additional hiccup though: the coarse estimator may fail to produce an estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In this case, we fall back to a different algorithm that exploits (0,훿)-DP and does not require a coarse estimate, as in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The following privacy and utility gurantee is the more general version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 35 Algorithm 2: Unbiased DP Estimator DPUMean휀,훿,푐,휎,푛1,푛2(푥) Input: Dataset 푥 = (푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ,푥푛1, 푥푛1+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ,푥푛1+푛2) ∈ R푛1+푛2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Privacy parameters 휀,훿 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Clipping parameter 푐 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Scale parameter 휎 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Dataset splitting parameters 푛1,푛2 ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Output: Estimate �휇 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' // Get a coarse unbiased symmetric estimate of the mean privately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' �휇 ← 휎 · DPUCoarse휀,훿 � 푥1 휎 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' , 푥푛1 휎 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' If �휇 = ⊥ // When the coarse estimator fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 휉1, 휉2, · · · , 휉푛2 ∈ {0, 1} be independent samples from Bernoulli(훿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let �휇 = 1 푛2훿 �푛2 푖=1 푥푛1+푖 · 휉푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Else // When the coarse estimator outputs �휇 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let �휇 = � 1 푛2 푛2� 푖=1 clip[�휇−푐,�휇+푐](푥푛1+푖) � + Lap � 2푐 푛2휀 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Return (�휇, �휇).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='7 (Unbiased DP Estimator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Fix 휀,훿 ∈ (0, 1), 푛2 ∈ N, 휓 ≥ 1, and 휆 ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Set 훾 = 훿2, 휎 = 10, 푐 = 휎 +휓 · (푛2휀)1/휆, 푛1 = 푂(log(푛1/훾훿)/휀) (as in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='6), and 푛 = 푛1 + 푛2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Algorithm 2 (DPUMean휀,훿,푐,휎,푛1,푛2) satisfies (휀,훿)-DP and the following bias and accuracy properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푃 be a symmetric distribution over R with center 휇(푃), variance E 푋←푃 � (푋 − 휇(푃))2� ≤ 1, and E 푋←푃 � |푋 − 휇(푃)|휆� ≤ 휓휆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푋 = (푋1, · · · ,푋푛) ← 푃푛 and (�휇, �휇) ← DPUMean휀,훿,푐,휎,푛1,푛2(푋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then E[�휇] = 휇(푃), E � (�휇 − 휇(푃))2� ≤ 1 푛2 + 푂 � 휓2 (푛2휀)2−2/휆 + 훿 · 휇(푃)2 푛2 + 훿2−4/휆 · (푛1 + 푛2휀)2/휆 · 휓2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Furthermore, P[�휇 ≠ ⊥] ≥ 1 − 훾 = 1 − 훿2, E[�휇 | �휇 ≠ ⊥] = E[�휇 | �휇 ≠ ⊥] = 휇(푃), E � (�휇 − 휇(푃))2 | �휇 ≠ ⊥ � ≤ 1 푛2 + 푂 � 휓2 (푛2휀)2−2/휆 + 훿2−4/휆 · (푛1 + 푛2휀)2/(휆−1) · 휓2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In particular, we can apply Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='7 to Gaussians (or subgaussians).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Using the bound on Gaus- sian moments E 푋←N(0,1) � |푋 |휆� = 푂( � log휆)휆 and setting 휆 = Θ(log푛) yields the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Note that we also restrict |휇| ≤ 훿−1/2 to remove the 훿 · 휇2/푛 term from the bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='8 (Unbiased Gaussian Mean Estimation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let휀 ∈ (0, 1),훿 ∈ (0, 1/푛), and푛 ≥ 푂(log(1/훿)/휀).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푀 = DPUMean휀,훿,푐,휎,푛1,푛2 be as in Algorithm 2 with appropriate settings of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then, for all 휇 ∈ [−훿−1/2, +훿1/2], E 푋←N(휇,1)푛,푀[푀(푋)] = 휇 and E 푋←N(휇,1)푛,푀 � (푀(푋) − 휇)2� ≤ 푂 � 1 푛 + log log푛 푛2휀2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 36 To prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='7, we use the following lemma that characterizes the symmetry of a clipped random variable from a symmetric distribution under special circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푃 and 푄 be symmetric distributions with the same center 휇(푃) = 휇(푄).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푐 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Define a distribution 푅 to be clip[푌−푐,푌+푐](푋) where 푋 ← 푃 and 푌 ← 푄 are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then 푅 is symmetric with the same center 휇(푅) = 휇(푃) = 휇(푄).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Assume, without loss of generality, that 휇(푃) = 휇(푄) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let푋 ← 푃 and푌 ← 푄 be independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푍 = clip[푌−푐,푌+푐](푋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We claim that ∀푥,푦 ∈ R clip[(−푦)−푐,(−푦)+푐] (−푥) = −clip[푦−푐,푦+푐](푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' This can be verified by analyzing the following cases: (1) 푥 < 푦 −푐;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' (2) 푥 ∈ [푦 −푐,푦 +푐];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' and (3) 푥 > 푦 +푐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Since 푃 and 푄 are symmetric, clip[(−푌)−푐,(−푌)+푐] (−푋) has the same distribution as 푍.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' By the claim, this is simply −푍.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Ergo, the distribution of 푍 is symmetric and centered at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' □ Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The privacy of Algorithm 2 follows from parallel composition, as we split the dataset in two, and apply (휀,훿)-DP algorithms to each half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Computing �휇 is (휀,훿)-DP by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' If �휇 = ⊥, then we compute �휇 in a (0,훿)-DP manner by sampling a 훿 fraction of the data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' If �휇 ≠ ⊥, then we compute �휇 in a (휀, 0)-DP manner using clipping and Laplace noise addition (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Note that �휇 is independent from 푋푛1+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ,푋푛1+푛2, which are the data points used to compute �휇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' If �휇 = ⊥, then we compute �휇 in an unbiased manner: E[�휇 | �휇 = ⊥] = E � 1 푛2훿 푛1+푛2 � 푖=푛1+1 푋푖휉푖 � = 1 푛2훿 푛1+푛2 � 푖=푛1+1 E[푋푖]E[휉푖] = 1 푛2훿 푛1+푛2 � 푖=푛1+1 휇(푃)훿 = 휇(푃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Now, condition on �휇 ≠ ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' By Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='3, �휇 has a symmetric distribution with center 휇(푃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='9, E � clip[�휇−푐,�휇+푐] (푋푖) | �휇 ≠ ⊥ � = 휇(푃), which implies that E[�휇 | �휇 ≠ ⊥] = 휇(푃) because the Laplace noise has expected value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Combining these two cases implies E[�휇] = 휇(푃).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Finally, we analyze the variance: E � (�휇 − 휇(푃))2� = P[�휇 = ⊥] · E � (�휇 − 휇(푃))2 | �휇 = ⊥ � + P[�휇 ≠ ⊥] · E � (�휇 − 휇(푃))2 | �휇 ≠ ⊥ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We bound the two terms for �휇 = ⊥ and �휇 ≠ ⊥ separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For the first term, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='6 gives us P[�휇 = ⊥] ≤ 훾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' P[�휇 = ⊥] · E � (�휇 − 휇(푃))2 | �휇 = ⊥ � = P[�휇 = ⊥] · E \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 � 1 푛2훿 푛1+푛2 � 푖=푛1+1 푋푖휉푖 − 휇(푃) �2\uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb = P[�휇 = ⊥] · 1 푛2 2훿2 푛1+푛2 � 푖=푛1+1 E � (푋푖휉푖 − 휇(푃))2� ≤ P[�휇 = ⊥] · 1 푛2 2훿2 푛1+푛2 � 푖=푛1+1 E � (푋푖휉푖)2� = P[�휇 = ⊥] · 휇(푃)2 + E 푋←푃 � (푋 − 휇(푃))2� 푛2훿 ≤ 훾 · 휇(푃)2 + 1 푛2훿 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 37 Now, we bound the second term: P[�휇 ≠ ⊥] · E � (�휇 − 휇(푃))2 | �휇 ≠ ⊥ � = E � I[�휇 ≠ ⊥] · (�휇 − 휇(푃))2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We split this into two cases: 퐴 ≔ [�휇 ∈ [휇(푃) − 휎, 휇(푃) + 휎]] and 퐵 ≔ [�휇 ∈ R \\ [휇(푃) − 휎, 휇(푃) + 휎]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Note that the event 퐴∧�휇 ≠ ⊥ is equivalent to 퐴 because 퐴 cannot happen if �휇 = ⊥, because ⊥ ∉ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Similarly, 퐵 ∧ �휇 ≠ ⊥ is equivalent to 퐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Note that �휇 ≠ ⊥ =⇒ 퐴 ∨ 퐵.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' we have ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='P[�휇 ≠ ⊥] · E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='(�휇 − 휇(푃))2 | �휇 ≠ ⊥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='= P[�휇 ≠ ⊥] · P[퐴 | �휇 ≠ ⊥] · E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='(�휇 − 휇(푃))2 | �휇 ≠ ⊥ ∧ 퐴 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='+ P[�휇 ≠ ⊥] · P[퐵 | �휇 ≠ ⊥] · E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='(�휇 − 휇(푃))2 | �휇 ≠ ⊥ ∧ 퐵 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='= P[�휇 ≠ ⊥ ∧ 퐴] · E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='(�휇 − 휇(푃))2 | �휇 ≠ ⊥ ∧ 퐴 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='+ P[�휇 ≠ ⊥ ∧ 퐵] · E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='(�휇 − 휇(푃))2 | �휇 ≠ ⊥ ∧ 퐵 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='= P[�휇 ≠ ⊥ ∧ 퐴] · E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='(�휇 − 휇(푃))2 | �휇 ≠ ⊥ ∧ 퐴 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='+ E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='I[�휇 ≠ ⊥] · I[퐵] · (�휇 − 휇(푃))2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='≤ E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='(�휇 − 휇(푃))2 | �휇 ≠ ⊥ ∧ 퐴 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='+ E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='I[�휇 ≠ ⊥] · I[퐵] · (�휇 − 휇(푃))2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='= E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='(�휇 − 휇(푃))2 | 퐴 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='+ E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='I[퐵] · (�휇 − 휇(푃))2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' (9) If 퐴 holds (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=', �휇 ∈ [휇(푃) − 휎, 휇(푃) + 휎]), then 휇(푃) ∈ [�휇 − 휎, �휇 + 휎], so we can bound the first term of the last line in Inequality 9 as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' E � (�휇 − 휇(푃))2 | 퐴 � = E \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 � 1 푛2 푛1+푛2 � 푖=푛1+1 clip[�휇−푐,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='�휇+푐](푋푖) + Lap � 2푐 푛2휀 � − 휇(푃) �2 | 퐴 \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb (A)= E \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 � 1 푛2 푛1+푛2 � 푖=푛1+1 clip[�휇−푐,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='�휇+푐] (푋푖) − 휇(푃) �2 | 퐴 \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb + 2 � 2푐 푛2휀 �2 (B) ≤ E \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 E 푋←푃 � (푋 − 휇(푃))2� 푛2 | 퐴 \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb + E \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 �� � E 푋←푃 � |푋 − 휇(푃)|휆� 휆 · (min{휇(푃) − (�휇 − 푐),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' (�휇 + 푐) − 휇(푃)})휆−1 �� � 2 | 퐴 \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb + 8푐2 푛2 2휀2 ≤ E 푋←푃 � (푋 − 휇(푃))2� 푛2 + �� � 1 휆 · E 푋←푃 � |푋 − 휇(푃)|휆� (푐 − 휎)휆−1 �� � 2 + 8푐2 푛2 2휀2 ≤ 1 푛2 + 휓2휆 휆2 · (푐 − 휎)2(휆−1) + 8푐2 푛2 2휀2 (C)= 1 푛2 + 휓2휆 휆2 · 휓2(휆−1) · (푛2휀)2−2/휆 + 8푐2 푛2 2휀2 = 1 푛2 + 8푐2 +휓2 · (푛2휀)2/휆 · 휆−2 푛2 2휀2 ≤ 1 푛2 + 8푐2 +휓2 · (푛2휀)2/휆 푛2 2휀2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In the above: Equality A follows from the fact that the Laplace noise is independent from everything else;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Inequality B follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2 and linearity of expectations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' and Equality C follows from 38 the setting of 푐 = 휎 +휓 · (푛2휀)1/휆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Next, we bound the second term in the last line of Inequality 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We use the fact that ∀푥 ∈ R |clip[�휇−푐,�휇+푐](푥) − 휇(푃)| ≤ |�휇 − 휇(푃)| + 푐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We have E � I[퐵] · (�휇 − 휇(푃))2� = E \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 I[퐵] · � 1 푛2 푛1+푛2 � 푖=푛1+1 clip[�휇−푐,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='�휇+푐] (푋푖) + Lap � 2푐 푛2휀 � − 휇(푃) �2\uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb (D)= E \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 I[퐵] · � 1 푛2 푛1+푛2 � 푖=푛1+1 clip[�휇−푐,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='�휇+푐](푋푖) − 휇(푃) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='�2\uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='+ 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='� 2푐 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='푛2휀 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='P[퐵] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='≤ E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='I[퐵] · (|�휇 − 휇(푃)| + 푐)2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='+ 8푐2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='푛2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2휀2 · P[퐵] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='(E) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='≤ E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='I[퐵] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='휆 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='휆−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='� 휆−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='휆 · E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='I[�휇 ≠ ⊥] · (|�휇 − 휇(푃)| + 푐)휆�2/휆 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='+ 8푐2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='푛2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2휀2 · P[퐵] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='(F) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='≤ E[I[퐵]] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='휆−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='휆 · E ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='I[�휇 ≠ ⊥] · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='|�휇 − 휇(푃)|휆 + 푐휆� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2휆−1�2/휆 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='+ 8푐2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='푛2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2휀2 · P[퐵] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='(G) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='≤ 훾 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='휆−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='휆 · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='�1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2 + 푛1 · 2휆−1 ·휓휆 + 푐휆 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='�2/휆 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='22−2/휆 + 8푐2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='푛2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2휀2 · 훾 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='(H) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='≤ 훾 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='휆−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='휆 · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2−2/휆 + 푛2/휆 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='22−2/휆 · 휓2 + 푐2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='22−2/휆 + 8푐2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='푛2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2휀2 · 훾 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='≤ 훾 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='휆−2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='휆 · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 + 푛2/휆 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='4 ·휓2 + 푐2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='4 + 8푐2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='푛2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2휀2 · 훾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In the above: Equality D follows from the independence of the Laplace noise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Inequality E follows from Hölder’s inequality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Inequality F holds because ∀푝 ≥ 1 ∀푥,푦 ≥ 0 (푥 +푦)푝 ≤ (푥푝 +푦푝) · 2푝−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Inequality G follows from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' and Inequality H holds because ∀푝 ∈ (0, 1] ∀푥,푦 ≥ 0 (푥 +푦)푝 ≤ 푥푝 +푦푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Finally, we can combine all the pieces, and use our parameter settings 훾 = 훿2 ≤ 1 and 푐2 = (10 +휓 · (푛2휀)1/휆)2 ≤ 2휓2(푛2휀)2/휆 + 200, to get the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' E � (�휇 − 휇(푃))2� ≤ � 훾 · 휇(푃)2 + 1 푛2훿 � + � 1 푛2 + 8푐2 + 휓2 · (푛2휀)2/휆 푛2 2휀2 � + � 훾 휆−2 휆 · � 1 + 푛2/휆 1 4 · 휓2 + 푐2� 4 + 8푐2 푛2 2휀2 · 훾 � ≤ 1 푛2 + 16푐2 + 휓2 · (푛2휀)2/휆 푛2 2휀2 + 훿 · 1 + 휇(푃)2 푛2 + 훿2−4/휆 · 4 � 4푛2/휆 1 휓2 + 푐2 + 1 � ≤ 1 푛2 + 33휓2(푛2휀)2/휆 + 3200 푛2 2휀2 + 훿 · 1 + 휇(푃)2 푛2 + 훿2−4/휆 · � 16휓2푛2/휆 1 + 8휓2(푛2휀)2/휆 + 804 � = 1 푛2 + 푂 � 휓2 (푛2휀)2−2/휆 + 훿 · 휇(푃)2 푛2 + 훿2−4/휆 · (푛1 + 푛2휀)2/휆 ·휓2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Our proof is now complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' □ 39 6 An Impossibility Result for Exponential Families Under Pure DP In Section 5, we showed that it is possible to perform unbiased mean estimation for symmetric distri- butions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' However, this result only provides approximate DP (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=', (휀,훿)-DP with 훿 > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We now show that this is inherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We show that unbiased estimation is impossible under pure DP (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=', (휀, 0)-DP) when the data comes from an exponential family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Exponential families include a wide range of distributions, including Gaus- sians, exponential distributions, Laplace distributions with fixed mean, and Gamma distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We begin by formally defining exponential families: Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 (Exponential Family).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푈 ⊆ R푑 be open, ℎ: 푈 → [0, ∞), and 푇 : 푈 → R푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Given 휂 ∈ R푘, we say that the probability measure 푃휂 belongs to the exponential family E(푇,ℎ) if its density can be written as: 푓푇,ℎ,휂(푥) = ℎ(푥) exp�휂⊤푇 (푥) − 푍 (휂)�, ∀푥 ∈ 푈, where: 푍 (휂) = log�� � ∫ 푆 ℎ(푥) exp�휂⊤푇 (푥)� 푑푥�� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The functions ℎ and 푇 are referred to as the carrier measure and the sufficient statistics of the family, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Additionally, 푍 is known as the log-partition function and 휂 is the natural parameter vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Finally, we denote the range of natural parameters by H ⊆ R푘, which is the set of values of 휂 for which the log-partition function is well-defined (푍 (휂) < ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The density of a Gaussian N (휇, 1) at 푥 is exp(−(푥 − 휇)2/2) = exp(−푥2/2) · exp(휇 · 푥 − 휇2/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Thus we see that Gaussians are an exponential family with natural parameter 휇, carrier measure ℎ(푥) = exp(−푥2/2), sufficient statistics 푇 (푥) = 푥, and log-partition function 푍 (휇) = 휇2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Now we can state our main result: Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2 (Impossibility of Pure DP Unbiased Estimation for Exponential Families ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푈 ⊆ R be an interval of infinite length, let {푃휂 : 휂 ∈ 푈 } be an exponential family, and let 퐼 ⊆ 푈 be any interval of positive length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then, for any 휀 ≥ 0 and 푛 ≥ 0, there exists no (휀, 0)-DP algorithm 푀 : R푛 → R satisfying E푋←푃푛휂 ,푀 [푀(푋)] = 휂 for all 휂 ∈ 퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For instance, this theorem shows that there is no pure DP estimator that is unbiased for Gaussians N (휇, 1), even if the algorithm is only required to be unbiased when, say, 휇 ∈ [2, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Similarly, there can be no pure-DP estimator that gives an unbiased estimate for the mean of Exponential(휆) for all 휆 ∈ (0, 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We remark that in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2, the interval where the algorithm is guaranteed to be unbiased 퐼 having positive length is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' It is easy to construct a pathological estimator that is unbiased at a single point 휂0 ∈ 푈 but not anywhere else, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' by setting 푀(푥) = 휂0 for all 푥 ∈ R푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The interval on which the parameter 휂 is well-defined 푈 having infinite length is also essential to our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We emphasize that this is a property of the distribution and not of the algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' that is, the algorithm does not need to “know” about 푈 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Note that the family of distributions {Bernoulli(푝) : 푝 ∈ [0, 1]} is an exponential family3 and it is possible to estimate the mean 푝 under pure DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In this case 푈 = [0, 1] has finite length, so we see that the assumption that 푈 has infinite length is also necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The proof of this result relies on tools and language from complex analysis and measure theory, a review of which can be found in Appendices A and B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We first show that, for an estimator 3Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 is stated in terms of densities, but it can be extended to discrete distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 40 휙 : R푛 → R and an exponential family {푃휂 : 휂}, the expected value of the estimator E푋←푃푛휂 [휙(푋)] is an analytic function in 휂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We then apply the identity theorem for analytic functions to argue that if 휙 is locally unbiased, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=', unbiased when 휂 lies in some small set, then 휙 must also be globally-unbiased, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=', unbiased for all choices of 휂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' On the other hand, we will argue that global unbiasedness over an infinite interval is impossible for pure DP estimators as a consequence of the strong group privacy properties of pure DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 Locally Unbiased Estimators Are Globally Unbiased The following result shows that the the expectation function of any estimator for the parameters of distributions from an exponential family is analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='3 (Analyticity under Exponential Families).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let {푃휂 : 휂 ∈ 푈 } be an exponential family on R푛 in canonical form (recall Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1) and let 휙 : X푛 → R be any well-defined estimator for {푃휂 : 휂 ∈ 푈 }, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=', E푋←푃휂 [|휙(푋)|] is finite for all휂 ∈ 푈 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then푔 : 푈 → R defined by푔(휂) ≔ E푋←푃휂 [휙(푋)] is an analytic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' A function being analytic means that its Taylor series provides an exact representation of the func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Thus, if an analytic function is linear in some nontrivial interval, we can compute the Taylor series at an interior point of that interval to deduce that the funciton is linear globally, which yields the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='4 (Local Unbiasedness Implies Global Unbiasedness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let {푃휂 : 휂 ∈ 푈 } be an exponential family on R푛 and let 휙 : X푛 → R be any well-defined estimator for {푃휂 : 휂 ∈ 푈 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 퐼 ⊂ 푈 be an interval of nonzero length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' If E푋←푃휂 [휙(푋)] = 휂 for all 휂 ∈ 퐼, then E푋←푃휂 [휙(푋)] = 휂 for all 휂 ∈ 푈 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The main idea behind the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='3 is that analyticity is preserved under integration under certain circumstances, which we show next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Although a proof for the real plane is possible, it will be technically convenient to pass to the complex plane where we can wield Morera’s theorem (Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Before delving into the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='3, we show that the product distribution where each marginal has the same distribution from an exponential family is also an exponential family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let E(푇,ℎ) be an exponential family with support 푈 ⊆ R, with the range of natural parameters H ⊆ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then for any 푛 ∈ N, the family of distributions {푃푛 : 푃 ∈ E(푇,ℎ)} is an exponential family over 푈 푛( denoted by E(푇푛,ℎ푛), where 푇푛 : 푈 푛 → R and ℎ푛 : 푈 푛 → [0, ∞)) with the range of natural parameters H푛 = H, such that for every 푃 ∈ E(푇,ℎ), if the natural parameter of 푃 is some 휂 ∈ H, then the natural parameter of 푃푛 is also 휂, and for all 푥 = (푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ,푥푛) ∈ 푈 푛, 푇푛(푥) = 푛 � 푖=1 푇 (푥푖) and ℎ푛(푥) = 푛 � 푖=1 ℎ(푥푖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 휂 ∈ H, and 푃휂 ∈ E(푇,ℎ) with natural parameter 휂 and density function 푓푇,ℎ,휂 as described in Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Suppose 푓 : 푈 푛 → R is the density function of 푃푛 휂 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then for any 푥 = (푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' , 푥푛) ∈ 푈 푛, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 푓 (푥) = 푛 � 푖=1 푓푇,ℎ,휂(푥푖) 41 = 푛 � 푖=1 ℎ(푥푖) exp(휂 · 푇 (푥푖) − 푍 (휂)) = � 푛 � 푖=1 ℎ(푥푖) � exp � 휂 푛 � 푖=1 푇 (푥푖) − 푛푍 (휂) � This gives us: 푇푛(푥) = 푛� 푖=1 푇 (푥푖);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ℎ푛(푥) = 푛� 푖=1 ℎ(푥푖);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' and the natural parameter of 푃푛 휂 being 휂푛 = 휂 ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' One can easily verify that the log-partition function (푍푛(휂푛)) of 푃푛 휂 equals 푛푍 (휂).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' □ Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let Ω be a 휎-finite measure space with measure 휈, let 푉 ⊆ C be open, and let 푓 : Ω ×푉 → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Assume that 푓 (휔,휂) is analytic in 휂 for every fixed 휔 ∈ Ω and that, for every compact 퐾 ⊆ 푉 , there is a 휈-integrable function (see Section B) 퐺 : Ω → [0, ∞) for which |푓 (휔,휂)| ≤ 퐺(휔) for all 휂 ∈ 퐾.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then 푔(휂) ≔ ∫ Ω 푓 (휔,휂) 푑휈(휔) is analytic, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Our plan is to apply Morera’s theorem (Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' To that end, we must first show that 푔 is continuous, so let (휂푛)푛∈N be any sequence with 휂푛 → 휂 as 푛 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' By our assumption, there is a 휈-integrable퐺 : Ω → [0, ∞) such that |푓 (휔,휂푛)| ≤ 퐺(휔) for all 푛 ∈ N and 휔 ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' So, by the dominated convergence theorem (Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1), 푔(휂푛) → 푔(휂) as 푛 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Now, let 훾 : [0, 1] → C be any closed contour lying in a simply connected (see Section A) subset of 푉 , and let 훾 ′ denote its first derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then, |훾 ′| must be bounded by some 퐶 > 0, so 1 ∫ 0 ∫ Ω |푓 (휔,훾(푡))훾 ′(푡)|푑휈(휔) 푑푡 ≤ 1 ∫ 0 ∫ Ω 퐺(휔)퐶 푑휈(휔) 푑푡 = 퐶 ∫ Ω 퐺 푑휈 < ∞ and thus Fubini’s theorem (Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2) implies that ∮ 훾 푔(휂) 푑휂 = 1 ∫ 0 ∫ Ω 푓 (휔,훾(푡))훾 ′(푡) 푑휈(휔) 푑푡 = ∫ Ω 1 ∫ 0 푓 (휔,훾(푡))훾 ′(푡) 푑푡 푑휈(휔) = ∫ Ω ∮ 훾 푓 (휔,휂) 푑휂 푑휈(휔) = ∫ Ω 0푑휈(휔) (Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' As 훾 was arbitrary, 푔 must be analytic by Morera’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' □ Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Our main goal is to show that 푔(휂) ≔ E푋←푃휂 [휙(푋)] is analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' To that end, let ℎ,푇, and 푍 be the carrier measure, the sufficient statistic, and the log-partition function of {푃휂 : 휂 ∈ 푈 }, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We first show that 푒푍 (휂) is analytic by way of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Indeed, 푟 (푥,휂) ≔ ℎ(푥)푒휂푇 (푥) is entire (see Section A) in 휂 ∈ C for each fixed 푥 ∈ R푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 퐾 ⊆ C be an arbitrary compact set, and let 푚 and 푀 42 be the minimum and the maximum real coordinates among the points within 퐾, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then for any 푥 ∈ R푛 and 휂 ∈ 퐾, 푇 (푥) < 0 =⇒ |푟 (푥,휂)| = ℎ(푥)푒Re(휂)푇 (푥) ≤ ℎ(푥)푒푚푇 (푥) and 푇 (푥) ≥ 0 =⇒ |푟 (푥,휂)| ≤ ℎ(푥)푒푀푇 (푥), so we have |푟 (푥,휂)| ≤ ℎ(푥)푒푚푇 (푥) + ℎ(푥)푒푀푇 (푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' But ∫ R푛 ℎ(푥)푒푚푇 (푥) +ℎ(푥)푒푀푇 (푥) 푑푥 = 푒푍 (푚) +푒푍 (푀) < ∞, so, since 퐾 was arbitrary, 푒푍 (휂) = ∫ R푛 푟 (푥,휂) 푑푥 must be entire by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' As a consequence, ℎ(푥)푒휂푇 (푥)−푍 (휂) is analytic in 휂 for every fixed 푥 ∈ R푛, so we can apply nearly the same argument to ℎ(푥)푒휂푇 (푥)−푍 (휂) in order to conclude that 푔(휂) = E 푋←푃휂[휙(푋)] = ∫ Ω 휙(푥)ℎ(푥)푒휂푇 (푥)−푍 (휂) 푑푥 is analytic, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2 Pure DP Estimators Are Uniformly Bounded We now exploit the strong group privacy property of pure DP (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='4) to show that a pure DP estimator that is bounded locally is uniformly bounded globally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='7 (Pure DP Estimators Are Uniformly Bounded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 퐴 : X푛 → R be a randomized algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' If 퐴 is (휀, 0)-DP, then for all 푥, 푥∗ ∈ X푛, ����E 퐴[퐴(푥)] ���� ≤ 푒휀푛 · E 퐴[|퐴(푥∗)|].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We emphasize that the above result holds for any 푥, 푥∗ (not just neighbouring datasets or ones drawn from an appropriate distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Thus the bound on ����E 퐴[퐴(푥)] ���� is uniform – i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=', it does not depend on 푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Recall that E[푌] = ∞ ∫ 0 P[푌 ≥ 푡] 푑푡 for any non-negative random variable 푌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For any 푥 ∈ X푛, we have ����E 퐴[퐴(푥)] ���� (a) ≤ E 퐴[|퐴(푥)|] = ∞ ∫ 0 P 퐴[|퐴(푥)| ≥ 푡] 푑푡 (b) ≤ 푒휀푛 · ∞ ∫ 0 P 퐴[|퐴(푥∗)| ≥ 푡] 푑푡 = 푒휀푛 · E 퐴[|퐴(푥∗)|], where inequalities (a) and (b) follow from Jensen’s inequality and group privacy (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='4), respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' □ We remark that Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='7 does not hold for approximate DP unless we make further assump- tions about 퐴(푥∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Our impossibility result for exponential families now follows by stringing together the tools we have collected so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 43 Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Suppose, for the sake of contradiction, there exist 휀 ≥ 0, 푛 ≥ 0, and an 휀-DP algorithm 푀 : R푛 → R for which E푋←푃푛휂 ,푀 [푀(푋)] = 휂 when 휂 ∈ 퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' By Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='5, {푃푛 휂 : 휂} is an exponential family, such that for every 푃 ∈ {푃휂 : 휂}, the natural parameter of 푃푛 is the same as that of 푃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Therefore, by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='3, E푋←푃푛휂 ,푀 [푀(푋)] is an analytic function in 휂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' But E푋←푃푛휂 ,푀 [푀(푋)] = 휂 for 휂 ∈ 퐼, so it follows from the identity theorem (Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='4) that E푋←푃푛휂 ,푀 [푀(푋)] = 휂 for all 휂 ∈ 푈 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In particular, sine 푈 is unbounded, E푋←푃푛휂 ,푀 [푀(푋)] must be an unbounded function of 휂, which contradicts Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' □ Acknowledgements GK was supported by an NSERC Discovery Grant, an unrestricted gift from Google, an unrestricted gift from Apple, and a University of Waterloo startup grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' AM was supported by an NSERC Discovery Grant and a David R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Cheriton Graduate Scholarship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' MR was supported by a Vector Scholarship in AI and an NSERC CGS-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' VS was supported 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' arXiv preprint arXiv:2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='02800, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' [RJC22] Kelly Ramsay, Aukosh Jagannath, and Shoja’eddin Chenouri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Concentration of the ex- ponential mechanism and differentially private multivariate medians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' arXiv preprint arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} 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Computer Science, FOCS ’17, pages 552–563, Washington, DC, USA, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' IEEE Computer Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' [Tar08] Gabor Tardos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Optimal probabilistic fingerprint codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Journal of the ACM, 55(2), 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' [TCK+22] Eliad Tsfadia, Edith Cohen, Haim Kaplan, Yishay Mansour, and Uri Stemmer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Friendly- core: Practical differentially private aggregation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In Proceedings of the 39th International Conference on Machine Learning, ICML ’22, pages 21828–21863.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' JMLR, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=', 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' [TVGZ20] Christos Tzamos, Emmanouil-Vasileios Vlatakis-Gkaragkounis, and Ilias Zadik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Optimal private median estimation under minimal distributional assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In Advances in Neu- ral Information Processing Systems 33, NeurIPS ’20, pages 3301–3311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' [Vad17] Salil Vadhan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The complexity of differential privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In Yehuda Lindell, editor, Tutorials on the Foundations of Cryptography: Dedicated to Oded Goldreich, chapter 7, pages 347–450.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Springer International Publishing AG, Cham, Switzerland, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' [WXDX20] Di Wang, Hanshen Xiao, Srinivas Devadas, and Jinhui Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' On differentially private stochas- tic convex optimization with heavy-tailed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In Proceedings of the 37th International Conference on Machine Learning, ICML ’20, pages 10081–10091.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' JMLR, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' [ZKKW20] Huanyu Zhang, Gautam Kamath, Janardhan Kulkarni, and Zhiwei Steven Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Privately learning Markov random fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In Proceedings of the 37th International Conference on Ma- chine Learning, ICML ’20, pages 11129–11140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' JMLR, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=', 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' [ZVHF21] Keyu Zhu, Pascal Van Hentenryck, and Ferdinando Fioretto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Bias and variance of post- processing in differential privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In Proceedings of the Thirty-Fifth AAAI Conference on Artificial Intelligence, volume 35 of AAAI ’21, pages 11177–11184, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' A Background on Complex Analysis The primary objects of interest in complex analysis are the holomorphic functions in the complex plane, namely those functions 푓 : 푈 → C that are differentiable at every point 푧 ∈ 푈 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Many familiar functions, such as the polynomials, are in fact holomorphic or may be extended to a holomorphic function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Note that when 푈 = C, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=', 푓 is differentiable on the whole complex plane, we say that 푓 is an entire function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' A basic result of complex analysis asserts that a function 푓 : 푈 → C is holomorphic exactly when it is analytic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=', its Taylor series expansion around any point 푧0 ∈ 푈 converges to 푓 in some neighbor- hood of 푧0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For this reason, holomorphic functions are typically referred to as analytic functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We consider analyticity in our work as there exist useful mathematical tools to check when functions are analytic, and even more useful tools for constraining functions that we have established to be analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For our purposes, we define a closed contour in a region 퐷 ⊆ C to be a continuously differentiable map 훾 : [0, 1] → 퐷 with 훾(0) = 훾(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Informally, we say that a region in the plane is simply connected if it contains no holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' For instance, the disk {푧 ∈ C : |푧| ≤ 3} is simply connected, whereas the “donut” {푧 ∈ C : |푧| ∈ [1, 3]} is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 50 A thorough review of the language of complex analysis with the precise definitions of the above (which are not necessary for the understanding of our application) is outside the scope of this work, so we recommend the textbook by Ahlfors [Ahl53] for a more comprehensive background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' A useful property of analytic functions is that their closed contour integrals vanish in simply con- nected regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The following theorem characterises this more formally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 (Cauchy’s Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푈 be an open, simply connected subset of C and let 푓 : 푈 → C be analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then, for any closed contour 훾 in 푈 , we have ∮ 훾 푓 (푧) 푑푧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The converse is true, as well, and is a convenient technique for establishing analyticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2 (Morera’s Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푈 ⊆ C be open and let 푓 : 푈 → C be continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Suppose that, for all simply connected 퐷 ⊆ 푈 and any closed contour 훾 in 퐷, we have ∮ 훾 푓 (푧) 푑푧 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then 푓 is analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Next, for functions 푓1, 푓2 : 푈 → C and any 퐿 ⊆ 푈 , we write 푓1|퐿 ≡ 푓2|퐿, if for all 푥 ∈ 퐿, 푓1(푥) = 푓2(푥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Additionally, we write 푓1 ≡ 푓2, if 푓1|푈 ≡ 푓2|푈 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Finally, we define the limit points of a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='3 (Limit Point of a Set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Given a topological space X and 푆 ⊆ X, we say that 푥 ∈ X is a limit point of 푆, if for every neighbourhood 퐵 ⊆ X of 푥 (with respect to the topology of X), there exists a point 푦 ∈ 퐵, such that 푦 ∈ 푆 and 푦 ≠ 푥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In other words, a limit point 푥 of 푆 can be “approximated by points in 푆.” The main property of analytic functions that we exploit is the fact that any two analytic functions that agree locally must, in fact, agree globally, as we show next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='4 (Identity Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푈 ⊆ C be open, and 푓1, 푓2 : 푈 → C be analytic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Suppose there is a set 퐿 ⊆ 푈 with a limit point in 푈 , such that 푓1|퐿 ≡ 푓2|퐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then 푓1 ≡ 푓2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' B Background on Measure Theory Recall that a measure space is the combination of a set X with a collection Σ of subsets of X, which are closed under complement and countable unions, as well as a function 휇 : Σ → [0, ∞] satisfying 휇(∅) = 0 and 휇��∞ 푖=1 퐴푖 � = �∞ 푖=1 휇(퐴푖) for disjoint 퐴1,퐴2, · · · ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The subsets making up Σ are called the measurable subsets of X and 휇 is called a measure on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We say that X is 휎-finite when it can be decomposed as X = �∞ 푖=1 퐴푖 where 퐴1, 퐴2, · · · ∈ Σ are all of finite measure 휇(퐴푖) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' A function 푓 : X → C is said to be measurable if 푓 −1(푈 ) is a measurable subset of X for any open 푈 ⊆ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' In this case, we say that 푓 : X → R is 휇-integrable if ∫ X|푓 | 푑휇, the Lebesgue integral of |푓 | with respect to 휇, exists and is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Now, in order to apply Morera’s theorem, we will require some standard integral-limit interchange theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The first is the dominated convergence theorem, which asserts that pointwise convergence of a sequence of functions may be interchanged with integration, provided that the sequence is uniformly bounded by an integrable function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 (Dominated Convergence Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let X be a measure space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Suppose that (푓푛)푛∈N is a sequence of measurable functions X → C converging pointwise to some 푓 , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=', 푓푛(푥) → 푓 (푥) for all 푥 ∈ X as 푛 → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Suppose further that there is some measurable 퐺 : X → [0, ∞) such that ∫ X 퐺 푑휇 < ∞ and |푓푛(푥)| ≤ 퐺(푥) for all 푥 ∈ X and 푛 ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then 푓 is integrable such that lim 푛→∞ ∫ X 푓푛 푑휇 = ∫ X 푓 푑휇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 51 Switching the order of integration is a very useful operation that is permitted under fairly general measure-theoretic conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We describe it as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2 (Fubini’s Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let X and Y be 휎-finite measure spaces and suppose that 푓 : X×Y → R is measurable such that ∫ X ∫ Y |푓 (푥,푦)| 푑푦 푑푥 < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then ∫ X ∫ Y 푓 (푥,푦) 푑푦 푑푥 = ∫ Y ∫ X 푓 (푥,푦) 푑푥 푑푦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' C Impossibility Result for Concentrated DP We extend the impossibility result for unbiased estimation under pure DP in Section 6 to concentrated DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' But first we briefly introduce concentrated DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Concentrated DP is a variant of DP that has particularly nice composition properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' It was intro- duced by Dwork and Rothblum [DR16], but we use a slightly different definition due to Bun and Steinke [BS16] (see also [Ste22]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Definition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='1 (Concentrated DP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' A randomized algorithm 푀 : X푛 → Y satisfies 휌-zCDP if, for all neighboring inputs 푥, 푥 ′ ∈ X, ∀푡 > 0 D푡+1(푀(푥)∥푀(푥 ′)) ≔ 1 푡 log�� � E 푌←푀(푥) \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 �� � P 푀[푀(푥) = 푌] P 푀[푀(푥 ′) = 푌] �� � 푡\uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb �� � ≤ (푡 + 1)휌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The quantity D푡+1(·∥·) is the Rényi divergence of order 푡 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' The above definition applies when the distributions of 푀(푥) and 푀(푥 ′) are discrete;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' in the continuous case, we replace P 푀 [푀(푥)=푌 ] P 푀 [푀(푥′)=푌 ] with the Radon-Nikodym derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Concentrated DP is intermediate between pure DP and approximate DP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Specifically, (휀, 0)-DP im- plies 1 2휀2-zCDP and 휌-zCDP implies (휌 + 2 � 휌 · log(1/훿),훿)-DP for all 훿 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Concentrated DP cap- tures most common DP algorithms, including Laplace and Gaussian noise addition and the exponential mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Thus our impossibility result for concentrated DP provides a barrier against a wide range of techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Concentrated DP also has strong group privacy properties (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='4), which forms the basis of our impossibility result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2 (Group Privacy for Concentrated DP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 퐴 : X푛 → Y be 휌-zCDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then for any integer 푘 ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' ,푛} and pairs of datasets 푥, 푥 ′ ∈ X푛 differing in 푘 entries, ∀푡 > 0 D푡+1(퐴(푥)∥퐴(푥 ′)) ≤ (푡 + 1)푘2휌.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' We can also use Rényi divergences to bound expectations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='3 ([BS16, Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푋 and 푌 be random variables with D2(푋 ∥푌) and E � 푌 2� being finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then |E[푋]| ≤ � E[푌 2] · (exp(D2(푋 ∥푌)) − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 52 Thus, we can give an analog of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='4 (Concentrated DP Estimators Are Uniformly Bounded).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 퐴 : X푛 → R be a ran- domized algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' If 퐴 is 휌-zCDP, then for all 푥, 푥∗ ∈ X푛, ����E 퐴[퐴(푥)] ���� ≤ � (푒2푛2휌 − 1) · E 퐴 � 퐴(푥∗)2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' This yields an extension of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content='5 (Impossibility of Concentrated DP Unbiased Estimation for Exponential Families ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Let 푈 ⊆ R be an interval of infinite length, let {푃휂 : 휂 ∈ 푈 } be an exponential family, and let 퐼 ⊆ 푈 be any interval of positive length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' Then, for any 휌 ≥ 0 and 푛 ≥ 0, there exists no 휌-zCDP algorithm 푀 : R푛 → R satisfying E푋←푃푛휂 ,푀 [푀(푋)] = 휂 and E푋←푃푛휂 ,푀 [푀(푋)2] < ∞ for all 휂 ∈ 퐼.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} +page_content=' 53' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFQT4oBgHgl3EQfeDaR/content/2301.13334v1.pdf'} diff --git a/YtFRT4oBgHgl3EQfOzfW/content/tmp_files/2301.13515v1.pdf.txt b/YtFRT4oBgHgl3EQfOzfW/content/tmp_files/2301.13515v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f746f57e484c28cb0640503b54a52eda3528fa27 --- /dev/null +++ b/YtFRT4oBgHgl3EQfOzfW/content/tmp_files/2301.13515v1.pdf.txt @@ -0,0 +1,2998 @@ +Calculating Point Spread Functions +Calculating Point Spread Functions: Methods, Pitfalls and Solutions +R. Holinirina Dina Miora,1, 2 E. Rohwer,1 M. Kielhorn,2 C.J.R. Sheppard,3, 4 G. Bosman,1 and R. Heintzmann∗2, 5 +1)Physics Department, Stellenbosch University, Private Bag X1, Matieland 7602, South Africa +2)Institute of Physical Chemistry and Abbe Center of Photonics, Friedrich-Schiller-Universität Jena, Helmholtzweg 4, 07743 Jena, +Germany +3)Nanoscopy & NIC@IIT, Istituto Italiano di Tecnologia, Via Enrico Melen, 83 Edificio B, 16152 Genova, +Italy +4)Molecular Horizons, School of Chemistry and Molecular Biosciences, University of Wollongong, Wollongong, NSW 2522, +Australia +5)Leibniz Institute of Photonic Technology, Albert-Einstein Str. 9, 07745 Jena, Germany +(*Electronic mail: heintzmann@gmail.com) +(Dated: 1 February 2023) +We discuss advantages and disadvantages of various ways of calculating an optical Point Spread Function (PSF) +and present novel Fourier-based techniques for computing vector PSF. The knowledge of the exact structure of the +PSF of a given optical system is of interest in fluorescence microscopy to be able to perform high-quality image +reconstructions. Even, if we know how an aberrant optical path deviates from the original design, the corresponding +PSF is often hard to calculate, as the phase and amplitude modifications need to be modelled in detail. Accurate PSF +models need to account for the vector nature of the electric fields in particular for high numerical apertures. Compared +to the computation of a commonly used scalar PSF model, the vectorial model is computationally more expensive, +yet more accurate. State-of-the-art scalar and vector PSF models exist, but they all have their pros and cons. Many +real-space-based models fall into the sampling pitfall near the centre of the image, yielding integrated plane intensities +which are not constant near the nominal focus position, violating energy conservation. +This and other problems +which typically arise when calculating PSFs are discussed and their shortfalls are quantitatively compared. A highly +oversampled Richards and Wolf model is chosen as the gold standard for our quantitative comparison due its ability +to represent the ideal field accurately, albeit being practically very slow in the calculation. Fourier-based methods are +shown to be computationally very efficient and radial symmetry assumption are not needed making it easy to include +non-centro-symmetric aberrations. For this reason newly presented methods such as the SincR and the Fourier-Shell +method are essentially based on multidimensional Fourier-transformations. +Keywords: point spread function, Fourier optics, fluorescence microscopy +I. +INTRODUCTION +Aplanatic imaging systems strive to image a plane in the +object to the detector plane (e.g. +a CCD camera)1. +To +achieve this, any optical system needs to fulfill the Abbe sine +condition2. This condition relates any angle of a beam emit- +ted by the sample (θem) to its corresponding angle reaching +the detector (θdet), and is given by sin(θem)/sin(θdet) = M, +with M being the magnification of the optical system in terms +of geometric optics1,2. As shown in the thin lens approxi- +mation in Fig. 1, a simple 4f-imaging microscope imaging +system with a magnification of M = 2 would not fulfill the +above-mentioned Abbe sine condition. +The simplicity of such drawings can be maintained by intro- +ducing the concept of the Gaussian reference sphere (see Fig. +2) describing the equivalent refractive loci for aplanatic imag- +ing systems. A Gaussian reference sphere is a sphere centred +where the object plane (the plane to image) or at the focus po- +sition (in image plane) intersect the optical axis. Its radius is +equal to the focal length f1 or f2 in each of the corresponding +planes1. The rays propagate to the Gaussian reference sphere +and get teleported without acquiring any extra phase to a refer- +ence plane, which is a plane surface parallel to the pupil plane +in the back focal plane (BFP), indicated by the positions of +the planes of lens 1 and lens 2 in Fig. 1. The same principle is +𝜃em +𝜃det +𝑓1 +𝑓1 +BFP +𝑓2 +lens 1 +lens 2 +ℎ +𝑓2 +FIG. 1: Geometrical representation of a 4f-imaging system +using the thin lens approximation. Note that here +tan(θem)/tan(θdet) = f2/ f1, which violates Abbe's sine +condition. BFP stands for back focal plane. Here lens 1 +corresponds to the microscope objective lens while lens 2 +represents the tube lens. +applied in reverse to the tube lens (lens 2 in Fig. 1), where the +effect can often be neglected due to the usually large magnifi- +cation of the objective and thus small angles (see right side of +Fig. 2). +If this task of imaging a plane to a plane is performed well +by a careful design, the structure of an image of point object, +called the point spread function (PSF) does, to a very good ap- +arXiv:2301.13515v1 [physics.optics] 31 Jan 2023 + +Calculating Point Spread Functions +2 +Gaussian +reference sphere +Gaussian +reference sphere +''teleportation'' +𝜃em +𝜃det +ℎ +FIG. 2: Introduction refraction at the Gaussian reference sphere warrants that sin(θem)/sin(θdet) = f2/f1 is a constant and the +system therefore satisfies Abbe's sine condition. "Teleportation" means that the beams are continued at the connected surface +without acquiring any phase for the space in between the plane reference plane and Gaussian reference sphere. +proximation, not change with location over the field of view +of the optical system. The image of an incoherently emitting +sample can well be described by a convolution of the object +emitter density with the PSF. Such imaging systems are typi- +cally called linear shift invariant imaging systems3. +Here we discuss calculating PSFs using various computa- +tional tools. An accurate PSF model is an important require- +ment for a successful image reconstruction4. Various stud- +ies have been conducted to model the PSF of a given optical +system5–10. The computation of each of these models has pros +and cons, some of which we will discuss below. In Section II, +we firstly introduce the readers to the state-of-the-art scalar +and vector PSF models. Secondly, we present ingredients for +computing our novel methods using Fourier optics III. The +fast Fourier-transformation is a very handy tool to speed up +PSF calculations, but its pitfalls need to be carefully avoided. +We therefore present the pitfalls that one may encounter in +the calculation and ways around them. In Section IV, we de- +scribe in detail novel Fourier-based techniques for computing +vector PSFs. Finally, in Section V the various methods are +compared quantitatively in terms of their accuracy and com- +putation time. A further aim of this manuscript is to release a +toolbox to the scientific community, which others can benefit +from for calculating PSFs using uniform or modified (aber- +rated) apertures11. +II. +EXISTING PSFS MODELS +A scalar PSF model is computed from one integral per +point, (x,y,z). It is considerably cheap computationally. How- +ever, it does account for the vector nature of the electric field +describing the light which will lead to a wrong estimation of +the field at higher aperture angles. A vector PSF, on the other +hand, requires a calculation of all the three spatial components +of the electric field6. It is more accurate as it carries more in- +formation about the field such its polarization state12. The +Richards and Wolf (RW) vector PSF model has been shown +to represent a vector field to a very high accuracy7,10,13. Yet, +the implementation of the RW model still has its limitations +in terms of sampling and computation time as we will discuss +below. +A. +Scalar PSF model +A well-known scalar PSF model was developed by Gibson +and Lanni (GL)6. The model is valid within the Kirchhoff +boundary conditions for scalar diffraction theory6. The ex- +pression of the PSF as per the GL model at a position (x,y,z) +is given by: +PSF(x,y,z;p) = +����A +� 1 +0 exp(ikΦ(ρ,z,p))J0(krNAρ)ρdρ +���� +2 +, +(1) +where r = +� +x2 +y2, A is a constant complex amplitude, ρ is +the normalized radius in the back focal plane, Φ is aberration +function and, p is a vector summarizing optical characteristic +such as refractive indices and thicknesses of of aberrant sur- +faces in the system. In our calculation, we consider a case +where the actual condition is assumed to meet the design con- +dition of the imaging system14. +B. +The Richards and Wolf Model (RW) +1. +Description of the model +A scalar electric field model is limited to only a single com- +plex amplitude over the image space. It does not give any in- +formation about the polarization state of the image field, the +direction of the energy flow and it is also not applicable to +imaging at high numerical apertures12. Richards and Wolf +described the focusing of electromagnetic waves for low and +high numerical aperture using an angular spectrum of plane +waves in an integral representation7. The model is a vector +formulation of the scalar Debye model. The expressions of +the electric field from a point source S0 at time t at a position +(x,y,z) in the image space is as follows: +⃗E(x,y,z,t) = R{ ⃗ +AE(x,y,z)exp(iϖt)} +(2) +where ⃗ +AE is the amplitude vector, ϖ the angular frequency of +the point source and R{} denotes taking the real part. The +amplitude vector satisfies the independent wave equation and + +Calculating Point Spread Functions +3 +is given by: +⃗ +AE(x,y,z) = +� +� +� +� +� +−ia(I0 +I2 cos2φxy) +−iaI2 sin2φxy +−2aI1 cos2φxy +(3) +with the constant a and +� +� +� +� +� +I0 = +� α +0 cos1/2 θ sinθ(1+cosθ)J0(krsinθ)eikzcosθdθ +I1 = +� α +0 cos1/2 θ sin2 θJ1(krsinθ)eikzcosθdθ +I2 = +� α +0 cos1/2 θ sinθ(1−cosθ)J2(krsinθ)eikzcosθdθ +(4) +α is the maximum angular aperture, Jj is the Bessel function +of order j, the integration variable θ correspond to the angle +a point on the Gaussian reference sphere has to the optical +axis, r = +� +x2 +y2 as well as φxy relate to the point where +the field is evaluated, with φxy being the azimuth at (x,y,z)7. +The term cos1/2 θ in Eq. 4 is the aplanatic factor for energy +conservation, illuminating the objective with a plane wave. +These equations rely on the Kirchhoff boundary conditions +and only consider homogeneous waves12. An inhomogeneous +wave corresponds to a field which decays for a large propaga- +tion distance whereas the Kirchhoff boundary conditions im- +ply that only incident field within the opening aperture of the +exit pupil contribute to the field at (x,y,z) in the image space. +2. +Computation +There exist different ways for computing the three integrals +in Eq. (4). One way could consist of evaluating the integration +numerically. To improve the accuracy, finer steps of integra- +tion are needed which can lead to an expensive computation. +One technique that we denote by RW1 exploits a cylindrical +coordinate system to perform the integration. This has com- +putational advantages, since only a two-dimensional rz-map +(i.e. a centered radial axis versus axial z) has to be calculated +for I0,I1 and I2, since their definition is independent of the az- +imuth. However, this procedure may have the disadvantage +that we cannot easily include arbitrary aperture modifications +which do not possess circular symmetry into the model. If +there is no need for calculating the electric field but only the +intensity, this technique has a shorter path to compute the in- +tensity. We denote this shorter path by RW0. +The second method that has been developed in this work, +denoted by RW2, uses a Fourier transform along the optical z +axis (see Section IV B 3) to compute the (x,y,z) components +I0,I1 and I2 in Eq. (4). We choose as a variable of integration +ξ = cosθ. RW2 is on average 1.45 times slower than RW1. +However, it does not require in its computation the radial sym- +metry property of a PSF while RW0 and RW1 do. +C. +State-of-the-art PSFs models +Four state-of-the-art commonly used PSF models were cho- +sen to compare our methods to. The first model is a scalar PSF +based on the work of Gibson and Lanni 6, and further devel- +oped by Li, XUE, and BLU 8. This technique calculates the +PSF fast by using a combination of Bessel series. The sec- +ond and third models are a scalar and vector PSF as described +in Aguet et al. 9. The PSFs are computed using a numerical +integration based on Simpson’s rule. The last state-of-the-art +PSF that we compare with is a vector PSF calculated with the +Richards and Wolf 3D optical model from the PSFGenerator +toolbox in10. Each of those models have their advantages, as +well as their limitations as we discuss in Section V. +III. +TOOLS FOR COMPUTING A VECTOR PSF USING +FOURIER OPTICS +A. +Scalar high numerical aperture (NA) model +To start with the PSF calculation using a Fourier optic +formulation, let us first consider the calculation of the in- +tensity distribution near the nominal focus when focusing +a monochromatic coherent plane wave by a high-NA mi- +croscopy objective (Fig. 3). The beam is entering the ob- +jective system from the left side and is spatially limited by the +entrance pupil of the optical system, corresponding to the im- +age of the aperture stop as created by the optics to the right of +the aperture (see Fig. 3). An “aperture stop”, is in practice ei- +ther intentionally introduced to warrant the linear shift invari- +ant performance of the system and to avoid aberrations from +unwanted beams, or effectively provided by the inner geome- +try of the objective. The limitation of the beams is therefore +approximated to be at the limit of that aperture stop. At this +pupil plane, every point PW on the wavefront is considered as +a source of a Huygens wavelet, denoted by W 15. +PW W +aperture stop +entrance pupil +exit pupil +optical +S +plane +wave +z +𝜃em +𝜃det +axis +𝑘W +FIG. 3: Visualization of the pupils and representation of the +Huygens Wavelet becoming a plane wave. +Firstly, we limit ourselves to a scalar electric field, where +the field is a directionless value and only a function of its posi- +tion in space. The vector nature of the electric field will be in- + +Calculating Point Spread Functions +4 +troduced further down below. The aperture plane can be seen +as a superposition of spherical wavelets, each of which gives +rise to a plane wave after the objective directed towards the +nominal focus point S (see the wavelet labelled “W” in Fig. +3). According to the Huygens-Fresnel principle, the spheri- +cally converging wave is obtained by superimposing all these +wavelets15. The superimposed wavelets have to acquire ex- +actly the same optical path length and constructively interfere +at S. In other words, the phase at the nominal focus is iden- +tical for all such wavelets and can thus be set to zero in our +simulation. For convenience, we choose S as the center of our +real-space coordinate system. The wavelet W giving rise to +a plane wave in focus (Fig. 3) can now conveniently be de- +scribed in Fourier space as a single point PW (Fig. 4b), i.e. +a single 3-dimensional vector (⃗kW) in Fourier space. Such +vectors have all to necessarily reside on a sphere of a radius +k0 = 2π/λem,λem = λ/n, λem and λ being the emission and +the corresponding vacuum wavelengths respectively, and n the +refractive index of the embedding medium. +A pupil position in real space corresponds to the lateral +component ⃗kx of the wave-vector ⃗k. This linear correspon- +dence is forced by the Abbe sine condition between the pupil +plane coordinate and the k-vector position of the wave near +the focal plane. +The pupil plane aperture stop thus gives rise to a three- +dimensional cap residing on the k-sphere in Fourier space (see +solid cap in Fig. 4b in the Fourier space representation). As +the aperture is limited by the NA of the objective, the 3D fre- +quency spectrum in Fourier space is represented in a segment +of the k-sphere sphere. This segment is called “generalized +aperture” or “McCutchen pupil”16. +To now calculate the (complex-valued) amplitude distribu- +tion in real space near the focus S, we need to generate a +three-dimensional McCutchen pupil of uniform amplitude and +perform an inverse three-dimensional Fourier transformation. +The Fourier-based PSF models that are presented in this work +are based on this understanding. The four different methods +differ in how the amplitude on the McCutchen pupil is cac- +ulcated and how the field is propagated in the homogeneous +medium. To the best of our knowledge, these approaches have +not been previously described elsewhere. Each method has its +own advantages, pitfalls and drawbacks as discussed in the +next section. +It is instructive to first limit ourselves to the calculation of +the xy-plane at the nominal focus position z = 0. We can in- +terpret this plane as being a slice of the three-dimensional fo- +cus volume, i.e. a multiplication with δ(z = 0). In Fourier +space such a z-slice in real space corresponds to an integral of +the amplitude over kz according to the Fourier-slice theorem3. +Since the McCutchen pupil is infinitely thin, and only waves +along the positive z direction, i.e. positive kz, contribute to the +focus, there is a bijective mapping from the 3D McCutchen +pupil to its 2D projection. This establishes a direct corre- +spondence between the pupil plane amplitude and the two- +dimensional Fourier transformation of the xy-amplitude at the +focus. +Slicing at a different z-position, z1, can be seen as a trans- +lation by z1 along z which amounts to a phase-change by +exp(iz1kz) which can be written as exp +� +iz1 +� +k2 +0 −k2r +� +. In this +projection over kz we may need to account for the effect of the +local orientation of the McCutchen pupil via a projection fac- +tor and for possible other factors as given below in a more +detailed analysis regarding the Gaussian reference sphere. +B. +Vector electric field +The above discussion assumed a scalar field (e.g. as com- +mon in acoustics). However, the electric field is described +by a spatially varying amplitude field vector with three com- +plex components ⃗E = (Ex,Ey,Ez) of the form Ex,y,z(r) = +Ax,y,z(r)exp(iϖt) where the amplitude, Ax,y,z(r), is a func- +tion of the position r where the field is evaluated and exp(iϖt) +is the time dependent phasor, ϖ being the angular frequency +and t the time. Since we stay in the realm of linear optics, +where the interaction between the excitation light and matter +is achieved linearly, we continue our considerations with the +assumption that the PSF (i.e. the excitation of a weakly ex- +cited fluorophore) is given by a linear dependence to the local +irradiance: I = |⃗E|2. All of the Fourier-space considerations +as stated for the scalar field above can now be applied individ- +ually to each of the components of the amplitude field vector +as long as its strength and phase on the k-sphere are accounted +for. +C. +Aplanatic correction +By definition, an aplanatic system is a system which is free +from aberration and any small displacement of the system +does not induce aberration17. The energy of light propagating +through layers of such a focusing system must be conserved. +Integrating over the radial position r in the BFP must yield +to the same quantity of energy as integrating over the corre- +sponding angle θ in the equivalent refractive locus. +In a system where the equivalent refractive locus corre- +sponds to a Gaussian reference sphere, the system satisfies +the Abbe's sine condition. The energy conservation yields to +the notion of an apodization or aplanatic factor (AF), which is +equal to +� +cos(θ)18. +The concept can also be understood by using a simple +geometric figure (see Fig. +5a). +Firstly, let us assume an +isotropic emitter placed at the centre of the Gaussian refer- +ence sphere, S0. Since the emitter is emitting uniformly in +all directions, the strength of the amplitudes on the Gaussian +reference sphere is uniform. In Fourier space, the spectrum +on the McCutchen pupil is as well uniform. Let us consider a +small parallel ray which is redirected from the Gaussian ref- +erence sphere at a given angle θ. If we attribute a given ir- +radiance I0 as a power P0 per unit area A0 to such a beam- +let from S0, the same power will have to be distributed to +a smaller area A1 = A0 cos(θ) after the teleportation of the +beam from the Gaussian reference sphere to the plane parallel +to the pupil plane in the BFP (see Fig. 5a). This means that +the irradiance measured perpendicular to the local direction of + +Calculating Point Spread Functions +5 +𝑘𝑥 +𝑘𝑧 +𝑘W +plane wave +𝜃 +𝑘0 +2 = 𝑘𝑥2 + 𝑘𝑦2 + 𝑘𝑧2, 𝑘0 = 2π 𝜆em +𝑘𝑥 = 𝑘0 sin𝜃 cos 𝜙 +𝑘𝑦 = 𝑘0 sin 𝜃 sin 𝜙 +𝑘𝑧 = 𝑘0 cos𝜃 +𝑘0 +𝑘z +𝑘y +𝑘𝑥 +𝜃 +𝜙 +(a) +(b) +FIG. 4: (a) Coordinate system in Fourier space: θ and φ are the elevation and azimuthal angles respectively. (b) Fourier space +representation of a plane wave from a wavelet W at a point source PW in the pupil plane called McCutchen pupil (bold black +line). The ky−axis in (b) is oriented towards the front plane of the paper. +A1 +A0 +"teleportation" +Objective +lens +BFP +r +S1 +S0 +𝜃 +𝜃 +𝜃 +z +Ξ1 +Ξ0 +(a) +0 +0.2 +0.4 +0.6 +0.8 +1 +sin +0 +0.5 +1 +1.5 +2 +2.5 +Radial mean irradiance +\arb. unit +Experimental + = -1 + = -1 with TC included +{(ns, ts) = (1.4, 0); +(ncs, tcs) = (1.518, 170); +(noil, toil) = (1.518, 150)} + = 0 + = 1 +cos( ) +sin max +(b) +(c) +(d) +(e) +(f) +(g) +FIG. 5: Aplanatic correction for energy conservation. (a) Schematics illustrating the “aplanatism effect”. (b) Radial mean of the +normalized BFP intensities. The normalization is done in such a way that the power remains constant. We have here the image +of the BFP (c) recorded experimentally using a 70 nm microtom-sectioned thin layer of fluorescent Chroma slide attached to +the coverslip and embedded in immersion oil. Figure (c) is cropped from the original experimental BFP image for display. The +corresponding BFP computed theoretically use different scaling factors +� +cosη(θ) of the electric field. (d) η = −1, (e) η = −1 +and included transmission coefficients (TC) due to the layer system {sample of refractive index ns placed exactly at the surface +of the coverslip, coverslip of refractive index ncs and thickness tcs, oil immersion of refractive index noil and thickness toil }. +The thicknesses of each layers are expressed in µm and are assumed to be the same as in the design condition of the system. (f) +η = 0 and (g) η = 1. +propagation changes to I0/cos(θ), and each of the field vec- +tors for this beamlet thus needs to change by +� +cos−1(θ) to +be consistent with this intensity change. This factor corre- +sponds to 1/AF. The inverse of this effect, which is spreading +irradiance to a larger area, is somewhat akin to what happens +to radiative power measured on the surface of the earth dur- +ing winter. It is sometimes called “natural vignetting”. In +this manner, the energy spread over the 2D pupil equals to +the energy of a point source over the active Gaussian sphere. +This corresponds to the imaging of an emission PSF where +we have our emitter placed at the focal position of the objec- +tive lens, which is at S0 in this illustration. If a large enough +magnification is assumed, the vectorial and aplanatic facgtor +effects caused by the tube lens can be neglected. To describe +the Fourier-transformation of the focussed field in this scalar +model, we expect a correspondence to the amplitude on the + +1 mmCalculating Point Spread Functions +6 +BFP and therefore a +√ +cos−1 θ dependence. +To now understand the factor that needs to be applied to +conserve the energy when focussing a uniformly illuminated +2D pupil (BFP) with a high-NA objective (excitation PSF), +let us consider the same Fig. +5a. +The point S1 denotes +a point in the BFP from which a ray emerges. +S2 being +the focal point of the objective corresponds to S0. We can +consider the reciprocity theorem: a lossless (non-magnetic) +monochromatic optical system in which a field (Ex0,Ey0,Ez0) +(here an isotropic emitter) at position S0 gives rise to a field +(Ex1,Ey1,Ez1) at position S1 in the image plane, warrants that +placing the isotropic emitter (Ex0,Ey0,Ez0) as a source at the +former image plane, generates the field (Ex1,Ey1,Ez1) at the +focus position S0 of the objective18. The latter situation, a +uniform emitter at the focus of the tube lens, leads, due to +the low NA of the tubelense to the aforementioned uniform +illumination of the pupil plane (BFP). Therefore the excita- +tion PSF should be equal to the emission PSF as long as we +can neglect the NA of the tubelens. This is confirmed also +by considering the +√ +cosθ dependence on the Gaussian ref- +erence sphere and thus on the McCutchen pupil. To arrive at +the 2D Fourier-transform of the in-focus excitation field, we +need to project the McCutchen pupil and thus apply the pro- +jection factor cos−1θ which leads to an overall amplitude of +cos−1θ +√ +cosθ = +√ +cos−1 θ confirming the above reciprocity +argument. +1. +Emission PSF +To test the theory described above, we measure experimen- +tally the intensity of the BFP of emitting Flourophores. A +large collection of randomly orientated emitters (e.g. fluo- +rescent) quickly rotating emitters will emit with the same in- +tensity along all directions. The detection objective is thus +expected to concentrate the light it receives at high angles. +This effect is visible when imaging the back focal plane of +an objective imaging a fluorescent plane sample of uniform +emitters. +In Fig. 5c, we show the image of the BFP of a high-NA ob- +jective imaging a 70 nm thick layer of a Chroma slide. For the +imaging, we have to take great care with our measurements +to avoid supercritical angle fluorescence effect as we did not +account for this in our calculation. Fluorophores which are +directly at a dielectric surface emit fluorescence into the cov- +erslip and thus the objective with a totally different angular +characteristic. This affects the pupil plane distribution and +completely changes the expected measured PSF if not cor- +rected appropriately. To avoid this effect, we embedded the +thin layer of Chroma slide in oil with refractive index of 1.518 +and place it onto a coverslip. A microscope slide is used to +support the sample. +The image of the BFP is recorded by removing the eye- +piece through the observation tube of the microscope [Zeiss +Axio Observer, Objective Plan-Apochromat 63 × /1.4 Oil +DIC M27] and replacing the eyepiece with a system composed +of a converging lens of 50 mm focal length coupled with a +Basler camera [acA4024-29um] (see Fig. 6). The Basler cam- +era is placed at 50 mm after the lens. An x-cite lamp is used as +illumination light source for this particular measurement. Due +to missing information on the optics inside the microscope ob- +servation unit, we were not able to determine the theoretical +magnification of our pupil plane re-imaging but estimate it to +be 0.446 as given by the ratio of the measured over the the- +oretical pupil radius. The pixel size of the recorded BFP is +estimated to be 4.148 µm. The pupil diameter is calculated to +be 4.816 mm and is represented by 1160 pixels on the detector. +In Fig. 5b, the radial mean profile of the BFPs is normalized +such that the intensity at the angle of incidence equal to 0 +corresponds to 1. +Theoretical models are calculated using the slice propaga- +tion method described in Section IV B at higher sampling un- +der different η parameter in the amplitude factor +� +cosη(θ) . +Those models are computed to match the experimental pupil +size. In Fig. 5d, we have the BFP of an emission PSF (η = −1 +thus predicting an intensity scaled by cos−1(θ)). We observe +that the qualitative expectation (an increase towards the edge +of the pupil) is experimentally confirmed but the theory pre- +dicts a stronger rise of intensity with θ than was experimen- +tally found. +This difference could be caused by the photoselection of +preferably in-plane transition-dipoles which leads to a less +pronounced emission along Ez components and thus less pro- +nounced higher angle contributions. The experiments should +be repeated with a thin volume of freely rotating dipoles with a +rotational correlation time far below the fluorescence lifetime. +A further effect could be due to residual back-reflections at +higher angles effectively dimming the light at high angles. +To have a realistic model to compare with the experimental +result, we use the theoretical BFP with η = −1 and include +the transmission coefficient (TC) due to refractive mismatch +of different layers (see Fig. 5e). The TC is calculated using +Eq. (17) in13. We assumed the 70 nm thick polymer sam- +ple matrix having a refractive index of 1.4 placed at the sur- +face of the coverslip. For the coverslip we assumed a 170 µm +thick glass with a refractive index matching the immersion oil +1.518. We observe in Fig. 5b the radial profiles of each of the +BFP models. It may also be worth considering that the spec- +trum of fluorescence emission from the thin slab of fluorescent +plastic is a band rather than a single wavelength19, but we did +not consider this any further. These preliminary experiments +qualitatively confirmed the model of the aplanatic factor. As +a result, high angular amplitudes in detection get enhanced by +� +cos−1(θ). +2. +Intensity +As a matter of perspective, if our interest resides in calcu- +lating a three-dimensional intensity distribution defining the +rate at which randomly oriented molecules will be excited, +the scaling factor ( +� +cos−1(θ) in the Fourier transform of the +amplitude in the focal plane) described in the previous section +should yield the correct values. The randomly oriented fluo- +rophores have on average the same probability of excitation +independent of the angle under which the light comes from. + +Calculating Point Spread Functions +7 +x-cite lamp +NA = 1.4 +n = 1.518 +63x +tube lens +thin layer of +chroma slide +image +plane +BFP +re-imaged BFP +Basler +camera +Lens +50 mm +50 mm +FIG. 6: Back focal plane re-imaging system using a thin layer of chroma slide as a sample emitter. +Therefore, the excitation probability is proportional to the sum +of the absolute squares of all three field components, Ex,Ey, +and Ez. This quantity is proportional to the “radiant intensity” +measured in steradians per area which is different from the ra- +diant flux (called “irradiance”), which is often called “inten- +sity”. However, the physically relevant quantity can also be +irradiance (e.g. when projecting onto a camera) which quan- +tifies the flux through a unit area oriented perpendicular to the +optical axis. In this later case, i.e measuring the focal inten- +sity with a pixelated detector, we would need to account for +another Lambertian factor (cos2(θ) for intensity or cos(θ) for +amplitude) in addition to the aforementioned factor (see Fig. +5a). +D. +Sampling condition +The imaging of a PSF, which is ultimately detected on a pix- +elated imaging device, can commonly be done by a CCD or +CMOS camera. Those devices integrate, in each of their recti- +linearly spaced pixels, over the signal weighted by a pixel sen- +sitivity function. A PSF calculation typically samples the con- +tinuous mathematical function at infinitely thin (delta-shaped) +points. Luckily the local integration of the PSF in every pixel +by the camera can be rewritten as first convolving the PSF +with the pixel sensitivity function and then sampling it at reg- +ularly spaced points. Due to the convolution theorem, the +effect of detector integration can be represented by a simple +multiplication of the Fourier transform of the PSF, the optical +transfer function (OTF) with the Fourier transform of the pixel +sensitivity function. If we assume square pixels with uniform +sensitivity, the OTF gets modified by a multiplication with a +sinc +� +πkx/ksamp +� +sinc +� +πky/ksamp +� +. This means that at twice +the current sampling frequency ksamp = 1/dsamp the overall +transfer would cross zero, dsamp being the pixel pitch. To sam- +ple the PSF free of aliasing, the highest frequency has to be at +most at the half of the inverse of the sampling size. This fre- +quency is called the Nyquist frequency, kNyq = ksamp/2, and +the requirement constitutes what is called the Nyquist Shan- +non theorem20. If the PSF is not sampled under this require- +ment, i.e. dsamp > 1/(2kNyq), there is a presence of aliasing +in the signal (see Fig. ??) leading not only to a potential loss +of signal but also to wrong results at frequencies within the +frequency band. +It is worth emphasizing that a PSF with perfect circular +symmetry (e.g. using circular or random polarization) gets +modified by the pixel sensitivity form factor and loses its sym- +metry, with decreased sensitivity, especially along the diago- +nal directions connecting the corners of the pixels. +In a confocal microscope, the detection is performed by an +integrating detector. However, the data is typically acquired +by integrating in each pixel over the pixel dwell time, with the +scan not being stopped. As an effect, the confocal excitation +and detection PSFs both are modified by a single-directional +sinc +� +πkx/ksamp +� +term if the scanning is performed along x. +1. +Resolution limit and Nyquist Shannon theorem +For a wide-field microscope imaging fluorescence, the min- +imum resolvable spatial structure (periodicity) observed in +lateral axis is given by the Abbe diffraction limit dxy = +λem/(2NA), with NA = nsin(θ) the numerical aperture, θ the +angular aperture of the objective, n the refractive index of the +medium and λem the vacuum emission wavelength. Therefore, +the maximal in-plane spatial frequency is given by kxy,max = +2NA/λem20. Similarly, the axial limit in real space for a wide- +field microscope is given by dz = λem/(n(1−cos(θ))). In our +calculations the electric field has to be sampled according to +the desired PSF, which is sampled twice as fine as the Nyquist- +Shannon theorem would require, since resampling when cal- +culating the intensity was not performed in our calculations. +Thus the highest frequency of the intensity result has to be +sampled with at least two positions per shortes period that +can be transmitted by the system20. This requires the pupil +to fit into half the digital Fourier-space representation such + +2 mmCalculating Point Spread Functions +8 +that its autocorrelation (i.e. the incoherent OTF) fits in the +digital Fourier space. The maximal pupil radius in Fourier +space along x or y should be lower than half the maximally +represented frequency along kx or ky in our Fourier-space rep- +resentation. +2. +The Fourier sampling pitfall +Other potential sources of error that have to be considered +in addition to the aforementioned sampling are as follows. A +digitization of the usually round pupil in Fourier space as a +hard aperture onto a rectilinear grid may induce severe arte- +facts. We consider a field distribution of a high-NA PSF with +numerical aperture NA = 1.4, refractive index n = 1.518 and +emission wavelength λem = 580 nm. The pupil radius, which +also corresponds to the Nyquist frequency, is calculated us- +ing the theory stated in the previous section. We denote Rmax +as this pupil radius. We generate a hard aperture with radius +equal to Rmax/8 and calculate the corresponding field distri- +bution in real space by generating the Fourier transform of the +hard aperture. The window size for this first experiment is +128×128 pixels. +For symmetry reasons, a perfect circularly symmetric PSF +should expected. In Fig. 7a, a significant deviation from circu- +lar symmetry is clearly visible. By repeating the same calcula- +tion for 1024×1024 pixels, the discrepancy is significantly re- +duced even though it is still not totally spherically symmetric +(see Fig. 7c). However, calculating on such large grids causes +a significantly computational overhead (0.178 s vs 0.037 s i.e. +more than 4 times slower) which can be unnecessary as the +user may not need quite so many pixels of the PSF far away +from its center. +Interpolation in Fourier-space to obtain a better represen- +tation of the pupil may be one route to reestablish spherical +symmetry. However, this is a tricky business21 and we there- +fore choose a slightly different route. We calculate the two- +dimensional (2D) Fourier-transform of the uniform pupil disk, +for which the analytical solution in real space is well known: +jinc(R) = J1(R)/R,J1 being the Bessel function of the first +kind. We therefore obtain an “ideally” representation of a disk +in Fourier-space by Fourier-transforming a two-dimensional +jinc function. This “interpolated” disk can then be appropri- +ately modified with k-space dependent phase and magnitude +alterations. +The computation time of the 1024×1024 pixels ideal repre- +sentation of disk in Fourier-space using the jinc trick is 0.078 +s on average. The calculation is done with MATLAB R2018a +on Windows 10 with Intel(R) Core(TM) i5-6200U CPU @ +2.30 GHz 2.40 GHz. +As the jinc-function possesses first order discontinuities in +real space at the border, whos Fourier-transformation causes +unwanted high-frequencies22. To avoid this, the jinc-function +was modified at the outer rim by appropriately smoothing the +15 % of its edges towards zero (“DampEdge” function in the +PSFToolbox11). +As seen in Fig. 7b, the real-space representation of the field +distribution is perfectly symmetric and spherical by design +even for images with relatively few pixels. We refer to this +method of generating an interpolated disc in Fourier space as +the FT(jinc)-pupil trick. +IV. +NOVEL FOURIER-BASED METHODS FOR PSF +CALCULATION +A. +The electric field on the k-sphere +To calculate the electric field amplitude distribution near +the focus, we first need to understand the electric field prop- +erties of each Huygens wavelet as previously described. To +this aim, we associate each plane wave arriving at the focus +with the refractive effect that the “bending” at the Gaussian +reference has at the point where this “ray” would hit it. We +assume a perfect anti–reflection coated objective lens and all +the energy is transmitted for such a ray and exploit the fact that +the electrical field of the plane wave needs to be a transversal +wave. We thus have to project the electric field at the pupil +(Ex,Ey) to the electrical field (Ex,Ey,Ez) of the plane waves +(i.e. the McCutchen pupil). +Let us consider Fig. 8. The incident wave ⃗Ei is incident +from infinity at the left side in Fig. 8a and focuses at a the +focal point of the objective lens. The system is assumed to +fulfil the Abbe sine condition, requiring the beams to change +direction at the Gaussian reference sphere. At the entrance +pupil, the incident wave Ei can be described by two compo- +nents (Ex,Ey), but it is very useful to here consider a locally +varying coordinate system along azimuthal (⃗es) and radial (⃗ep) +directions respectively. +Let ⃗Et denote the field amplitude transmitted along the +wave vector⃗et towards a point (x,y,z) near the focus where the +field is evaluated (see Fig. 8a). The unit vector correspond- +ing to the radial component⃗ep is refracted by θ and becomes +⃗er while the azimuthal component oriented along ⃗es remains +unchanged. +⃗es = +� +� +−sinφ +cosφ +0 +� +�, +⃗ep = +� +� +cosφ +sinφ +0 +� +� ⇒⃗er = +� +� +cosφ cosθ +sinφ cosθ +sinθ +� +� +(5) +The new coordinate system is illustrated in Fig. 8. +The field amplitude distribution at a point (x,y,z) is there- +fore given by: +⃗Et(x,y,z) = (⃗Ei ·⃗ep)⃗er +(⃗Ei ·⃗es)⃗es. +(6) +For a given polarization state of the incident wave field ⃗P, +the pupil plane amplitude distribution can be calculated using +the directional change of the electric field described in the pre- +vious paragraphs where the incident electric field is given by +⃗Ei = E0⃗P, E0 is a constant factor which includes the conserva- +tion of energy such as aplanatic factor or apodization. There- +fore, the amplitude field on the McCutchen pupil is given by +⃗Et = E0[(⃗P·⃗eP)⃗er +(⃗P·⃗es)⃗es]. +(7) + +Calculating Point Spread Functions +9 +-5 +0 +5 +x/µm +-5 +0 +5 +y/µm +(a) +-5 +0 +5 +x/µm +-5 +0 +5 +y/µm +(b) +-5 +0 +5 +x/µm +-5 +0 +5 +y/µm +(c) +-5 +0 +5 +x/µm +-5 +0 +5 +y/µm +-1 +-0.5 +0 +0.5 +1 +(d) +FIG. 7: Field distribution calculated from the Fourier transform of (a) a hard aperture of size 128×128 pixels, (b) a jinc +aperture aperture of size 128×128 pixels, (c) a hard aperture of size 1024×1024 pixels cropped to 128×128 pixels size for +display and, (d) a jinc aperture aperture of size 1024×1024 pixels cropped to 128×128 pixels size for display. A DampEdge +of 15 % is applied to the generated field (full size) using the jinc-trick. Figures are displayed at tan−1(γE) and centered at the +zero of the display, E being the field distribution and γ = 20. +𝜙 +x +y +z +ep +es +y' +x' +oriented toward the front of the plane. +𝜃 +x' +z +er +et +ep +es +y' +(a) +(b) +FIG. 8: Schematic diagram of the coordinate systems for the +directional change of a scalar wave field Ei through an +objective lens. (a) Adaptive coordinates on the McCutchen +pupil. (b) 2D pupil plane. +At this point, it seems appropriate to comment on the ver- +satility of this approach. If we want to include any additional +linear shift-invariant effect into our calculation, such as the ef- +fect of an additional slab of glass, which was not considered in +the design of the objective or a wrong medium of sample em- +bedding, it is fairly simple to work out the magnitude, phase +and even polarization effect that such a modification would +have on each field vector component on the generalized Mc- +Cutchen pupil. +Likewise, we can also easily consider the effect any in- +tentional change of the complex amplitude transmittance at +the pupil plane will have, for example to calculate Bessel +beam23, spiral phase modification24, the doughnut-shaped +STED (Stimulated Emission Depletion Microscopy) beam25, +apodizations26 or other PSF modifications27. +B. +The slice propagation method (SP) +With the various considerations described above, the com- +plete problem of calculating the in-focus field distribution for +excitation and emission PSFs can also be achieved using a +method called the “slice propagation”. This method derives +from the vectorial Debye model and the propagation of the +field is based on the angular spectrum method28. The steps to +follow are: +1. Choose a 2D pupil plane amplitude and polarization +distribution; +2. Propagate it to the Generalized McCutchen pupil as a +projection of a two-dimensional pupil onto three field +amplitude components; +3. Apply to each component the same aplanatic factor ac- +cording to the desired calculation in Section III C (exci- +tation or emission PSF, angle-independent flux or flux +through a reference surface); +4. Perform a separate two-dimensional inverse Fourier- +transformation for each of the three field components +to go from the McCutchen pupil to the focal field in +real space; +5. Calculate the PSF as a probability of excitation or de- +tection for a collection of a randomly oriented fluo- +rophores, PSF(x,y) = ExE∗ +x +EyE∗ +y +EzE∗ +z , such that +each of the field components depends on the spatial +(x,y) coordinates. +1. +Free space propagation of the field +In order to calculate a defocussed PSF or 3D PSF as a +stack of defocussed PSFs, we can modify the slice propaga- +tion method by including a defocus phase in the generalized +McCutchen pupil. Given the field in the McCutchen pupil +as illustrated in Fig. 4b, the in-focus lateral field distribu- +tion components can be calculated as a projection along kz +of the corresponding McCutchen pupil. Each projection cor- +responds to the 2D in-focus slice of the 3D field distribution +represented by the 3D McCutchen pupil. To move to a differ- +ent z position, the Fourier-shift theorem needs to be applied, +which states that a translation by ∆z in real space corresponds + +OOOCalculating Point Spread Functions +10 +to a multiplication with exp(ikz∆z) in Fourier space3. Shifting +an xy-amplitude by ∆z thus means: +1. Calculate the 2D fast Fourier transformation (FFT) +from real space to Fourier space of the 2D pupil plane +amplitude distribution or use the already calculated pro- +jections of the McCutchen field components from the +result of step 3 above; +2. Project onto the k-sphere to obtain the field on the Mc- +Cutchen pupil; +3. Apply the phase modification; +4. Project back (sum along kz) onto the kxky-plane; +5. Inverse Fourier transform to obtain the amplitude at ∆z. +These steps follow the angular spectrum method. Luckily, +steps 2 to 4 do not actually need to be calculated individu- +ally, since each pixel exactly ends up where it was but only +having accumulated a phase modification which only depends +on |kz| = +� +k2 +0 −k2x −k2y. This well-known phase modifica- +tion (the homogeneous medium angular spectrum “propaga- +tor”) can thus simply be applied to the projected McCutchen +pupil(s) yielding the wanted defocus PSF in step 5. When +propagating a pupil that was generated by the jinc-FT pupil +trick (see Section III D 2), it is recommended for accuracy rea- +sons not to apply the pupil a second time during propagation. +2. +Fourier wrap-around pitfall +Even though the propagator with the slice propaga- +tion method allows a convenient calculation of the three- +dimensional PSF, a severe problem arises outside a ∆z-range +in the xy-cut through a calculated PSF. The ∆z-range is defined +as the axial region where the disk of defocus stays well within +the available lateral space provided by the real-space grid (see +Fig. 9a). Outside the ∆z-range, waves leaving on one side of +the xy−sampling grid and entering into the simulation from +the opposite side due to the periodic boundary conditions of +the Fourier-transform cause severe standing–wave effects (see +Fig. 9a and 9d). Three possible strategies can help to avoid +this: +A. By appropriately zero-padding the in-focus plane, the +∆z-range from whereon the standing wave patterns oc- +cur can be extended (Fig. 9b) and 9e. However, this +approach is computationally expensive, since twice the +initial image window size slows the calculation down +by a factor of 5. However, padding with zero to twice +the original size can still yield unacceptable artifacts for +typical z ranges used in 3D PSF calculations. +B. Establishing absorptive boundary conditions: At ev- +ery propagated slice, one can apply an ideal absorp- +tive boundary condition to the outside boundary and +continue propagation by re-projecting this filtered field +onto the pupil plane. This has the disadvantage of sac- +rificing a good PSF for a portion of pixels near the xy +sides of the calculation. In addition, every slice propa- +gation requires two Fourier transformations, instead of +only one. +C. Using the chirp Z-transform (CZT). With the help of +the CZT, also called zoomed Fourier transform29 it is +possible to calculate only a part of the field without the +need for periodic boundary conditions (Fig. 9c and 9f). +In this way the wrap-around artefacts can be partially +avoided at the expense of a roughly twice or more in- +crease of computation time. +Practically method C. seems to be the most appropriate +among the three options here. The use of CZT for Fourier +optics and PSF modelling is not new in the literature and has +been proven to be more efficient without loss of accuracy than +FFT30–32. +To demonstrate this, the xz-profiles of the lower part of the +PSFs which corresponds to the slice propagation method us- +ing FFT, by zero-padding as described previously as case A +and using the CZT as described as case C are displayed at +gamma 0.05 in Fig. 9d, 9e an 9f respectively. +The standing waves in Fig. 9d and clearly seen in Fig. 9a +are due to the Fourier-wrap around. These effects are reduced +considerably as the window size is doubled (see Fig. 9b). +A detail comparison of the slice propagation techniques with +FFT and CZT with a chosen gold standard is shown in Section +V. +3. +Chirp Z transform +The chirp Z transform (CZT) is a more generalized func- +tion converting a signal in real space into a frequency-domain +representation. For a 1D signal Xn,n ∈ [0,N − 1] ∩ N with N +being the number of points of the signal and N the set of nat- +ural numbers, the Z transform ˜Xz,z ∈ C is given as follows: +˜Xzm = CZT(Xn) = +N−1 +∑ +n=0 +XnZ−n +m , +(8) +where Zm = AW −m,m ∈ N is a spiral path in Z−path with +A being the starting point and W = exp(−i∆β) the ratio of +two consecutive points with a given angular increment phase +∆β. For A = 1 and ∆β = 2πm/N,Zm is computed over an +unit circle and the CZT operation becomes a discrete FFT. To +zoom the signal Xn in by a scalar factor c, A = exp(−iπ/c) and +W = exp(−i2π/Nc)29. Therefore, Eq. 8 can be expressed in +terms of convolution as follows30: +˜Xzm = W m2/2FFT−1 � +FFT +� +XnA−nW n2/2� +·FFT +� +W n2/2�� +. +(9) +The inverse CZT of a signal ˜Xzm in a frequency-domain rep- +resentation is defined as the complex conjugate of the CZT of +the complex conjugate ˜X∗ +zm of ˜Xzm within some scaling factor + +Calculating Point Spread Functions +11 +-10 +0 +10 +x/µm +-10 +-5 +0 +5 +10 +y/µm +(a) +-10 +0 +10 +x/µm +-10 +-5 +0 +5 +10 +y/µm +(b) +-10 +0 +10 +x/µm +-10 +-5 +0 +5 +10 +y/µm +0.42 +0.43 +0.44 +0.45 +0.46 +0.47 +0.48 +(c) +-10 +0 +10 +x/µm +4 +5 +6 +7 +z/µm +(d) +-10 +0 +10 +x/µm +4 +5 +6 +7 +z/µm +(e) +-10 +0 +10 +x/µm +4 +5 +6 +7 +z/µm +0.4 +0.42 +0.44 +0.46 +0.48 +0.5 +0.52 +(f) +FIG. 9: Profiles displayed at gamma 0.05 of the PSFs calculated from the slice propagation method. (a, b, c) xy-plane at +defocus position 7 µm. (d, e, f) xz-cut including to the optical axis starting at 3.5 µm away from the focus and with ∆z = 3.5 µm. +(a, d) Using the standard FFT. (b, e) By zero-padding the image window size to twice. (c, f) Using CZT. The parameters are NA += 1.4, immersion medium : water (n = 1.33), polarization : circular; emission wavelength λem = 580 nm, voxel size 80 nm × +80 nm × 140 nm and, window size : 256×256×25 pixel. +for a CZT operating on a unit circle33. The propagator func- +tion using CZT is: +1. Calculate an appropriate zoom-in factor c such that the +lateral window size of the calculated PSF is slightly big- +ger or equal to the lateral dimension of the PSF at the +position ∆z from the focus. The factor c is calculated +as c = (D+Nxy/2)/(Nxy/2), where tanθmax = D/∆z in +real space and tanθmax = kxy/kz in Fourier space with +θmax being the maximal angular aperture and Nxy the +number of pixels in the xy-plane (see Fig. 10); +2. Zoom in the pupil plane amplitude distribution with the +calculated factor c. If the factor c leads to a pupil radius +bigger than the given window size, an appropriate new +target image size must be chosen, and the pupil radius +is always zoomed to just fit in the image window; +3. Apply the angular spectrum propagator (see above) +4. Apply an inverse CZT and zoom out with the same +above-mentioned factor c to obtain the amplitude at ∆z; +5. Extract the field within the original window size. +C. +The sinc-shell method (SR) +This method is based on the fact that the three-dimensional +Fourier transform of a complete spherical shell has a +convenient solution in real space, which is sinc(k0|r|) = +Nxy/2 +real space in focus +real space at ∆z +D +𝜃max +∆z +x +z +FIG. 10: Illustration of the wave propagation in real space for +calculating the zoom factor c. +sin(k0|r|)/(k0|r|), k0 being the wavenumber in the medium +and r the radial position. Given this, the sinc-shell method is +described as follows: +1. Extend the border of the desired window size N×N by +25% to get a new window N'×N'; +2. Generate a sinc(k0|r|) amplitude distribution in three +dimensions in real space within the window size +N'×N'×z, z being the range along the axial axis. Mul- +tiply this distribution by a compact disk of radius equal +to N; +3. Generate a 3D spherical shell by Fourier transforming +the result from step 2; +4. Set all values at negative kz to zero (akin to a Hilbert- +transform) or and/or keeping only the kz-range which + +- +:OCalculating Point Spread Functions +12 +contains valid ⃗k vectors (yielding a change in z- +sampling and a phase ramp in real space, not affecting +intensity values). This half of the 3D spherical shell +corresponds to the propagator of the field in free space; +5. Generate 3 versions (for Ex, Ey, Ez) of the 2D Pupil as +obtained by the jinc-FT trick, each containing the apla- +natic factor and the appropriate electric field modifica- +tion factors. +6. make two additional copies of the 3D McCutchen pupil +and apply the 3 electiv field pupils to each of the 3 Mc- +Cutchen spheres by multiplying it with each kz slice. At +this point, only a kz-range required for the intensity PSF +is needed; +7. Perform a three-dimensional Fourier-transformation of +each of the three field component McCutchen spheres to +obtain the sought-after field components in real space; +8. Extract the field within the desired window N×N×z. +This method has the attractive property that it does not +suffer from the Fourier wrap-around effect. The only wrap- +around effect is suppressed by extending the window by only +25% and filtering the amplitude field with a disk (step 1 and +2). This filtering in real space corresponds to a convolution of +the 3D spherical shell with a Jinc function in Fourier space. It +increases the precision in the values of the shell and removes +any artefact that may arise during the computation. A disad- +vantage of this technique is that at least step 2,3 and 4 have +to be performed while observing the Nyquist sampling along +kz for the full field to include its z-propagation. This method +is also not readily applicable to a single slice (in or out-of- +focus). +D. +The Fourier-shell interpolation (VS) +This method aims at representing the useful part of the Mc- +Cutchen pupil directly in 3D-Fourier-space and projecting the +two-dimensional pupil functions onto this three-dimensional +shell. The difficulty is that the shell, at each integer [kx,ky] +position has a non-integer kz position which needs to be rep- +resented by interpolating along kz in Fourier space. +As a credible representation of such a non-integer kz would +require essentially the whole available kz-range, an appropri- +ate compromise to keep the computation efficient was made. +We aim to represent only the central part of the correspond- +ing real-space representation as faithfully as possible and label +the rest as “don't care” region (see Fig. 11b). The border of +this “don't care” region is limited by a chosen factor (here it +is chosen to be at the 8th pixel from both edges). To calcu- +late the necessary Fourier space interpolation kernel, the part +of real space near the border of the z-volume is iteratively up- +dated, while the central part is forced to the expected values +in each iteration in this iterative Fourier transformation algo- +rithm (IFTA). +In addition, a pre-defined cut-off frequency is chosen. This +cut-off frequency limits the number of interpolation coeffi- +cients, which can be used to fill the voxels along kz in Fourier- +space adjacent to the one nearest to the non-integer kz(kx,ky) +position of the McCutchen pupil. The cut-off frequency here +is set to ±8 Fourier space pixels yielding 17 interpolation co- +efficients to be determined. The required interpolation co- +efficients are generated with the help of the IFTA34. +An +interpolation table of 60 sup-pixel positions along kz was +pre-computed via IFTA. As initialization, ideal non-cyclic +exp(2πikzz) waves were generated in real space correspond- +ing to the respective sub-pixel frequencies in Fourier space. +The ideal waves are then Fourier-transformed and only Nz +(here 17) interpolator values are kept and all others are set to +zero. The result is transformed back to real space, where the +central area (here the inner is about 66 % of the given z-range) +is replaced by the original perfect waves, but the “don't care” +region is not touched. This is repeated (typically 500 times) +until convergence. The so-generated interpolation table (17 +complex valued coefficient as a function of 60 sub-pixel lo- +cations) is stored for later use (see Fig. 11c). Note that we +only need to calculate the residual non-integer part of an os- +cillation, leading to an interpolation table which only contains +less then one oscillation. The integer oscillations are taken +care of by the placement of the kernel in Fourier-space. A +typical example for the offset of 0.25 pixels is shown in real +and Fourier space in Fig. 11b and 11d respectively, overlayed +with the ideal subpixel wave (solid line which corresponds to +the legend ‘Original signal’). The border of the “don't care +region” is indicated by the dashed vertical lines. A real space +representation of the full interpolation table is shown in Fig. +11a with the “don’t care region” also indicated by the vertical +red dashed lines. +The size of the border factor (in pixels) and the cut-off +frequency defining the number of interpolation coefficients +should be roughly the same. If the “don’t care region” is far +bigger than the “do care region”, there tend to be less inter- +polation coefficients generated for the given region of support +frequencies in Fourier space, which is delimited by the blue +dashed lines in Fig. 11c. This leads to a large computational +overhead for a given region of interest, since the region re- +quires extensive padding . A small “don’t care region” on the +other hand can lead to inaccuracies inside the “do care region” +hence the region of support frequencies in Fourier space. +This PSF generation algorithm based on Fourier-shell inter- +polation works as follows: +1. Generate the three two-dimensional McCutchen pupil +projections as described above (using the jinc-FT trick +as described in Section III D 2) +2. Calculate kz(kx,ky) for every pixels within the pupil and +round it to the nearest 1/60 subpixel kz position; +3. Write these pupils into Fourier space by applying the +appropriate interpolation kernel for this sub-pixel posi- +tion; +4. Perform a three-dimensional Fourier transformation to + +Calculating Point Spread Functions +13 +"don't care +region" +"don't care +region" +0.25 +sub-pixel +-20 +-15 +-10 +-5 +0 +5 +10 +15 +20 +Axial position/pixel +10 +20 +30 +40 +50 +60 +Sub-pixel position +-0.15 +-0.1 +-0.05 +0 +0.05 +0.1 +0.15 +(a) +-20 +-10 +0 +10 +20 +Axial position/pixel +-0.1 +-0.05 +0 +0.05 +0.1 +Phase in real space +/arb. unit +border +border +Original signal +Final signal +"don't care +region" +updated +"don't care +region" +updated +"do care region" +re-enforced +(b) +"frequency +support +region" +0.25 sub-pixel +-20 +-15 +-10 +-5 +0 +5 +10 +15 +20 +Spatial frequency +10 +20 +30 +40 +50 +60 +Sub-pixel frequency +-0.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +(c) +(d) +FIG. 11: (a) Phase shift in the ideal wave exp(2πikzz). (b) Phase at a sub-pixel 0.25, indicated by the horizontal white line in +(a). (c) Interpolation table in Fourier space containing the interpolator coefficients at 60 different sub-pixels. (d) Interpolation +coefficients in Fourier space along the 0.25 sub-pixel indicated by the horizontal white line in (a) and (c). +obtain the three-dimensional field distributions (with +expected errors in the “don't care region”). +This method can be performed fast and memory efficient as +a single access operation in Matlab by exploiting its indexed +addressing capabilities. In this way, the complex-valued 2D +pupil can be rapidly filled into the appropriate Fourier space +region with the optimized interpolation coefficients as de- +scribed above and the “don't care” region can be later re- +moved. The required kz-range can be kept to a minimum. This +method was originally constructed to help with the reconstruc- +tion of coherent tomography data, where each entirely dif- +ferent phase projection can then directly written into Fourier- +space without the need of a full immediate propagation (e.g. +by the slice propagation method) for each projection, which +saves an enormous computational overhead. +E. +Dipole emission PSF +In reality, a fluorescent emitter can rotate freely at its place +or its orientation can be fixed. The PSF of an emitting dipole +in the focal plane depends on the polarization of the illumina- +tion light and the dipole orientation in space. To describe the +dipole emission, let ⃗µ(θdip,φdip) denotes its emission transi- +tion dipole moment which is a function of its elevation angle +θdip about the optical axis z and azimuthal angle φdip about +x−axis. The following steps describe how to accommodate a +fixed dipole into the PSF calculation: +1. Calculate the field amplitude distribution by taking only +the x−polarized part at the pupil plane, using the direc- +tional field changes as described above and the appro- +priate aplanatic factor for an emission PSF to obtain the +three-dimensional amplitude field ⃗E1 = [E1x,E1y,E1z] +as a function of lateral distance from the optical axis +in the image plane; +2. By projecting this amplitude field on the dipole orien- +tation and calculating the absolute square of the scalar +product, we obtain the detected intensity Ix = |⃗E1 ·⃗µ|2 +as measured through an analyzer oriented along x; +3. Steps 1 and 2 can be repeated for a y−oriented analyzer +yielding Iy and the average intensity without analyzer is +obtained as the PSF of the dipole emitter: +PSFdip = 1 +2(Ix +Iy). +(10) +To calculate the intensity in the back focal plane of a dipole +emitter with a given transition dipole orientation, the same +steps as described above are to follow but instead of taking +the amplitude field in the image plane, the user is to use the +three-dimensional amplitude field in the McCutchen pupil. + +-10 +-5 +S +patia0 +5 +l frequency10 +1520nterpol +0cut-off +freguenoVcut-offarb +ators +0.2-unit +0.4 is enfor +zeroregior +ced"zero is enfoDrced"spa +0.8 +erfrequen +suppoicy +rt-Original +Final sign1gnal +ale--20 +-15Calculating Point Spread Functions +14 +If the illumination light is unpolarized or circularly polar- +ized, averaging over all the dipole orientations or over three +different perpendicular dipole orientations leads to the same +PSF as obtained by calculating a corresponding illumination +PSF using circular polarized light, apart from a possible dif- +ference in the aplanatic factor. This leads back to the PSF of +an isotropic emitter. +PSFav = +1 +4π/3 +� 2π +0 +� π +0 PSFdip sinθdipdθdipdφdip. +(11) +If the illumination light is linearly polarized, averaging over +all the dipole orientations does not lead to a symmetric PSF. +Further study in polarization in fluorescence microscopy is +found in35. +V. +QUANTITATIVE PSF COMPARISON +The comparison of the models is performed in two steps. +The first step consists of comparing the theoretical models +with a chosen gold standard (GS) and the second one consists +of comparing all the models with experimental data. +We choose an emission PSF according to the RW0 model +(described in Section II B 2) with a very high sampling as our +GS for the theoretical models comparison. The GS is calcu- +lated with a sampling of 16 nm × 16 nm × 100 nm and win- +dow size of 1280×1280×64 pixel. The result is subsampled +to 80 nm × 80 nm × 100 nm by binning groups of five ad- +jacent pixels along x and y to correspond to what we choose +as a “normal sampling”. This normal sampling consists of a +voxel size of 80 nm × 80 nm × 100 nm and image window +size 256×256×64 pixel. +The z-range by which the VS is computed is set to be 1.5× +larger than the other models (96 vs 64 pixels) to compare only +the “do care region” in the quantitative comparison of the +PSFs. The computational cost of this enlarged window size +(256 × 256 × 96) is therefore taken as the computation time +for the (256×256×64) PSF for the case of the VS. +We choose simulation parameters corresponding to our ex- +periment. We have an oil immersion objective where the re- +fractive index of the oil is n = 1.518 and numerical aperture is +NA = 1.4. To mimic randomized dipole orientation, the polar- +ization state is assumed to be circular and the emission wave- +length is λ = 580 nm. The PSFs models are implemented and +compared in quality and computational speed. The quantita- +tive comparison is done for a single in-focus plane and for the +whole volume, excluding 5 % on each side near each image +border. Each PSF data is normalized as a 3D Volume to yield +an integrated intensity of one at the focal plane over the central +90 % in x and y. The radial mean intensity profile at the focus +position for each model is plotted in Fig. 12 in logarithmic +scale. +In Fig. 12 and the following sections, we denote by RW1 +the RW model calculated by the same technique as the GS +but performed at the same (normal) sampling grid as all other +techniques (SP, CZT, VS, etc.) to compare to. RW2 refers to +an alternative way for calculating the RW model which uses a +Fourier transform along z. +A. +Error analysis and computation time of the theoretical +models compared with the RW gold standard (GS) +1. +Model accuracy +To verify how accurate each model is compared to the +GS, we use two different techniques: the mean relative er- +ror (MRE) and the normalized cross correlation (NCC) be- +tween each model and the GS. The MRE has the advantage +to describe the average performance-error of each model in +comparison with the GS. However, for a data which is shifted +and the shifting parameter might be unknown or does not +have much importance, the use of normalized cross correla- +tion (NCC) for the comparison is advisable as it is not sensi- +tive to linear shifting. The formula used to compute the MRE +is given by +MRE(z) = ∑xy |GSxy(z)−PSFxy(z)| +∑xy |GSxy(z)| +. +If not specified otherwise, only intensity values bigger than +the 0.1 % of the maximum intensity value of GS at each +z−slice are considered in the calculation of the MRE. The +MRE results are displayed in Fig. 13. +We calculate the 2D NCC slice by slice between our mod- +els and the GS. For this, a built-in function normxcorr2 in +Matlab is used36. For two 2D input images, this function gen- +erates as an output a 2D image with double the size minus one +of the input images. A value m at a given position (i, j) in +the output NCC refers to the NCC of the two images at (i, j) +shift. A shift of (i, j) = (0,0) means the two images are on +top of each other. The value of m is between −1 and 1, 0 +for zero-correlation, 1 for maximum correlation and −1 for +anti-correlation. This technique therefore accounts for possi- +ble shifts between the model and the GS. A perfect correlation +corresponds to a NCC of 1 at a shift position (0,0). We report +the maximum of the NCC at each xy-slice for each axial po- +sition z as well as the NCC of the 3D volume PSF compared +with the 3D GS and have both parameters to check how close +the model is to the GS. The result is summarized in Fig. 14. +2. +The missing cone problem +As observed in Fig. 13 and Fig. 14, the errors between most +of the PSF models and the GS are higher at a larger depth and +decrease as the field is focusing. The same errors tend to be +more enhanced again near the focus. The same observation +can be made for the case of the NCC. This observation can +be interpreted by studying the missing cone problem of each +model. +In wide-field microscopy, the missing cone corresponds to +frequencies close to the kz-axis that prevent the OTF to trans- + +Calculating Point Spread Functions +15 +0 +50 +100 +150 +vradial/o.u. +0 +2 +4 +6 +8 +10 +12 +Radial Position/µm +-15 +-10 +-5 +Radial Mean Intensity +in log scale +GS +RW1 +RW2 +SR +180 185 190 +vradial/o.u. +-16 +-14 +-12 +(a) +0 +50 +100 +150 +vradial/o.u. +0 +2 +4 +6 +8 +10 +12 +Radial Position/µm +-15 +-10 +-5 +Radial Mean Intensity +in log scale +GS +SP +CZT +VS +180 185 190 +vradial/o.u. +-17 +-16.5 +(b) +0 +50 +100 +150 +vradial/o.u. +0 +2 +4 +6 +8 +10 +12 +Radial Position/µm +-15 +-10 +-5 +Radial Mean Intensity +in log scale +GS +GL +ScalarPSF +VectorPSF +180 185 190 +vradial/o.u. +-20 +-18 +(c) +0 +50 +100 +150 +vradial/o.u. +0 +2 +4 +6 +8 +10 +12 +Radial Position/µm +-15 +-10 +-5 +Radial Mean Intensity +in log scale +GS +PSFGen_Good +PSFGen_Best +180 185 190 +vradial/o.u. +-17.2 +-17 +(d) +FIG. 12: Radial mean intensity profiles at the focus position in logarithm scale. (a) GS: The gold standard computed using the +Richards and Wolf method7,12, RW1 and RW2: two ways for computing the RW models (Section II B 2), SR: the sinc-shell +method (Section IV C). (b) SP: Slice propagation method with FFT (Section IV B), CZT: SP technique using the chirp Z +transform (Section IV B 3), VS: Fourier shell interpolation technique (Section IV D). The state-of-the-art are: (c) A scalar PSF +based on the work of Gibson and Lanni 6 and developed by Li, XUE, and BLU 8, a scalar and vector PSF from9 and, (d) a +vector PSF calculated with the Richards and Wolf 3D optical model with good and best accuracy respectively from the +PSFGenerator toolbox in10. The variable vradial denotes the radial optical unit. +mit information about the object along that axis (Fig. 15a). +As a consequence, a uniform illumination fluorescence imag- +ing setup is incapable of focussing (or “sectioning”) a planar +sample. Out-of-focus light is distributed to different regions +but, the energy is still conserved. The integrated intensity at +each z position remains the same. This should be the case for +any wide-field PSF as long as the PSF remains confined well +within the calculation grid. Sufficient sampling turns out to be +a key factor, especially for the methods, where radial symme- +try is exploited. Since we are dealing with an approximation +of the exact field in a finite grid, it is required to have the right +amount of data points. An inappropriate choice of grid and +pixel size can lead to the violation of the missing cone in the +corresponding transfer function of the system. This problem +is demonstrated by integrating the PSF for each xy−plane at a +given axial z−position. Even though the radial profile, illus- +trated in Fig. 12, seems to look good and fits into the GS at the +focus position, there may still be significant sampling-related +violations of the missing cone. Note that for simulations to +accurately evaluate the performance of 3D deconvolution rou- +tines on widefield data, preserving the missing cone property +is paramount. +A more precise observation of the violation of the missing +cone can be made by zooming on the z−range around the fo- +cus and displaying the integrated intensity of each slice along +the axial position z (see Fig. 15). To quantify this effect, +the standard deviation (Std) of the integrated intensities within +20 % around the focus of the given z−range is calculated and +is plotted in Fig. 15b. This 20 % region is delimited by the +two dashed vertical lines in Fig. 15c, 15d, 15e and 15f. The +Std measures the non-uniformity of the laterally integrated in- +tensity over axial position and the importance of the peak of +the integrated intensity compared to the minimum within the +z-range that is considered. The order of magnitude of the Std +from the techniques CZT, RW2, VS, and, SP are in range of +10−4 which are fairly small and smaller than the Std of the +defined GS itself. Increasing the sampling of the models re- +duce the violation of the missing cone considerably. This is +seen in Fig. 15b for the case of the GS and RW1 where the +two PSFs are both generated from the same model but with +different sampling and window size. The Std has decreased +by about 5.41 % by oversampling the lateral grid five times. +However, higher sampling, i.e. finer step with a big windows, +may lead to an expensive computation (about 7× slower) and +requires a larger computer memory. + +Calculating Point Spread Functions +16 +-60 +-40 +-20 +0 +20 +40 +60 +uaxial/o.u. +-3 +-2 +-1 +0 +1 +2 +3 +Axial Position/µm +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +Mean Relative Error +RW1: MRE3D = 0.0163 +RW2: MRE3D = 0.2994 +SR: MRE3D = 0.0293 +(a) +-60 +-40 +-20 +0 +20 +40 +60 +uaxial/o.u. +-3 +-2 +-1 +0 +1 +2 +3 +Axial Position/µm +0.01 +0.02 +0.03 +0.04 +Mean Relative Error +SP: MRE3D = 0.0121 +CZT: MRE3D = 0.0088 +VS: MRE3D = 0.0117 +(b) +-60 +-40 +-20 +0 +20 +40 +60 +uaxial/o.u. +-3 +-2 +-1 +0 +1 +2 +3 +Axial Position/µm +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +Mean Relative Error +GL: MRE3D = 0.2089 +ScalarPSF: MRE3D = 0.2065 +VectorPSF: MRE3D = 0.0035 +(c) +-60 +-40 +-20 +0 +20 +40 +60 +uaxial/o.u. +-3 +-2 +-1 +0 +1 +2 +3 +Axial Position/µm +0.052 +0.054 +0.056 +0.058 +Mean Relative Error +PSFGen_Good: +MRE3D = 0.0561 +PSFGen_Best: +MRE3D = 0.0561 +(d) +FIG. 13: Mean Relative Error of (a) our RW methods and SR, (b) the SP, CZT and VS and the state-of-the-art (c) GL for the +Gibson and Lanni model, ScalarPSF and VectorPSF and (d) vector PSF from PSFGenerator for good and best accuracy with the +GS. +-60 +-40 +-20 +0 +20 +40 +60 +uaxial/o.u. +-3 +-2 +-1 +0 +1 +2 +3 +Axial Position/µm +0.96 +0.97 +0.98 +0.99 +NCC +RW1: NCC3D = 0.99986 +RW2: NCC3D = 0.9959 +SR: NCC3D = 0.99963 +(a) +-60 +-40 +-20 +0 +20 +40 +60 +uaxial/o.u. +-3 +-2 +-1 +0 +1 +2 +3 +Axial Position/µm +0.9996 +0.9997 +0.9998 +0.9999 +NCC +SP: NCC3D = 0.99965 +CZT: NCC3D = 0.99965 +VS: NCC3D = 0.99965 +(b) +-60 +-40 +-20 +0 +20 +40 +60 +uaxial/o.u. +-3 +-2 +-1 +0 +1 +2 +3 +Axial Position/µm +0.985 +0.99 +0.995 +NCC +GL: +NCC3D = 0.98863 +ScalarPSF: +NCC3D = 0.99158 +VectorPSF: +NCC3D = 1 +(c) +-60 +-40 +-20 +0 +20 +40 +60 +uaxial/o.u. +-3 +-2 +-1 +0 +1 +2 +3 +Axial Position/µm +0.9988 +0.999 +0.9992 +0.9994 +0.9996 +0.9998 +NCC +PSFGen_Good: +NCC3D = 0.99872 +PSFGen_Best: +NCC3D = 0.99872 +(d) +FIG. 14: Normalized cross correlation between the theoretical RW GS with the (a) RW1, RW2 and SR methods, (b) the SP, +CZT and SR and the state-of-the-art (c) GL for the Gibson and Lanni model, ScalarPSF and VectorPSF and (d) vector PSF +from PSFGenerator for good and best accuracy. + +Calculating Point Spread Functions +17 +Radial frequency +Axial frequency +0.25 +0.3 +0.35 +0.4 +0.45 +0.5 +0.55 +Missing cone +(a) +0.036 +0.66 +0.004 +0.011 +0.001 +0.009 +0.448 +0.678 +0.227 +0.169 +1.875 +1.875 +GS +RW1RW2SP +CZT +VS +SR +GL +ScalarPSF +VectorPSF +PSFGen_Good +PSFGen_Best +Methods +0 +0.5 +1 +1.5 +2 +Std of the Relative Integrated +Intensity/arb. unit + 10-2 +(b) +-50 +0 +50 +uaxial/o.u. +-2 +-1 +0 +1 +2 +Axial Position/µm +0.98 +0.99 +1 +1.01 +Integrated Intensity +/arb. unit +GS +RW1 +RW2 +SR +(c) +-50 +0 +50 +uaxial/o.u. +-2 +-1 +0 +1 +2 +Axial Position/µm +0.994 +0.996 +0.998 +1 +Integrated Intensity +/arb. unit +GS +SP +CZT +VS +(d) +-50 +0 +50 +uaxial/o.u. +-2 +-1 +0 +1 +2 +Axial Position/µm +0.98 +0.985 +0.99 +0.995 +1 +Integrated Intensity +/arb. unit +GS +GL +ScalarPSF +VectorPSF +(e) +-50 +0 +50 +uaxial/o.u. +-2 +-1 +0 +1 +2 +Axial Position/µm +0.94 +0.96 +0.98 +1 +Integrated Intensity +/arb. unit +GS +PSFGen_Good +PSFGen_Best +(f) +FIG. 15: The violation of the missing cone is observed by calculating the integrated intensity of 90 % of the total surface +xy−plane for each axial position z. (a) The OTF calculated from the GS and displayed at gamma of 0.05. The missing cone is +presented in yellow. (b) The standard deviation (Std) of the relative integrated intensity calculated within a z−range equal to +1.3 µm near the focus. The profiles of the integrated intensity along z from (c) our RW methods and SR, (d) the SP, CZT and +VS and the state-of-the-art (e) GL for the Gibson and Lanni model Li, XUE, and BLU 8, ScalarPSF and VectorPSF9 and (f) +vector PSF from PSFGenerator for good and best accuracy10. The variable uaxial denotes the axial optical unit. +3. +Wrap-around effect +In addition to the missing cone problem, the wrap-around +effect due to FFT-based convolutions can contribute to the ac- +curacy of the model. To investigate this effect, we define a ref- +erence window grid Wref where this wrap-around effect should +be minimal. The size of the window Wref is defined such that +the width of the PSF at the highest depth could still fit into it. +It is calculated to be: +Wref|x = 2×2× +� +� +zmax +kx|max +kz|max +1.3dres +sx +� +�; +(12) + +Calculating Point Spread Functions +18 +where the first applied factor 2 is to double the half window; +the second factor 2 is to sample the frequency space two times +finer; zmax is the maximum depth expressed in nm; kx|max and +kz|max are the maximal radial and axial frequency respectively; +dres is the resolution limit of the optical system; the factor +1.3 is an heuristic factor and, sx is the sampling size along x, +expressed in nm. The same formulation applies along y. +To quantify the wrap-around effect, we generate different +PSFs from each model at different windows denoted by W +and denote the PSF h. We denote h0 the reference PSF which +is calculated with the reference window Wref. If W is smaller +than Wref, the wrap-around effect in h relative to h0 is calcu- +lated within the window W. If W is bigger than Wref, no wrap- +around effect is expected. However, the energy is spread over +a larger grid so the difference between h and h0 is not expected +to be zero but is expected to converge to a constant. In this +case, we calculate the difference between h and h0 within the +window Wref. The three-dimensional MRE and mean square +error (MSE) between h and h0 for the given window for each +model are calculated and plotted in Fig. 16. +In this computation, Wref is calculated and rounded to be +300 × 300 pixel. The lateral window is square and the rest +of the parameters for the comparison are the same as those +described for calculating the model accuracy (see Section V). +As mentioned earlier, the scalar PSF based on Gibson and +Lanni denoted by GL in Fig. 16 uses Bessel series as a fast +approach to calculate the PSF8. As described in9, a numer- +ical integration based on Simpson’s rule is used to compute +ScalarPSF and the VectorPSF. Thus, those models do not pos- +sess any FFT wrap-around effect in their computation. They +can therefore be used as reference to measure the wrap-around +effect. +A modified CZT PSF model, denoted by mCZT, is in- +troduced in this particular PSFs comparison. +This model +works similarly as the CZT PSF model except the fact that +if the given window W as described above is smaller than +Wref, mCZT chooses Wref as window grid for the computa- +tion, crops the generated PSF to get the input size W and scale +the PSF by the integrated intensity in focus with a window +size W. The errors between the PSF h at a window W and +the reference PSF h0 in this case are small enough but are not +zeros. +As it is also observed in Fig. 16a and 16b, errors calculated +from the Fourier-based models decrease generally as the win- +dow grid becomes bigger except for the case of RW2. How- +ever, as it has been said computation of a PSF at a bigger win- +dow grid can lead to an expensive computation and requires a +larger computer memory. +The wrap-around effect can also be interpreted as under- +sampling the phase in Fourier-space, which leads to aliasing +effects in real space. Large propagation distances thus lead to +more aliasing, particularly for small pupils in Fourier space, +i.e. small pixel-sizes. +4. +Computational time +To investigate the computation time of each model, we use +MATLAB R2018a under the same operating system stated in +Section 1.4.2. We run each model with the regular sampling +80 nm × 80 nm × 100 nm, the same microscope parameters +in the previous section. We choose two different window size +256 × 256 × 32 and 256 × 256 × 256 pixel and generate the +computation time per voxel of each model. The results are +displayed in Fig. 17. +As mentioned previously, eight pixels are added on both +borders of the z-range (32 and 256) to account for the “don’t +care region” in the VS model. The addition of 8 pixels on both +sides of the z-range 32 pixels in the VS slows the computation +down by a few milliseconds. The first run of the VS requires +more computation time. It can go up to 20× or even slower +than the second run. The computation is faster in the second +run as the interpolators in the models are stored and used for +the next computation if the PSF and imaging parameters re- +main the same. +The computation time per voxel of a PSF using the CZT +technique is considerably higher compared to the other tech- +niques apart from the PSF generated from the PSFGenerator at +the best accuracy10. This computation cost is explained from +the fact that the CZT technique expands the lateral size of the +PSF to avoid any FFT-based wrap-around problems if the z- +range is large (See Section IV B). As the z-range increases, the +computation time per voxel of the CZT is growing because a +large lateral window is needed. A similar cost must be paid +when one requires higher sampling to generate a more accu- +rate model and to reduce as much as possible the missing cone +error near the focus and any wrap-around effect. +5. +Discussion +In summary all the vector models agree closely. VectorPSF +has a 3D NCC value equal to 1 while CZT, VS, SP and SR +differ deviate at the 4th and PSFGen_Good and PSFGen_Best +at the 3rd digit from the perfect NCC of one. The scalar PSFs +have the smallest 3D NCC values of 0.9916 for ScalarPSF and +0.9886 for GL. +Similarly, the 3D mean relative error between the Vec- +torPSF is the closest to the GS with a value of MRE ≈ 0.0035. +This is followed by the CZT model with MRE ≈ 0.0088, +VS and SP models with MRE ≈ 0.012. This high similar- +ity and accuracy is achieved at a high computation time cost +(in average 20.432 µs·voxel −1 for the CZT if the window size +256×256×256 voxels. +Slightly less accuracy can be achieved for a less expensive +computation time with the SP and VS techniques and espe- +cially if a repetitive computation of the PSF with the same +imaging parameters is required. However, one needs to ex- +pand the z-range to consider the “don’t care region”. A higher +z-range is therefore needed accordingly. Another disadvan- +tage of the VS is that the computation of a single slice PSF +is not possible with this technique. Similarly, as discussed in +Section IV B 1, SP suffers from a wrap around problem at a + +Calculating Point Spread Functions +19 +0 +100 +200 +300 +400 +500 +600 +window size +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +0.4 +0.45 +MRE / arb. unit +RW1 +RW2 +SP +CZT +mCZT +VS +SR +GL +ScalarPSF +VectorPSF +PSFGen_Good +50 +50.0001 50.0002 +0.016 +0.018 +0.02 +0.022 +0.024 +0.026 +0.028 +Zoom +(a) +0 +100 +200 +300 +400 +500 +600 +window size +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +MSE / arb. unit +10-7 +RW1 +RW2 +SP +CZT +mCZT +VS +SR +GL +ScalarPSF +VectorPSF +PSFGen_Good +49.995 +50 +50.005 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +10-10 +Zoom +(b) +FIG. 16: Quantification of the wrap-around effect in each of the models. (a) Three-Dimensional Mean Relative error. (b) +Three-Dimensional Mean Square Error. The right hand side windows entitled Zoom of (a) and (b) are the zoomed-view of the +red dashed rectangular in the left hand side. +higher depth and with a smaller window grid. Nevertheless, +SP, CZT and VS are less prone, compared to the other method, +to violate the missing cone properties of PSF in wide-field mi- +croscopy. +On the other hand, the SR method can beat SP, VS and even +CZT in terms of wrap-around effect. The still observed wrap- +around effect quantified and displayed in Fig. 16 are due to +the jinc aperture added to the pupil aperture stop to limit the +field. SR has a very high similarity to the GS especially near +the focus. Its missing cone problem is also considerably small +but not as small as in SP, VS and CZT. SR is faster to compute +than CZT. +PSFGen_Good and PSFGen_Best have relatively higher +MRE and lower NCC with the GS compared to the PSF +models developed under this project and the state-of-the-art +VectorPSF. PSFGen_Good yields to a lower MRE than PS- +FGen_Best. We speculate that the reason for this could be +that the PSFGen_Best is overestimated. The probability to +violate the missing cone is higher in the computation of this +model. The model is also very expensive in time compared to +the other models discussed in this manuscript. +Compared to the rest of the models, the scalar model (GL +and ScalarPSF) have the lowest 3D NCC and highest 3D MRE +(after RW2) with the GS. These values quantify the difference + +Calculating Point Spread Functions +20 +2.226 +2.213 +2.526 +2.94 +1.578 +1.644 +5.384 +20.432 +2 +1.421 +4.278 +5.665 +0.107 +0.106 +1.206 +1.686 +1.392 +1.897 +2.348 +3.01 +30.512 +44.023 +RW1 +RW2 +SP +CZT +VS +SR +GL +ScalarPSF +VectorPSF +PSFGen_Good +PSFGen_Best +0 +10 +20 +30 +40 +Computation time/µs voxel-1 +256 +256 +32 +256 +256 +256 +window size: +FIG. 17: Computational time of each technique per pixel in +µs. RW1 and RW2 are two ways for computing the Richards +and Wolf models7,12, SP: slice propagation technique +(Section IV B), CZT: SP technique using the chirp Z +transform (Section IV B 3), VS: Fourier shell interpolation +technique (Section IV D), SR: sinc-shell method (Section +IV C), GL: scalar PSF based on the work of Gibson and +Lanni 6 and developed by Li, XUE, and BLU 8, ScalarPSF +and VectorPSF: scalar and vector PSF from9 and, +PSFGen_Good and PSFGen_Best: vector PSFs calculated +with the Richards and Wolf 3D optical model with good and +best accuracy respectively from the PSFGenerator toolbox in +reference10. +between a vector PSF and scalar PSF. These scalar PSFs are +however the fastest models. Their precision and accuracy can +be sufficient for some application. Although RW2 is among +the models which have higher ability to satisfy the missing +cone properties, its accuracy tends to be close to the scalar +PSF models, the least accurate. +B. +Experimental validation +1. +Experimental PSF data +To validate the model experimentally, a PSF measurement +is performed by averaging the images of beads. For this, we +use Tetraspeak beads of 100 nm in diameter (Invitrogen by +Thermo Fisher Scientific, REF T7279). The beads fluoresce +with an emission wavelength of 580 nm at an excitation light +of 561 nm. The beads are first diluted with distilled water to +get 1/400 concentration and vortexed. We drop some droplets +of the solution on a coverslip and let it dry for over 4 h. Our +imaging system is composed of a 63× Zeiss C-Apochromat +oil immersion (n = 1.518) objective lens of 1.4 NA (DIC +M27). The choice of the oil immersion for the microscope +is very important as this can contribute to any observed aber- +ration in the recorded image. Any contributions from back- +ground light is reduced and eliminated during the measure- +ment. The image of background is recorded before any mea- +surement and several measurements are averaged to reduce +the noise in the offset estimation. The exposure time of the +detector camera is set to be 100 ms and the output laser power +illuminating the sample is measured at 25 mW. +To obtain the experimental PSF data, an offset image is sub- +tracted from the raw data. The PSF is afterwards constructed +by averaging over 64 beads. This averaged PSF data is of size +40 × 40 × 159 pixel and voxel size 80 nm × 80 nm × 50 nm. +Let hexp denote the experimentally averaged PSF data. +To compare our theoretical models with hexp , two differ- +ent sets of comparison are conducted. The first set consists +of retrieving the phase aberration contained in hexp and add +it in the corresponding pupil plane of our theoretical model. +This comparison is only achieved with the PSFs in our toolbox +(RW1, RW2, SP, CZT, VS, SR) since we do not have access +to the amplitudes fields of GL, ScalarPSF, VectorPSF, PSF- +Gen_Good and PSFGen_Best. The phase retrieval technique +developed by Hanser et al.37 is conducted to retrieve the pupil +function hence phase aberration of hexp . By testing on a the- +oretically computed known phase aberration and pupil func- +tion with the same parameters as the imaging system here, the +phase retrieval algorithm is able to retrieve the phase with a +MRE of 0.0084×10−4 and the retrieved PSF has a 3D MRE +of 0.786×10−8 compared to the preprocessed measured PSF. +Only the z−range within the indicated blue dashed in Fig. 18a +are used for this set of comparison in order to avoid a compar- +ison with only noise. The size of hexp within this region is +40×40×80. +The second set of comparison consists of comparing a block +region near the focus with higher signal and discard those at +higher depth and compare the region with non-aberrant theo- +retical PSFs. +2. +Theoretical PSF data +The required window to avoid any wrap-around is calcu- +lated using Eq. +12. +Each theoretical model is computed +with a lateral size equal to the result from this computation +(242×242 pixel) and with a depth equal to 80 pixel. The com- +puted PSFs are cropped to get the same size 40 × 40 × 80 as +hexp . The xz−profile can be observed in Fig. 18. +3. +First set of comparison: aberrated PSFs +To retrieve the phase aberration Ψ contained in the aver- +aged experimental data, we take each second axial position of +hexp within the blue dashed rectangle indicated in Fig. 18a. +The retrieved phase is displayed in Fig. 19a. To generate +the theoretical aberrant PSFs, the complex-valued amplitude +PSFs are 2D Fourier transformed to obtain their pupil func- +tions and the retrieved phase is added by multiplying the pupil +by expiΨ. The aberrant pupil is inverse Fourier transformed +back to real space and we get the aberrant PSF by taking +the absolute square of the aberrant complex-valued amplitude +PSFs. The results are displayed in Fig. 19. +The similarity and difference between the theoretical aber- +rant PSFs and hexp are quantified by computing the NCC slice +by slice and the MRE respectively. The 2D NCC results are +plotted in Fig. 20a. + +Calculating Point Spread Functions +21 +(a) +(b) +(c) +(d) +(e) +(f) +(g) +(h) +(i) +(j) +(k) +FIG. 18: Meridional sections at gamma 0.05 of the PSFs of a wide-field microscope with an oil objective of a 1.4 NA, +wavelength emission of 580 nm, window size 40×40×80 and voxel size 80 nm × 80 nm × 50 nm. (a) Experimental PSF with +the blue dashed rectangle indicating the selected region to be the ground truth for the first set PSF comparison and the red +dashed rectangle delimiting the ground truth for the second set of PSF comparison. (b) The Richards and Wolf model RW1. (c) +RW2 using the RW model but computed in different way. (d) SP model. (e) CZT model. (f) Fourier-shell interpolation model +(VS). (g) Sinc-shell method (SR). (h) Gibson and Lanni model by Li, XUE, and BLU 8. (i,j) Scalar and vector models +respectively from9 (k) Vector model using the RW technique computed at the corresponding best accuracy from10 +-3 +-2 +-1 +0 +1 +2 +3 +(a) +(b) +(c) +(d) +(e) +(f) +(g) +(h) +FIG. 19: Comparison of aberrant PSFs with the experimentally averaged PSF data: (a) Retrieved phase aberration from the +experimental data; (b) Experimental averaged PSFdata; (c) aberrant RW1; (d) aberrant RW2; (e) aberrant SP; (f) aberrant CZT; +(g) aberrant VS; (h) aberrant SR. PSFs are displayed at gamma equal to 0.05. + +Calculating Point Spread Functions +22 +-40 +-30 +-20 +-10 +0 +10 +20 +30 +uaxial +-2 +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +Axial position/µm +0.5 +0.6 +0.7 +0.8 +0.9 +NCC between aberrated PSF + and experimental PSF data +RW1: 0.95411 +RW2: 0.9497 +SP: 0.98363 +CZT: 0.98365 +VS: 0.98363 +SR: 0.98356 +NCC3D +(a) +-40 +-30 +-20 +-10 +0 +10 +20 +30 +uaxial +-2 +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +Axial position/µm +0.4 +0.45 +0.5 +0.55 +MRE between aberrated PSF + and experimental PSF data +RW1: 0.56228 +RW2: 0.59195 +SP: 0.55861 +CZT: 0.55866 +VS: 0.55857 +SR: 0.55933 +MRE3D +(b) +FIG. 20: Evaluation of the similarity and difference between the experimental PSF data and the theoretical PSFs which are +aberrant by the retrieved phase from the experimental data. (a) Similarity quantified by the NCC. (b) Difference quantified by +the MRE. +4. +Second set of comparison: non-aberrant PSFs +The errors considered in the first set of comparison include +error due to phase retrieval. To not consider that error and to +include the state-of-art PSFs into the comparison, we compare +non-aberrant theoretical PSFs with the experimental data in a +region of interest where the aberration and noise in the exper- +imental data are minimal. Firstly, to check the region where +aberrations are minimal, we conduct a slice by slice compari- +son between the computed theoretical aberrant PSF described +in Section V B 3 and the corresponding non-aberrant PSFs. +The NCC results are displayed in Fig. 21. Secondly, as we +would want to compare our non-aberrant models with an ex- +perimental data having as little noise as possible, the standard +deviation (Std) of the experimental data is calculated at each +axial position (see Fig. 21). We fit this Std distribution into a +Gaussian function. The Std distribution is not originally cen- +tered to z = 0 so we shift it to zero accordingly. The same shift +in z is applied to the experimental measured PSF data before +the region of interest is extracted. To construct a comparison +metric, we thus chose a common region where the slicewise +NCC (after z-alignment) is high. This corresponds to posi- +tions near the focus where the Std is above half of its maxi- +mum as estimated by a Gaussian fit (Fig. 21). This region of +interest is indicated by the red dashed rectangle in Fig. 18a. +The NCC results between the experimental data and the +non-aberrant theoretical data within the region of interest are +displayed in Fig. 22. +5. +Discussion +By observing Fig. 20, on one side we can conclude that +there is a high similarity between the aberrant theoretical PSFs +with the experimental data. On the other side, the MRE results +are not symmetric about the focus. This asymmetry is ex- +plained by the fact that the shift of focus in the experimental +measured PSF was not perfectly retrieved through the phase +retrieval technique. +This induces the error detected in the +MRE since the computation of the MRE is conducted point +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +Axial position/µm +0.2 +0.4 +0.6 +0.8 +1 +Normalized std of + the experimental PSF data +Experimental data +Gaussian fit +std +-2 +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +Axial position/µm +0.5 +0.6 +0.7 +0.8 +0.9 +NCC between aberrated PSF + and non-aberrated PSF +RW1 +RW2 +SP +CZT +VS +SR +NCC +FWHM +FIG. 21: Selection of the region of interest (ROI) for +comparing the non-aberrant PSFs with experimental PSF +data. The ROI is the volume delimited by the full-width half +maximum (FWHM) of the Gaussian fit of the standard +deviation (Std) of the experimental data along the axial axis. +The Std distribution is normalized by its maximum for +display. +by point and slice by slice. However, we can conclude from +the graph that SP, CZT and VS are the models which have +higher similarity and less error compared to the experimental +data. +By considering the region of interest where the signal-to- +noise ratio is higher and the aberration is small, the quantifi- +cation of the NCC between the non-aberrant PSFs and the ex- +perimental data allows us to short the models in order of accu- +racy. The VectorPSF described in reference9 leads with a 3D +NCC equal to 0.98124. This lead is followed closely by PSF- +Gen_Good and PSFGen_Best in reference Kirshner et al. 10, +RW1, VS, CZT, SP, SR, ScalarPSF Aguet et al. 9, RW2 and +GL in reference Li, XUE, and BLU 8. The scalar PSFs are the +least accurate especially near the focus in this particular com- +parison. The difference in NCC between the models and with +the experimental data is however very close for the vector PSF +models. + +Calculating Point Spread Functions +23 +-8 +-6 +-4 +-2 +0 +2 +4 +6 +8 +uaxial +-0.4 +-0.3 +-0.2 +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +Axial position/µm +0.94 +0.95 +0.96 +0.97 +0.98 +NCC between non-aberrated PSF + and experimental PSF data +RW1 : 0.97888 +RW2 : 0.97673 +SP : 0.97873 +CZT : 0.97874 +VS : 0.97876 +SR : 0.97869 +GL : 0.96156 +ScalarPSF : 0.96725 +VectorPSF : 0.98124 +PSFGen_Good : 0.97933 +PSFGen_Best : 0.97933 +NCC3D +FIG. 22: NCC between experimental PSF and non-aberrant +theoretical PSFs within a particular region of interest near the +focus. +VI. +CONCLUSION +In this work, we provided a general approach for calculat- +ing the 3D PSF of a system satisfying the Abbe sine condi- +tion. We focused on Fourier based techniques and compared +the results of a variety of PSF calculations schemes with a +gold standard from a Richard and Wolf model computed at +higher sampling. We explained the algorithmic details of each +technique and potential advantages and pitfalls. The Fourier +models agree with high precision with the state-of-the-art and +are validated experimentally to have good accuracy around the +focus. We also showed in this work that vector PSFs are more +accurate than scalar models. The study of the PSFs at higher +depth of focus as well as the inclusion of refractive index mis- +matching in the theoretical model is not covered in this work. +This constitutes the next step for PSFs comparisons in addi- +tion to the test of each model in image reconstruction (de- +convolution). The Fourier based 3-D PSF models are already +fast enough given the fact that there is no radial symmetry in- +cluded in them. The models discussed in this manuscript are +under the condition that all the planes in the optical system are +perpendicular to the optical axis. The ability of our models to +accommodate radial asymmetry is advantageous compared to +the state-of-the-art because our models can accommodate any +non-circular aberration and tilted planes caused in the system +such as a tilt of a coverslip. We plan to combine some of the +models such that the computation is still faster without com- +promising the accuracy of the models at any axial depth z. The +models can be adjusted for confocal microscopy, STED (Stim- +ulated emission depletion microscopy) and PSF engineering. +FUNDING +This work was funded by the DAAD through the African +Institute for Mathematical Sciences and Stellenbosch Univer- +sity, and Friedrich Schiller University Jena. This work was +also supported by the German Research Foundation (DFG) +through the Collaborative Research Center PolyTarget 1278, +project number 316213987, subproject C04 and the Council +for Scientific and Industrial Research (CSIR), project number +LREQA03. +ACKNOWLEDGMENTS +The authors wish to acknowledge Herbert Gross and Nor- +man Girma Worku for the CZT function, Peter Verveer for the +first version of the DIPimage Library of the RW code, Colin +Sheppard for valuable discussions and, the nano-imaging re- +search group especially René Lachmann, Ronny Förster and +Jan Becker at the Leibniz Institute of Photonic Technology, +Jena, Germany for their contributions to this work. +1W. Singer, M. Totzeck, and H. Gross, Handbook of Optical Systems, Vol- +ume 2, Physical Image Formation, Vol. 2 (2005). +2M. Gu, Advanced optical imaging theory, Vol. 75 (Springer Science & +Business Media, 2000). +3J. W. 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Preza, “Fluorescence microscopy point spread function +model accounting for aberrations due to refractive index variability within +a specimen,” Journal of biomedical optics 20, 075003 (2015). + diff --git a/YtFRT4oBgHgl3EQfOzfW/content/tmp_files/load_file.txt b/YtFRT4oBgHgl3EQfOzfW/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..348d986aaa52b2d8b0f8a12ed0416da9678e6674 --- /dev/null +++ b/YtFRT4oBgHgl3EQfOzfW/content/tmp_files/load_file.txt @@ -0,0 +1,1430 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf,len=1429 +page_content='Calculating Point Spread Functions Calculating Point Spread Functions: Methods, Pitfalls and Solutions R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Holinirina Dina Miora,1, 2 E.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 16152 Genova,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Italy 4)Molecular Horizons,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' School of Chemistry and Molecular Biosciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' University of Wollongong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Wollongong,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' NSW 2522,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Australia 5)Leibniz Institute of Photonic Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Albert-Einstein Str.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 9, 07745 Jena, Germany (*Electronic mail: heintzmann@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='com) (Dated: 1 February 2023) We discuss advantages and disadvantages of various ways of calculating an optical Point Spread Function (PSF) and present novel Fourier-based techniques for computing vector PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The knowledge of the exact structure of the PSF of a given optical system is of interest in fluorescence microscopy to be able to perform high-quality image reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Even, if we know how an aberrant optical path deviates from the original design, the corresponding PSF is often hard to calculate, as the phase and amplitude modifications need to be modelled in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Accurate PSF models need to account for the vector nature of the electric fields in particular for high numerical apertures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Compared to the computation of a commonly used scalar PSF model, the vectorial model is computationally more expensive, yet more accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' State-of-the-art scalar and vector PSF models exist, but they all have their pros and cons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Many real-space-based models fall into the sampling pitfall near the centre of the image, yielding integrated plane intensities which are not constant near the nominal focus position, violating energy conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This and other problems which typically arise when calculating PSFs are discussed and their shortfalls are quantitatively compared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' A highly oversampled Richards and Wolf model is chosen as the gold standard for our quantitative comparison due its ability to represent the ideal field accurately, albeit being practically very slow in the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Fourier-based methods are shown to be computationally very efficient and radial symmetry assumption are not needed making it easy to include non-centro-symmetric aberrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' For this reason newly presented methods such as the SincR and the Fourier-Shell method are essentially based on multidimensional Fourier-transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Keywords: point spread function, Fourier optics, fluorescence microscopy I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' INTRODUCTION Aplanatic imaging systems strive to image a plane in the object to the detector plane (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' a CCD camera)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' To achieve this, any optical system needs to fulfill the Abbe sine condition2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This condition relates any angle of a beam emit- ted by the sample (θem) to its corresponding angle reaching the detector (θdet), and is given by sin(θem)/sin(θdet) = M, with M being the magnification of the optical system in terms of geometric optics1,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' As shown in the thin lens approxi- mation in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 1, a simple 4f-imaging microscope imaging system with a magnification of M = 2 would not fulfill the above-mentioned Abbe sine condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The simplicity of such drawings can be maintained by intro- ducing the concept of the Gaussian reference sphere (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 2) describing the equivalent refractive loci for aplanatic imag- ing systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' A Gaussian reference sphere is a sphere centred where the object plane (the plane to image) or at the focus po- sition (in image plane) intersect the optical axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Its radius is equal to the focal length f1 or f2 in each of the corresponding planes1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The rays propagate to the Gaussian reference sphere and get teleported without acquiring any extra phase to a refer- ence plane, which is a plane surface parallel to the pupil plane in the back focal plane (BFP), indicated by the positions of the planes of lens 1 and lens 2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The same principle is 𝜃em 𝜃det 𝑓1 𝑓1 BFP 𝑓2 lens 1 lens 2 ℎ 𝑓2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 1: Geometrical representation of a 4f-imaging system using the thin lens approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=" Note that here tan(θem)/tan(θdet) = f2/ f1, which violates Abbe's sine condition." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' BFP stands for back focal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Here lens 1 corresponds to the microscope objective lens while lens 2 represents the tube lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' applied in reverse to the tube lens (lens 2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 1), where the effect can often be neglected due to the usually large magnifi- cation of the objective and thus small angles (see right side of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' If this task of imaging a plane to a plane is performed well by a careful design, the structure of an image of point object, called the point spread function (PSF) does, to a very good ap- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='13515v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content="optics] 31 Jan 2023 Calculating Point Spread Functions 2 Gaussian reference sphere Gaussian reference sphere ''teleportation'' 𝜃em 𝜃det ℎ FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=" 2: Introduction refraction at the Gaussian reference sphere warrants that sin(θem)/sin(θdet) = f2/f1 is a constant and the system therefore satisfies Abbe's sine condition." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' "Teleportation" means that the beams are continued at the connected surface without acquiring any phase for the space in between the plane reference plane and Gaussian reference sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' proximation, not change with location over the field of view of the optical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The image of an incoherently emitting sample can well be described by a convolution of the object emitter density with the PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Such imaging systems are typi- cally called linear shift invariant imaging systems3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Here we discuss calculating PSFs using various computa- tional tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' An accurate PSF model is an important require- ment for a successful image reconstruction4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Various stud- ies have been conducted to model the PSF of a given optical system5–10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The computation of each of these models has pros and cons, some of which we will discuss below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' In Section II, we firstly introduce the readers to the state-of-the-art scalar and vector PSF models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Secondly, we present ingredients for computing our novel methods using Fourier optics III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The fast Fourier-transformation is a very handy tool to speed up PSF calculations, but its pitfalls need to be carefully avoided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' We therefore present the pitfalls that one may encounter in the calculation and ways around them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' In Section IV, we de- scribe in detail novel Fourier-based techniques for computing vector PSFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Finally, in Section V the various methods are compared quantitatively in terms of their accuracy and com- putation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' A further aim of this manuscript is to release a toolbox to the scientific community, which others can benefit from for calculating PSFs using uniform or modified (aber- rated) apertures11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' EXISTING PSFS MODELS A scalar PSF model is computed from one integral per point, (x,y,z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' It is considerably cheap computationally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' How- ever, it does account for the vector nature of the electric field describing the light which will lead to a wrong estimation of the field at higher aperture angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' A vector PSF, on the other hand, requires a calculation of all the three spatial components of the electric field6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' It is more accurate as it carries more in- formation about the field such its polarization state12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The Richards and Wolf (RW) vector PSF model has been shown to represent a vector field to a very high accuracy7,10,13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Yet, the implementation of the RW model still has its limitations in terms of sampling and computation time as we will discuss below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Scalar PSF model A well-known scalar PSF model was developed by Gibson and Lanni (GL)6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The model is valid within the Kirchhoff boundary conditions for scalar diffraction theory6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The ex- pression of the PSF as per the GL model at a position (x,y,z) is given by: PSF(x,y,z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='p) = ����A � 1 0 exp(ikΦ(ρ,z,p))J0(krNAρ)ρdρ ���� 2 , (1) where r = � x2 +y2, A is a constant complex amplitude, ρ is the normalized radius in the back focal plane, Φ is aberration function and, p is a vector summarizing optical characteristic such as refractive indices and thicknesses of of aberrant sur- faces in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' In our calculation, we consider a case where the actual condition is assumed to meet the design con- dition of the imaging system14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The Richards and Wolf Model (RW) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Description of the model A scalar electric field model is limited to only a single com- plex amplitude over the image space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' It does not give any in- formation about the polarization state of the image field, the direction of the energy flow and it is also not applicable to imaging at high numerical apertures12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Richards and Wolf described the focusing of electromagnetic waves for low and high numerical aperture using an angular spectrum of plane waves in an integral representation7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The model is a vector formulation of the scalar Debye model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The expressions of the electric field from a point source S0 at time t at a position (x,y,z) in the image space is as follows: ⃗E(x,y,z,t) = R{ ⃗ AE(x,y,z)exp(iϖt)} (2) where ⃗ AE is the amplitude vector, ϖ the angular frequency of the point source and R{} denotes taking the real part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The amplitude vector satisfies the independent wave equation and Calculating Point Spread Functions 3 is given by: ⃗ AE(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='z) = � � � � � −ia(I0 +I2 cos2φxy) −iaI2 sin2φxy −2aI1 cos2φxy (3) with the constant a and � � � � � I0 = � α 0 cos1/2 θ sinθ(1+cosθ)J0(krsinθ)eikzcosθdθ I1 = � α 0 cos1/2 θ sin2 θJ1(krsinθ)eikzcosθdθ I2 = � α 0 cos1/2 θ sinθ(1−cosθ)J2(krsinθ)eikzcosθdθ (4) α is the maximum angular aperture,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Jj is the Bessel function of order j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' the integration variable θ correspond to the angle a point on the Gaussian reference sphere has to the optical axis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' r = � x2 +y2 as well as φxy relate to the point where the field is evaluated,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' with φxy being the azimuth at (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='z)7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The term cos1/2 θ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 4 is the aplanatic factor for energy conservation, illuminating the objective with a plane wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' These equations rely on the Kirchhoff boundary conditions and only consider homogeneous waves12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' An inhomogeneous wave corresponds to a field which decays for a large propaga- tion distance whereas the Kirchhoff boundary conditions im- ply that only incident field within the opening aperture of the exit pupil contribute to the field at (x,y,z) in the image space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Computation There exist different ways for computing the three integrals in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' One way could consist of evaluating the integration numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' To improve the accuracy, finer steps of integra- tion are needed which can lead to an expensive computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' One technique that we denote by RW1 exploits a cylindrical coordinate system to perform the integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This has com- putational advantages, since only a two-dimensional rz-map (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' a centered radial axis versus axial z) has to be calculated for I0,I1 and I2, since their definition is independent of the az- imuth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' However, this procedure may have the disadvantage that we cannot easily include arbitrary aperture modifications which do not possess circular symmetry into the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' If there is no need for calculating the electric field but only the intensity, this technique has a shorter path to compute the in- tensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' We denote this shorter path by RW0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The second method that has been developed in this work, denoted by RW2, uses a Fourier transform along the optical z axis (see Section IV B 3) to compute the (x,y,z) components I0,I1 and I2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' We choose as a variable of integration ξ = cosθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' RW2 is on average 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='45 times slower than RW1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' However, it does not require in its computation the radial sym- metry property of a PSF while RW0 and RW1 do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' State-of-the-art PSFs models Four state-of-the-art commonly used PSF models were cho- sen to compare our methods to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The first model is a scalar PSF based on the work of Gibson and Lanni 6, and further devel- oped by Li, XUE, and BLU 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This technique calculates the PSF fast by using a combination of Bessel series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The sec- ond and third models are a scalar and vector PSF as described in Aguet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The PSFs are computed using a numerical integration based on Simpson’s rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The last state-of-the-art PSF that we compare with is a vector PSF calculated with the Richards and Wolf 3D optical model from the PSFGenerator toolbox in10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Each of those models have their advantages, as well as their limitations as we discuss in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' TOOLS FOR COMPUTING A VECTOR PSF USING FOURIER OPTICS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Scalar high numerical aperture (NA) model To start with the PSF calculation using a Fourier optic formulation, let us first consider the calculation of the in- tensity distribution near the nominal focus when focusing a monochromatic coherent plane wave by a high-NA mi- croscopy objective (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The beam is entering the ob- jective system from the left side and is spatially limited by the entrance pupil of the optical system, corresponding to the im- age of the aperture stop as created by the optics to the right of the aperture (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' An “aperture stop”, is in practice ei- ther intentionally introduced to warrant the linear shift invari- ant performance of the system and to avoid aberrations from unwanted beams, or effectively provided by the inner geome- try of the objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The limitation of the beams is therefore approximated to be at the limit of that aperture stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' At this pupil plane, every point PW on the wavefront is considered as a source of a Huygens wavelet, denoted by W 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' PW W aperture stop entrance pupil exit pupil optical S plane wave z 𝜃em 𝜃det axis 𝑘W FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 3: Visualization of the pupils and representation of the Huygens Wavelet becoming a plane wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Firstly, we limit ourselves to a scalar electric field, where the field is a directionless value and only a function of its posi- tion in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The vector nature of the electric field will be in- Calculating Point Spread Functions 4 troduced further down below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The aperture plane can be seen as a superposition of spherical wavelets, each of which gives rise to a plane wave after the objective directed towards the nominal focus point S (see the wavelet labelled “W” in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' According to the Huygens-Fresnel principle, the spheri- cally converging wave is obtained by superimposing all these wavelets15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The superimposed wavelets have to acquire ex- actly the same optical path length and constructively interfere at S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' In other words, the phase at the nominal focus is iden- tical for all such wavelets and can thus be set to zero in our simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' For convenience, we choose S as the center of our real-space coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The wavelet W giving rise to a plane wave in focus (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 3) can now conveniently be de- scribed in Fourier space as a single point PW (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 4b), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' a single 3-dimensional vector (⃗kW) in Fourier space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Such vectors have all to necessarily reside on a sphere of a radius k0 = 2π/λem,λem = λ/n, λem and λ being the emission and the corresponding vacuum wavelengths respectively, and n the refractive index of the embedding medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' A pupil position in real space corresponds to the lateral component ⃗kx of the wave-vector ⃗k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This linear correspon- dence is forced by the Abbe sine condition between the pupil plane coordinate and the k-vector position of the wave near the focal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The pupil plane aperture stop thus gives rise to a three- dimensional cap residing on the k-sphere in Fourier space (see solid cap in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 4b in the Fourier space representation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' As the aperture is limited by the NA of the objective, the 3D fre- quency spectrum in Fourier space is represented in a segment of the k-sphere sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This segment is called “generalized aperture” or “McCutchen pupil”16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' To now calculate the (complex-valued) amplitude distribu- tion in real space near the focus S, we need to generate a three-dimensional McCutchen pupil of uniform amplitude and perform an inverse three-dimensional Fourier transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The Fourier-based PSF models that are presented in this work are based on this understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The four different methods differ in how the amplitude on the McCutchen pupil is cac- ulcated and how the field is propagated in the homogeneous medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' To the best of our knowledge, these approaches have not been previously described elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Each method has its own advantages, pitfalls and drawbacks as discussed in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' It is instructive to first limit ourselves to the calculation of the xy-plane at the nominal focus position z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' We can in- terpret this plane as being a slice of the three-dimensional fo- cus volume, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' a multiplication with δ(z = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' In Fourier space such a z-slice in real space corresponds to an integral of the amplitude over kz according to the Fourier-slice theorem3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Since the McCutchen pupil is infinitely thin, and only waves along the positive z direction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' positive kz, contribute to the focus, there is a bijective mapping from the 3D McCutchen pupil to its 2D projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This establishes a direct corre- spondence between the pupil plane amplitude and the two- dimensional Fourier transformation of the xy-amplitude at the focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Slicing at a different z-position, z1, can be seen as a trans- lation by z1 along z which amounts to a phase-change by exp(iz1kz) which can be written as exp � iz1 � k2 0 −k2r � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' In this projection over kz we may need to account for the effect of the local orientation of the McCutchen pupil via a projection fac- tor and for possible other factors as given below in a more detailed analysis regarding the Gaussian reference sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Vector electric field The above discussion assumed a scalar field (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' as com- mon in acoustics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' However, the electric field is described by a spatially varying amplitude field vector with three com- plex components ⃗E = (Ex,Ey,Ez) of the form Ex,y,z(r) = Ax,y,z(r)exp(iϖt) where the amplitude, Ax,y,z(r), is a func- tion of the position r where the field is evaluated and exp(iϖt) is the time dependent phasor, ϖ being the angular frequency and t the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Since we stay in the realm of linear optics, where the interaction between the excitation light and matter is achieved linearly, we continue our considerations with the assumption that the PSF (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' the excitation of a weakly ex- cited fluorophore) is given by a linear dependence to the local irradiance: I = |⃗E|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' All of the Fourier-space considerations as stated for the scalar field above can now be applied individ- ually to each of the components of the amplitude field vector as long as its strength and phase on the k-sphere are accounted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Aplanatic correction By definition, an aplanatic system is a system which is free from aberration and any small displacement of the system does not induce aberration17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The energy of light propagating through layers of such a focusing system must be conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Integrating over the radial position r in the BFP must yield to the same quantity of energy as integrating over the corre- sponding angle θ in the equivalent refractive locus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=" In a system where the equivalent refractive locus corre- sponds to a Gaussian reference sphere, the system satisfies the Abbe's sine condition." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The energy conservation yields to the notion of an apodization or aplanatic factor (AF), which is equal to � cos(θ)18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The concept can also be understood by using a simple geometric figure (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Firstly, let us assume an isotropic emitter placed at the centre of the Gaussian refer- ence sphere, S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Since the emitter is emitting uniformly in all directions, the strength of the amplitudes on the Gaussian reference sphere is uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' In Fourier space, the spectrum on the McCutchen pupil is as well uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Let us consider a small parallel ray which is redirected from the Gaussian ref- erence sphere at a given angle θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' If we attribute a given ir- radiance I0 as a power P0 per unit area A0 to such a beam- let from S0, the same power will have to be distributed to a smaller area A1 = A0 cos(θ) after the teleportation of the beam from the Gaussian reference sphere to the plane parallel to the pupil plane in the BFP (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This means that the irradiance measured perpendicular to the local direction of Calculating Point Spread Functions 5 𝑘𝑥 𝑘𝑧 𝑘W plane wave 𝜃 𝑘0 2 = 𝑘𝑥2 + 𝑘𝑦2 + 𝑘𝑧2, 𝑘0 = 2π 𝜆em 𝑘𝑥 = 𝑘0 sin𝜃 cos 𝜙 𝑘𝑦 = 𝑘0 sin 𝜃 sin 𝜙 𝑘𝑧 = 𝑘0 cos𝜃 𝑘0 𝑘z 𝑘y 𝑘𝑥 𝜃 𝜙 (a) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 4: (a) Coordinate system in Fourier space: θ and φ are the elevation and azimuthal angles respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (b) Fourier space representation of a plane wave from a wavelet W at a point source PW in the pupil plane called McCutchen pupil (bold black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The ky−axis in (b) is oriented towards the front plane of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' A1 A0 "teleportation" Objective lens BFP r S1 S0 𝜃 𝜃 𝜃 z Ξ1 Ξ0 (a) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='8 1 sin 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='5 Radial mean irradiance \\arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' unit Experimental = -1 = -1 with TC included {(ns, ts) = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='4, 0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (ncs, tcs) = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='518, 170);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (noil, toil) = (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='518, 150)} = 0 = 1 cos( ) sin max (b) (c) (d) (e) (f) (g) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 5: Aplanatic correction for energy conservation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (a) Schematics illustrating the “aplanatism effect”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (b) Radial mean of the normalized BFP intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The normalization is done in such a way that the power remains constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' We have here the image of the BFP (c) recorded experimentally using a 70 nm microtom-sectioned thin layer of fluorescent Chroma slide attached to the coverslip and embedded in immersion oil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Figure (c) is cropped from the original experimental BFP image for display.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The corresponding BFP computed theoretically use different scaling factors � cosη(θ) of the electric field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (d) η = −1, (e) η = −1 and included transmission coefficients (TC) due to the layer system {sample of refractive index ns placed exactly at the surface of the coverslip, coverslip of refractive index ncs and thickness tcs, oil immersion of refractive index noil and thickness toil }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The thicknesses of each layers are expressed in µm and are assumed to be the same as in the design condition of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (f) η = 0 and (g) η = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' propagation changes to I0/cos(θ), and each of the field vec- tors for this beamlet thus needs to change by � cos−1(θ) to be consistent with this intensity change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This factor corre- sponds to 1/AF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The inverse of this effect, which is spreading irradiance to a larger area, is somewhat akin to what happens to radiative power measured on the surface of the earth dur- ing winter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' It is sometimes called “natural vignetting”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' In this manner, the energy spread over the 2D pupil equals to the energy of a point source over the active Gaussian sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This corresponds to the imaging of an emission PSF where we have our emitter placed at the focal position of the objec- tive lens, which is at S0 in this illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' If a large enough magnification is assumed, the vectorial and aplanatic facgtor effects caused by the tube lens can be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' To describe the Fourier-transformation of the focussed field in this scalar model, we expect a correspondence to the amplitude on the 1 mmCalculating Point Spread Functions 6 BFP and therefore a √ cos−1 θ dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' To now understand the factor that needs to be applied to conserve the energy when focussing a uniformly illuminated 2D pupil (BFP) with a high-NA objective (excitation PSF), let us consider the same Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The point S1 denotes a point in the BFP from which a ray emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' S2 being the focal point of the objective corresponds to S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' We can consider the reciprocity theorem: a lossless (non-magnetic) monochromatic optical system in which a field (Ex0,Ey0,Ez0) (here an isotropic emitter) at position S0 gives rise to a field (Ex1,Ey1,Ez1) at position S1 in the image plane, warrants that placing the isotropic emitter (Ex0,Ey0,Ez0) as a source at the former image plane, generates the field (Ex1,Ey1,Ez1) at the focus position S0 of the objective18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The latter situation, a uniform emitter at the focus of the tube lens, leads, due to the low NA of the tubelense to the aforementioned uniform illumination of the pupil plane (BFP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Therefore the excita- tion PSF should be equal to the emission PSF as long as we can neglect the NA of the tubelens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This is confirmed also by considering the √ cosθ dependence on the Gaussian ref- erence sphere and thus on the McCutchen pupil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' To arrive at the 2D Fourier-transform of the in-focus excitation field, we need to project the McCutchen pupil and thus apply the pro- jection factor cos−1θ which leads to an overall amplitude of cos−1θ √ cosθ = √ cos−1 θ confirming the above reciprocity argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Emission PSF To test the theory described above, we measure experimen- tally the intensity of the BFP of emitting Flourophores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' A large collection of randomly orientated emitters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' fluo- rescent) quickly rotating emitters will emit with the same in- tensity along all directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The detection objective is thus expected to concentrate the light it receives at high angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This effect is visible when imaging the back focal plane of an objective imaging a fluorescent plane sample of uniform emitters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 5c, we show the image of the BFP of a high-NA ob- jective imaging a 70 nm thick layer of a Chroma slide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' For the imaging, we have to take great care with our measurements to avoid supercritical angle fluorescence effect as we did not account for this in our calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Fluorophores which are directly at a dielectric surface emit fluorescence into the cov- erslip and thus the objective with a totally different angular characteristic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This affects the pupil plane distribution and completely changes the expected measured PSF if not cor- rected appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' To avoid this effect, we embedded the thin layer of Chroma slide in oil with refractive index of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='518 and place it onto a coverslip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' A microscope slide is used to support the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The image of the BFP is recorded by removing the eye- piece through the observation tube of the microscope [Zeiss Axio Observer, Objective Plan-Apochromat 63 × /1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='4 Oil DIC M27] and replacing the eyepiece with a system composed of a converging lens of 50 mm focal length coupled with a Basler camera [acA4024-29um] (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The Basler cam- era is placed at 50 mm after the lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' An x-cite lamp is used as illumination light source for this particular measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Due to missing information on the optics inside the microscope ob- servation unit, we were not able to determine the theoretical magnification of our pupil plane re-imaging but estimate it to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='446 as given by the ratio of the measured over the the- oretical pupil radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The pixel size of the recorded BFP is estimated to be 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='148 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The pupil diameter is calculated to be 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='816 mm and is represented by 1160 pixels on the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 5b, the radial mean profile of the BFPs is normalized such that the intensity at the angle of incidence equal to 0 corresponds to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Theoretical models are calculated using the slice propaga- tion method described in Section IV B at higher sampling un- der different η parameter in the amplitude factor � cosη(θ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Those models are computed to match the experimental pupil size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 5d, we have the BFP of an emission PSF (η = −1 thus predicting an intensity scaled by cos−1(θ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' We observe that the qualitative expectation (an increase towards the edge of the pupil) is experimentally confirmed but the theory pre- dicts a stronger rise of intensity with θ than was experimen- tally found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This difference could be caused by the photoselection of preferably in-plane transition-dipoles which leads to a less pronounced emission along Ez components and thus less pro- nounced higher angle contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The experiments should be repeated with a thin volume of freely rotating dipoles with a rotational correlation time far below the fluorescence lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' A further effect could be due to residual back-reflections at higher angles effectively dimming the light at high angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' To have a realistic model to compare with the experimental result, we use the theoretical BFP with η = −1 and include the transmission coefficient (TC) due to refractive mismatch of different layers (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 5e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The TC is calculated using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (17) in13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' We assumed the 70 nm thick polymer sam- ple matrix having a refractive index of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='4 placed at the sur- face of the coverslip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' For the coverslip we assumed a 170 µm thick glass with a refractive index matching the immersion oil 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='518.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' We observe in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 5b the radial profiles of each of the BFP models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' It may also be worth considering that the spec- trum of fluorescence emission from the thin slab of fluorescent plastic is a band rather than a single wavelength19, but we did not consider this any further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' These preliminary experiments qualitatively confirmed the model of the aplanatic factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' As a result, high angular amplitudes in detection get enhanced by � cos−1(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Intensity As a matter of perspective, if our interest resides in calcu- lating a three-dimensional intensity distribution defining the rate at which randomly oriented molecules will be excited, the scaling factor ( � cos−1(θ) in the Fourier transform of the amplitude in the focal plane) described in the previous section should yield the correct values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The randomly oriented fluo- rophores have on average the same probability of excitation independent of the angle under which the light comes from.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Calculating Point Spread Functions 7 x-cite lamp NA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='4 n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='518 63x tube lens thin layer of chroma slide image plane BFP re-imaged BFP Basler camera Lens 50 mm 50 mm FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 6: Back focal plane re-imaging system using a thin layer of chroma slide as a sample emitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Therefore, the excitation probability is proportional to the sum of the absolute squares of all three field components, Ex,Ey, and Ez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This quantity is proportional to the “radiant intensity” measured in steradians per area which is different from the ra- diant flux (called “irradiance”), which is often called “inten- sity”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' However, the physically relevant quantity can also be irradiance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' when projecting onto a camera) which quan- tifies the flux through a unit area oriented perpendicular to the optical axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' In this later case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='e measuring the focal inten- sity with a pixelated detector, we would need to account for another Lambertian factor (cos2(θ) for intensity or cos(θ) for amplitude) in addition to the aforementioned factor (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 5a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Sampling condition The imaging of a PSF, which is ultimately detected on a pix- elated imaging device, can commonly be done by a CCD or CMOS camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Those devices integrate, in each of their recti- linearly spaced pixels, over the signal weighted by a pixel sen- sitivity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' A PSF calculation typically samples the con- tinuous mathematical function at infinitely thin (delta-shaped) points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Luckily the local integration of the PSF in every pixel by the camera can be rewritten as first convolving the PSF with the pixel sensitivity function and then sampling it at reg- ularly spaced points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Due to the convolution theorem, the effect of detector integration can be represented by a simple multiplication of the Fourier transform of the PSF, the optical transfer function (OTF) with the Fourier transform of the pixel sensitivity function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' If we assume square pixels with uniform sensitivity, the OTF gets modified by a multiplication with a sinc � πkx/ksamp � sinc � πky/ksamp � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This means that at twice the current sampling frequency ksamp = 1/dsamp the overall transfer would cross zero, dsamp being the pixel pitch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' To sam- ple the PSF free of aliasing, the highest frequency has to be at most at the half of the inverse of the sampling size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This fre- quency is called the Nyquist frequency, kNyq = ksamp/2, and the requirement constitutes what is called the Nyquist Shan- non theorem20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' If the PSF is not sampled under this require- ment, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' dsamp > 1/(2kNyq), there is a presence of aliasing in the signal (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=') leading not only to a potential loss of signal but also to wrong results at frequencies within the frequency band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' It is worth emphasizing that a PSF with perfect circular symmetry (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' using circular or random polarization) gets modified by the pixel sensitivity form factor and loses its sym- metry, with decreased sensitivity, especially along the diago- nal directions connecting the corners of the pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' In a confocal microscope, the detection is performed by an integrating detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' However, the data is typically acquired by integrating in each pixel over the pixel dwell time, with the scan not being stopped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' As an effect, the confocal excitation and detection PSFs both are modified by a single-directional sinc � πkx/ksamp � term if the scanning is performed along x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Resolution limit and Nyquist Shannon theorem For a wide-field microscope imaging fluorescence, the min- imum resolvable spatial structure (periodicity) observed in lateral axis is given by the Abbe diffraction limit dxy = λem/(2NA), with NA = nsin(θ) the numerical aperture, θ the angular aperture of the objective, n the refractive index of the medium and λem the vacuum emission wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Therefore, the maximal in-plane spatial frequency is given by kxy,max = 2NA/λem20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Similarly, the axial limit in real space for a wide- field microscope is given by dz = λem/(n(1−cos(θ))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' In our calculations the electric field has to be sampled according to the desired PSF, which is sampled twice as fine as the Nyquist- Shannon theorem would require, since resampling when cal- culating the intensity was not performed in our calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Thus the highest frequency of the intensity result has to be sampled with at least two positions per shortes period that can be transmitted by the system20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This requires the pupil to fit into half the digital Fourier-space representation such 2 mmCalculating Point Spread Functions 8 that its autocorrelation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' the incoherent OTF) fits in the digital Fourier space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The maximal pupil radius in Fourier space along x or y should be lower than half the maximally represented frequency along kx or ky in our Fourier-space rep- resentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The Fourier sampling pitfall Other potential sources of error that have to be considered in addition to the aforementioned sampling are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' A digitization of the usually round pupil in Fourier space as a hard aperture onto a rectilinear grid may induce severe arte- facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' We consider a field distribution of a high-NA PSF with numerical aperture NA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='4, refractive index n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='518 and emission wavelength λem = 580 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The pupil radius, which also corresponds to the Nyquist frequency, is calculated us- ing the theory stated in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' We denote Rmax as this pupil radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' We generate a hard aperture with radius equal to Rmax/8 and calculate the corresponding field distri- bution in real space by generating the Fourier transform of the hard aperture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The window size for this first experiment is 128×128 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' For symmetry reasons, a perfect circularly symmetric PSF should expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 7a, a significant deviation from circu- lar symmetry is clearly visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' By repeating the same calcula- tion for 1024×1024 pixels, the discrepancy is significantly re- duced even though it is still not totally spherically symmetric (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 7c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' However, calculating on such large grids causes a significantly computational overhead (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='178 s vs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='037 s i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' more than 4 times slower) which can be unnecessary as the user may not need quite so many pixels of the PSF far away from its center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Interpolation in Fourier-space to obtain a better represen- tation of the pupil may be one route to reestablish spherical symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' However, this is a tricky business21 and we there- fore choose a slightly different route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' We calculate the two- dimensional (2D) Fourier-transform of the uniform pupil disk, for which the analytical solution in real space is well known: jinc(R) = J1(R)/R,J1 being the Bessel function of the first kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' We therefore obtain an “ideally” representation of a disk in Fourier-space by Fourier-transforming a two-dimensional jinc function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This “interpolated” disk can then be appropri- ately modified with k-space dependent phase and magnitude alterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The computation time of the 1024×1024 pixels ideal repre- sentation of disk in Fourier-space using the jinc trick is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='078 s on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The calculation is done with MATLAB R2018a on Windows 10 with Intel(R) Core(TM) i5-6200U CPU @ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='30 GHz 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='40 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' As the jinc-function possesses first order discontinuities in real space at the border, whos Fourier-transformation causes unwanted high-frequencies22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' To avoid this, the jinc-function was modified at the outer rim by appropriately smoothing the 15 % of its edges towards zero (“DampEdge” function in the PSFToolbox11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' As seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 7b, the real-space representation of the field distribution is perfectly symmetric and spherical by design even for images with relatively few pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' We refer to this method of generating an interpolated disc in Fourier space as the FT(jinc)-pupil trick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' NOVEL FOURIER-BASED METHODS FOR PSF CALCULATION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The electric field on the k-sphere To calculate the electric field amplitude distribution near the focus, we first need to understand the electric field prop- erties of each Huygens wavelet as previously described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' To this aim, we associate each plane wave arriving at the focus with the refractive effect that the “bending” at the Gaussian reference has at the point where this “ray” would hit it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' We assume a perfect anti–reflection coated objective lens and all the energy is transmitted for such a ray and exploit the fact that the electrical field of the plane wave needs to be a transversal wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' We thus have to project the electric field at the pupil (Ex,Ey) to the electrical field (Ex,Ey,Ez) of the plane waves (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' the McCutchen pupil).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Let us consider Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The incident wave ⃗Ei is incident from infinity at the left side in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 8a and focuses at a the focal point of the objective lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The system is assumed to fulfil the Abbe sine condition, requiring the beams to change direction at the Gaussian reference sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' At the entrance pupil, the incident wave Ei can be described by two compo- nents (Ex,Ey), but it is very useful to here consider a locally varying coordinate system along azimuthal (⃗es) and radial (⃗ep) directions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Let ⃗Et denote the field amplitude transmitted along the wave vector⃗et towards a point (x,y,z) near the focus where the field is evaluated (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 8a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The unit vector correspond- ing to the radial component⃗ep is refracted by θ and becomes ⃗er while the azimuthal component oriented along ⃗es remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' ⃗es = � � −sinφ cosφ 0 � �, ⃗ep = � � cosφ sinφ 0 � � ⇒⃗er = � � cosφ cosθ sinφ cosθ sinθ � � (5) The new coordinate system is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The field amplitude distribution at a point (x,y,z) is there- fore given by: ⃗Et(x,y,z) = (⃗Ei ·⃗ep)⃗er +(⃗Ei ·⃗es)⃗es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (6) For a given polarization state of the incident wave field ⃗P, the pupil plane amplitude distribution can be calculated using the directional change of the electric field described in the pre- vious paragraphs where the incident electric field is given by ⃗Ei = E0⃗P, E0 is a constant factor which includes the conserva- tion of energy such as aplanatic factor or apodization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' There- fore, the amplitude field on the McCutchen pupil is given by ⃗Et = E0[(⃗P·⃗eP)⃗er +(⃗P·⃗es)⃗es].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (7) Calculating Point Spread Functions 9 5 0 5 x/µm 5 0 5 y/µm (a) 5 0 5 x/µm 5 0 5 y/µm (b) 5 0 5 x/µm 5 0 5 y/µm (c) 5 0 5 x/µm 5 0 5 y/µm 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='5 1 (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 7: Field distribution calculated from the Fourier transform of (a) a hard aperture of size 128×128 pixels, (b) a jinc aperture aperture of size 128×128 pixels, (c) a hard aperture of size 1024×1024 pixels cropped to 128×128 pixels size for display and, (d) a jinc aperture aperture of size 1024×1024 pixels cropped to 128×128 pixels size for display.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' A DampEdge of 15 % is applied to the generated field (full size) using the jinc-trick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Figures are displayed at tan−1(γE) and centered at the zero of the display, E being the field distribution and γ = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=" 𝜙 x y z ep es y' x' oriented toward the front of the plane." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=" 𝜃 x' z er et ep es y' (a) (b) FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 8: Schematic diagram of the coordinate systems for the directional change of a scalar wave field Ei through an objective lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (a) Adaptive coordinates on the McCutchen pupil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (b) 2D pupil plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' At this point, it seems appropriate to comment on the ver- satility of this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' If we want to include any additional linear shift-invariant effect into our calculation, such as the ef- fect of an additional slab of glass, which was not considered in the design of the objective or a wrong medium of sample em- bedding, it is fairly simple to work out the magnitude, phase and even polarization effect that such a modification would have on each field vector component on the generalized Mc- Cutchen pupil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Likewise, we can also easily consider the effect any in- tentional change of the complex amplitude transmittance at the pupil plane will have, for example to calculate Bessel beam23, spiral phase modification24, the doughnut-shaped STED (Stimulated Emission Depletion Microscopy) beam25, apodizations26 or other PSF modifications27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The slice propagation method (SP) With the various considerations described above, the com- plete problem of calculating the in-focus field distribution for excitation and emission PSFs can also be achieved using a method called the “slice propagation”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This method derives from the vectorial Debye model and the propagation of the field is based on the angular spectrum method28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The steps to follow are: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Choose a 2D pupil plane amplitude and polarization distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Propagate it to the Generalized McCutchen pupil as a projection of a two-dimensional pupil onto three field amplitude components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Apply to each component the same aplanatic factor ac- cording to the desired calculation in Section III C (exci- tation or emission PSF, angle-independent flux or flux through a reference surface);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Perform a separate two-dimensional inverse Fourier- transformation for each of the three field components to go from the McCutchen pupil to the focal field in real space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Calculate the PSF as a probability of excitation or de- tection for a collection of a randomly oriented fluo- rophores, PSF(x,y) = ExE∗ x +EyE∗ y +EzE∗ z , such that each of the field components depends on the spatial (x,y) coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Free space propagation of the field In order to calculate a defocussed PSF or 3D PSF as a stack of defocussed PSFs, we can modify the slice propaga- tion method by including a defocus phase in the generalized McCutchen pupil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Given the field in the McCutchen pupil as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 4b, the in-focus lateral field distribu- tion components can be calculated as a projection along kz of the corresponding McCutchen pupil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Each projection cor- responds to the 2D in-focus slice of the 3D field distribution represented by the 3D McCutchen pupil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' To move to a differ- ent z position, the Fourier-shift theorem needs to be applied, which states that a translation by ∆z in real space corresponds OOOCalculating Point Spread Functions 10 to a multiplication with exp(ikz∆z) in Fourier space3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Shifting an xy-amplitude by ∆z thus means: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Calculate the 2D fast Fourier transformation (FFT) from real space to Fourier space of the 2D pupil plane amplitude distribution or use the already calculated pro- jections of the McCutchen field components from the result of step 3 above;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Project onto the k-sphere to obtain the field on the Mc- Cutchen pupil;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Apply the phase modification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Project back (sum along kz) onto the kxky-plane;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Inverse Fourier transform to obtain the amplitude at ∆z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' These steps follow the angular spectrum method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Luckily, steps 2 to 4 do not actually need to be calculated individu- ally, since each pixel exactly ends up where it was but only having accumulated a phase modification which only depends on |kz| = � k2 0 −k2x −k2y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This well-known phase modifica- tion (the homogeneous medium angular spectrum “propaga- tor”) can thus simply be applied to the projected McCutchen pupil(s) yielding the wanted defocus PSF in step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' When propagating a pupil that was generated by the jinc-FT pupil trick (see Section III D 2), it is recommended for accuracy rea- sons not to apply the pupil a second time during propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Fourier wrap-around pitfall Even though the propagator with the slice propaga- tion method allows a convenient calculation of the three- dimensional PSF, a severe problem arises outside a ∆z-range in the xy-cut through a calculated PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The ∆z-range is defined as the axial region where the disk of defocus stays well within the available lateral space provided by the real-space grid (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 9a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Outside the ∆z-range, waves leaving on one side of the xy−sampling grid and entering into the simulation from the opposite side due to the periodic boundary conditions of the Fourier-transform cause severe standing–wave effects (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 9a and 9d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Three possible strategies can help to avoid this: A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' By appropriately zero-padding the in-focus plane, the ∆z-range from whereon the standing wave patterns oc- cur can be extended (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 9b) and 9e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' However, this approach is computationally expensive, since twice the initial image window size slows the calculation down by a factor of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' However, padding with zero to twice the original size can still yield unacceptable artifacts for typical z ranges used in 3D PSF calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Establishing absorptive boundary conditions: At ev- ery propagated slice, one can apply an ideal absorp- tive boundary condition to the outside boundary and continue propagation by re-projecting this filtered field onto the pupil plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This has the disadvantage of sac- rificing a good PSF for a portion of pixels near the xy sides of the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' In addition, every slice propa- gation requires two Fourier transformations, instead of only one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Using the chirp Z-transform (CZT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' With the help of the CZT, also called zoomed Fourier transform29 it is possible to calculate only a part of the field without the need for periodic boundary conditions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 9c and 9f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' In this way the wrap-around artefacts can be partially avoided at the expense of a roughly twice or more in- crease of computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Practically method C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' seems to be the most appropriate among the three options here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The use of CZT for Fourier optics and PSF modelling is not new in the literature and has been proven to be more efficient without loss of accuracy than FFT30–32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' To demonstrate this, the xz-profiles of the lower part of the PSFs which corresponds to the slice propagation method us- ing FFT, by zero-padding as described previously as case A and using the CZT as described as case C are displayed at gamma 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='05 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 9d, 9e an 9f respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The standing waves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 9d and clearly seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 9a are due to the Fourier-wrap around.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' These effects are reduced considerably as the window size is doubled (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 9b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' A detail comparison of the slice propagation techniques with FFT and CZT with a chosen gold standard is shown in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Chirp Z transform The chirp Z transform (CZT) is a more generalized func- tion converting a signal in real space into a frequency-domain representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' For a 1D signal Xn,n ∈ [0,N − 1] ∩ N with N being the number of points of the signal and N the set of nat- ural numbers, the Z transform ˜Xz,z ∈ C is given as follows: ˜Xzm = CZT(Xn) = N−1 ∑ n=0 XnZ−n m , (8) where Zm = AW −m,m ∈ N is a spiral path in Z−path with A being the starting point and W = exp(−i∆β) the ratio of two consecutive points with a given angular increment phase ∆β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' For A = 1 and ∆β = 2πm/N,Zm is computed over an unit circle and the CZT operation becomes a discrete FFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' To zoom the signal Xn in by a scalar factor c, A = exp(−iπ/c) and W = exp(−i2π/Nc)29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Therefore, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 8 can be expressed in terms of convolution as follows30: ˜Xzm = W m2/2FFT−1 � FFT � XnA−nW n2/2� FFT � W n2/2�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (9) The inverse CZT of a signal ˜Xzm in a frequency-domain rep- resentation is defined as the complex conjugate of the CZT of the complex conjugate ˜X∗ zm of ˜Xzm within some scaling factor Calculating Point Spread Functions 11 10 0 10 x/µm 10 5 0 5 10 y/µm (a) 10 0 10 x/µm 10 5 0 5 10 y/µm (b) 10 0 10 x/µm 10 5 0 5 10 y/µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='48 (c) 10 0 10 x/µm 4 5 6 7 z/µm (d) 10 0 10 x/µm 4 5 6 7 z/µm (e) 10 0 10 x/µm 4 5 6 7 z/µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='46 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='52 (f) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 9: Profiles displayed at gamma 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='05 of the PSFs calculated from the slice propagation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (a, b, c) xy-plane at defocus position 7 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (d, e, f) xz-cut including to the optical axis starting at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='5 µm away from the focus and with ∆z = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (a, d) Using the standard FFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (b, e) By zero-padding the image window size to twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (c, f) Using CZT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The parameters are NA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='4, immersion medium : water (n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='33), polarization : circular;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' emission wavelength λem = 580 nm, voxel size 80 nm × 80 nm × 140 nm and, window size : 256×256×25 pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' for a CZT operating on a unit circle33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The propagator func- tion using CZT is: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Calculate an appropriate zoom-in factor c such that the lateral window size of the calculated PSF is slightly big- ger or equal to the lateral dimension of the PSF at the position ∆z from the focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The factor c is calculated as c = (D+Nxy/2)/(Nxy/2), where tanθmax = D/∆z in real space and tanθmax = kxy/kz in Fourier space with θmax being the maximal angular aperture and Nxy the number of pixels in the xy-plane (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 10);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Zoom in the pupil plane amplitude distribution with the calculated factor c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' If the factor c leads to a pupil radius bigger than the given window size, an appropriate new target image size must be chosen, and the pupil radius is always zoomed to just fit in the image window;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Apply the angular spectrum propagator (see above) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Apply an inverse CZT and zoom out with the same above-mentioned factor c to obtain the amplitude at ∆z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Extract the field within the original window size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The sinc-shell method (SR) This method is based on the fact that the three-dimensional Fourier transform of a complete spherical shell has a convenient solution in real space, which is sinc(k0|r|) = Nxy/2 real space in focus real space at ∆z D 𝜃max ∆z x z FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 10: Illustration of the wave propagation in real space for calculating the zoom factor c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' sin(k0|r|)/(k0|r|), k0 being the wavenumber in the medium and r the radial position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Given this, the sinc-shell method is described as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=" Extend the border of the desired window size N×N by 25% to get a new window N'×N';" metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=" Generate a sinc(k0|r|) amplitude distribution in three dimensions in real space within the window size N'×N'×z, z being the range along the axial axis." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Mul- tiply this distribution by a compact disk of radius equal to N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Generate a 3D spherical shell by Fourier transforming the result from step 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Set all values at negative kz to zero (akin to a Hilbert- transform) or and/or keeping only the kz-range which :OCalculating Point Spread Functions 12 contains valid ⃗k vectors (yielding a change in z- sampling and a phase ramp in real space, not affecting intensity values).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This half of the 3D spherical shell corresponds to the propagator of the field in free space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Generate 3 versions (for Ex, Ey, Ez) of the 2D Pupil as obtained by the jinc-FT trick, each containing the apla- natic factor and the appropriate electric field modifica- tion factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' make two additional copies of the 3D McCutchen pupil and apply the 3 electiv field pupils to each of the 3 Mc- Cutchen spheres by multiplying it with each kz slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' At this point, only a kz-range required for the intensity PSF is needed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Perform a three-dimensional Fourier-transformation of each of the three field component McCutchen spheres to obtain the sought-after field components in real space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Extract the field within the desired window N×N×z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This method has the attractive property that it does not suffer from the Fourier wrap-around effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The only wrap- around effect is suppressed by extending the window by only 25% and filtering the amplitude field with a disk (step 1 and 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This filtering in real space corresponds to a convolution of the 3D spherical shell with a Jinc function in Fourier space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' It increases the precision in the values of the shell and removes any artefact that may arise during the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' A disad- vantage of this technique is that at least step 2,3 and 4 have to be performed while observing the Nyquist sampling along kz for the full field to include its z-propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This method is also not readily applicable to a single slice (in or out-of- focus).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The Fourier-shell interpolation (VS) This method aims at representing the useful part of the Mc- Cutchen pupil directly in 3D-Fourier-space and projecting the two-dimensional pupil functions onto this three-dimensional shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The difficulty is that the shell, at each integer [kx,ky] position has a non-integer kz position which needs to be rep- resented by interpolating along kz in Fourier space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' As a credible representation of such a non-integer kz would require essentially the whole available kz-range, an appropri- ate compromise to keep the computation efficient was made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=" We aim to represent only the central part of the correspond- ing real-space representation as faithfully as possible and label the rest as “don't care” region (see Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 11b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=" The border of this “don't care” region is limited by a chosen factor (here it is chosen to be at the 8th pixel from both edges)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' To calcu- late the necessary Fourier space interpolation kernel, the part of real space near the border of the z-volume is iteratively up- dated, while the central part is forced to the expected values in each iteration in this iterative Fourier transformation algo- rithm (IFTA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' In addition, a pre-defined cut-off frequency is chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This cut-off frequency limits the number of interpolation coeffi- cients, which can be used to fill the voxels along kz in Fourier- space adjacent to the one nearest to the non-integer kz(kx,ky) position of the McCutchen pupil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The cut-off frequency here is set to ±8 Fourier space pixels yielding 17 interpolation co- efficients to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The required interpolation co- efficients are generated with the help of the IFTA34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' An interpolation table of 60 sup-pixel positions along kz was pre-computed via IFTA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' As initialization, ideal non-cyclic exp(2πikzz) waves were generated in real space correspond- ing to the respective sub-pixel frequencies in Fourier space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The ideal waves are then Fourier-transformed and only Nz (here 17) interpolator values are kept and all others are set to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=" The result is transformed back to real space, where the central area (here the inner is about 66 % of the given z-range) is replaced by the original perfect waves, but the “don't care” region is not touched." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This is repeated (typically 500 times) until convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The so-generated interpolation table (17 complex valued coefficient as a function of 60 sub-pixel lo- cations) is stored for later use (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 11c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Note that we only need to calculate the residual non-integer part of an os- cillation, leading to an interpolation table which only contains less then one oscillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The integer oscillations are taken care of by the placement of the kernel in Fourier-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' A typical example for the offset of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='25 pixels is shown in real and Fourier space in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 11b and 11d respectively, overlayed with the ideal subpixel wave (solid line which corresponds to the legend ‘Original signal’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=" The border of the “don't care region” is indicated by the dashed vertical lines." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' A real space representation of the full interpolation table is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 11a with the “don’t care region” also indicated by the vertical red dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The size of the border factor (in pixels) and the cut-off frequency defining the number of interpolation coefficients should be roughly the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' If the “don’t care region” is far bigger than the “do care region”, there tend to be less inter- polation coefficients generated for the given region of support frequencies in Fourier space, which is delimited by the blue dashed lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 11c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This leads to a large computational overhead for a given region of interest, since the region re- quires extensive padding .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' A small “don’t care region” on the other hand can lead to inaccuracies inside the “do care region” hence the region of support frequencies in Fourier space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This PSF generation algorithm based on Fourier-shell inter- polation works as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Generate the three two-dimensional McCutchen pupil projections as described above (using the jinc-FT trick as described in Section III D 2) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Calculate kz(kx,ky) for every pixels within the pupil and round it to the nearest 1/60 subpixel kz position;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Write these pupils into Fourier space by applying the appropriate interpolation kernel for this sub-pixel posi- tion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Perform a three-dimensional Fourier transformation to Calculating Point Spread Functions 13 "don\'t care region" "don\'t care region" 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='25 sub-pixel 20 15 10 5 0 5 10 15 20 Axial position/pixel 10 20 30 40 50 60 Sub-pixel position 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='15 (a) 20 10 0 10 20 Axial position/pixel 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='1 Phase in real space /arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' unit border border Original signal Final signal "don\'t care region" updated "don\'t care region" updated "do care region" re-enforced (b) "frequency support region" 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='25 sub-pixel 20 15 10 5 0 5 10 15 20 Spatial frequency 10 20 30 40 50 60 Sub-pixel frequency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='8 1 (c) (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 11: (a) Phase shift in the ideal wave exp(2πikzz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (b) Phase at a sub-pixel 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='25, indicated by the horizontal white line in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (c) Interpolation table in Fourier space containing the interpolator coefficients at 60 different sub-pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (d) Interpolation coefficients in Fourier space along the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='25 sub-pixel indicated by the horizontal white line in (a) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=" obtain the three-dimensional field distributions (with expected errors in the “don't care region”)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This method can be performed fast and memory efficient as a single access operation in Matlab by exploiting its indexed addressing capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=" In this way, the complex-valued 2D pupil can be rapidly filled into the appropriate Fourier space region with the optimized interpolation coefficients as de- scribed above and the “don't care” region can be later re- moved." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The required kz-range can be kept to a minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This method was originally constructed to help with the reconstruc- tion of coherent tomography data, where each entirely dif- ferent phase projection can then directly written into Fourier- space without the need of a full immediate propagation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' by the slice propagation method) for each projection, which saves an enormous computational overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Dipole emission PSF In reality, a fluorescent emitter can rotate freely at its place or its orientation can be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The PSF of an emitting dipole in the focal plane depends on the polarization of the illumina- tion light and the dipole orientation in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' To describe the dipole emission, let ⃗µ(θdip,φdip) denotes its emission transi- tion dipole moment which is a function of its elevation angle θdip about the optical axis z and azimuthal angle φdip about x−axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The following steps describe how to accommodate a fixed dipole into the PSF calculation: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Calculate the field amplitude distribution by taking only the x−polarized part at the pupil plane, using the direc- tional field changes as described above and the appro- priate aplanatic factor for an emission PSF to obtain the three-dimensional amplitude field ⃗E1 = [E1x,E1y,E1z] as a function of lateral distance from the optical axis in the image plane;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' By projecting this amplitude field on the dipole orien- tation and calculating the absolute square of the scalar product, we obtain the detected intensity Ix = |⃗E1 ·⃗µ|2 as measured through an analyzer oriented along x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Steps 1 and 2 can be repeated for a y−oriented analyzer yielding Iy and the average intensity without analyzer is obtained as the PSF of the dipole emitter: PSFdip = 1 2(Ix +Iy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (10) To calculate the intensity in the back focal plane of a dipole emitter with a given transition dipole orientation, the same steps as described above are to follow but instead of taking the amplitude field in the image plane, the user is to use the three-dimensional amplitude field in the McCutchen pupil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 10 5 S patia0 5 l frequency10 1520nterpol 0cut-off freguenoVcut-offarb ators 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='2-unit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='4 is enfor zeroregior ced"zero is enfoDrced"spa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='8 erfrequen suppoicy rt-Original Final sign1gnal ale--20 15Calculating Point Spread Functions 14 If the illumination light is unpolarized or circularly polar- ized, averaging over all the dipole orientations or over three different perpendicular dipole orientations leads to the same PSF as obtained by calculating a corresponding illumination PSF using circular polarized light, apart from a possible dif- ference in the aplanatic factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This leads back to the PSF of an isotropic emitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' PSFav = 1 4π/3 � 2π 0 � π 0 PSFdip sinθdipdθdipdφdip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (11) If the illumination light is linearly polarized, averaging over all the dipole orientations does not lead to a symmetric PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Further study in polarization in fluorescence microscopy is found in35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' QUANTITATIVE PSF COMPARISON The comparison of the models is performed in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The first step consists of comparing the theoretical models with a chosen gold standard (GS) and the second one consists of comparing all the models with experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' We choose an emission PSF according to the RW0 model (described in Section II B 2) with a very high sampling as our GS for the theoretical models comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The GS is calcu- lated with a sampling of 16 nm × 16 nm × 100 nm and win- dow size of 1280×1280×64 pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The result is subsampled to 80 nm × 80 nm × 100 nm by binning groups of five ad- jacent pixels along x and y to correspond to what we choose as a “normal sampling”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This normal sampling consists of a voxel size of 80 nm × 80 nm × 100 nm and image window size 256×256×64 pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The z-range by which the VS is computed is set to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='5× larger than the other models (96 vs 64 pixels) to compare only the “do care region” in the quantitative comparison of the PSFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The computational cost of this enlarged window size (256 × 256 × 96) is therefore taken as the computation time for the (256×256×64) PSF for the case of the VS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' We choose simulation parameters corresponding to our ex- periment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' We have an oil immersion objective where the re- fractive index of the oil is n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='518 and numerical aperture is NA = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' To mimic randomized dipole orientation, the polar- ization state is assumed to be circular and the emission wave- length is λ = 580 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The PSFs models are implemented and compared in quality and computational speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The quantita- tive comparison is done for a single in-focus plane and for the whole volume, excluding 5 % on each side near each image border.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Each PSF data is normalized as a 3D Volume to yield an integrated intensity of one at the focal plane over the central 90 % in x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The radial mean intensity profile at the focus position for each model is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 12 in logarithmic scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 12 and the following sections, we denote by RW1 the RW model calculated by the same technique as the GS but performed at the same (normal) sampling grid as all other techniques (SP, CZT, VS, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=') to compare to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' RW2 refers to an alternative way for calculating the RW model which uses a Fourier transform along z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Error analysis and computation time of the theoretical models compared with the RW gold standard (GS) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Model accuracy To verify how accurate each model is compared to the GS, we use two different techniques: the mean relative er- ror (MRE) and the normalized cross correlation (NCC) be- tween each model and the GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The MRE has the advantage to describe the average performance-error of each model in comparison with the GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' However, for a data which is shifted and the shifting parameter might be unknown or does not have much importance, the use of normalized cross correla- tion (NCC) for the comparison is advisable as it is not sensi- tive to linear shifting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The formula used to compute the MRE is given by MRE(z) = ∑xy |GSxy(z)−PSFxy(z)| ∑xy |GSxy(z)| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' If not specified otherwise, only intensity values bigger than the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='1 % of the maximum intensity value of GS at each z−slice are considered in the calculation of the MRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The MRE results are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' We calculate the 2D NCC slice by slice between our mod- els and the GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' For this, a built-in function normxcorr2 in Matlab is used36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' For two 2D input images, this function gen- erates as an output a 2D image with double the size minus one of the input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' A value m at a given position (i, j) in the output NCC refers to the NCC of the two images at (i, j) shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' A shift of (i, j) = (0,0) means the two images are on top of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The value of m is between −1 and 1, 0 for zero-correlation, 1 for maximum correlation and −1 for anti-correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This technique therefore accounts for possi- ble shifts between the model and the GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' A perfect correlation corresponds to a NCC of 1 at a shift position (0,0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' We report the maximum of the NCC at each xy-slice for each axial po- sition z as well as the NCC of the 3D volume PSF compared with the 3D GS and have both parameters to check how close the model is to the GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The result is summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The missing cone problem As observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 13 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 14, the errors between most of the PSF models and the GS are higher at a larger depth and decrease as the field is focusing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The same errors tend to be more enhanced again near the focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The same observation can be made for the case of the NCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This observation can be interpreted by studying the missing cone problem of each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' In wide-field microscopy, the missing cone corresponds to frequencies close to the kz-axis that prevent the OTF to trans- Calculating Point Spread Functions 15 0 50 100 150 vradial/o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 0 2 4 6 8 10 12 Radial Position/µm 15 10 5 Radial Mean Intensity in log scale GS RW1 RW2 SR 180 185 190 vradial/o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 16 14 12 (a) 0 50 100 150 vradial/o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 0 2 4 6 8 10 12 Radial Position/µm 15 10 5 Radial Mean Intensity in log scale GS SP CZT VS 180 185 190 vradial/o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 17 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='5 (b) 0 50 100 150 vradial/o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 0 2 4 6 8 10 12 Radial Position/µm 15 10 5 Radial Mean Intensity in log scale GS GL ScalarPSF VectorPSF 180 185 190 vradial/o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 20 18 (c) 0 50 100 150 vradial/o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 0 2 4 6 8 10 12 Radial Position/µm 15 10 5 Radial Mean Intensity in log scale GS PSFGen_Good PSFGen_Best 180 185 190 vradial/o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='2 17 (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 12: Radial mean intensity profiles at the focus position in logarithm scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (a) GS: The gold standard computed using the Richards and Wolf method7,12, RW1 and RW2: two ways for computing the RW models (Section II B 2), SR: the sinc-shell method (Section IV C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (b) SP: Slice propagation method with FFT (Section IV B), CZT: SP technique using the chirp Z transform (Section IV B 3), VS: Fourier shell interpolation technique (Section IV D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The state-of-the-art are: (c) A scalar PSF based on the work of Gibson and Lanni 6 and developed by Li, XUE, and BLU 8, a scalar and vector PSF from9 and, (d) a vector PSF calculated with the Richards and Wolf 3D optical model with good and best accuracy respectively from the PSFGenerator toolbox in10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The variable vradial denotes the radial optical unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' mit information about the object along that axis (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 15a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' As a consequence, a uniform illumination fluorescence imag- ing setup is incapable of focussing (or “sectioning”) a planar sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Out-of-focus light is distributed to different regions but, the energy is still conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The integrated intensity at each z position remains the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This should be the case for any wide-field PSF as long as the PSF remains confined well within the calculation grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Sufficient sampling turns out to be a key factor, especially for the methods, where radial symme- try is exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Since we are dealing with an approximation of the exact field in a finite grid, it is required to have the right amount of data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' An inappropriate choice of grid and pixel size can lead to the violation of the missing cone in the corresponding transfer function of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This problem is demonstrated by integrating the PSF for each xy−plane at a given axial z−position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Even though the radial profile, illus- trated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 12, seems to look good and fits into the GS at the focus position, there may still be significant sampling-related violations of the missing cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Note that for simulations to accurately evaluate the performance of 3D deconvolution rou- tines on widefield data, preserving the missing cone property is paramount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' A more precise observation of the violation of the missing cone can be made by zooming on the z−range around the fo- cus and displaying the integrated intensity of each slice along the axial position z (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' To quantify this effect, the standard deviation (Std) of the integrated intensities within 20 % around the focus of the given z−range is calculated and is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 15b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This 20 % region is delimited by the two dashed vertical lines in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 15c, 15d, 15e and 15f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The Std measures the non-uniformity of the laterally integrated in- tensity over axial position and the importance of the peak of the integrated intensity compared to the minimum within the z-range that is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The order of magnitude of the Std from the techniques CZT, RW2, VS, and, SP are in range of 10−4 which are fairly small and smaller than the Std of the defined GS itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Increasing the sampling of the models re- duce the violation of the missing cone considerably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This is seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 15b for the case of the GS and RW1 where the two PSFs are both generated from the same model but with different sampling and window size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The Std has decreased by about 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='41 % by oversampling the lateral grid five times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' However, higher sampling, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' finer step with a big windows, may lead to an expensive computation (about 7× slower) and requires a larger computer memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Calculating Point Spread Functions 16 60 40 20 0 20 40 60 uaxial/o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 3 2 1 0 1 2 3 Axial Position/µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='3 Mean Relative Error RW1: MRE3D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='0163 RW2: MRE3D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='2994 SR: MRE3D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='0293 (a) 60 40 20 0 20 40 60 uaxial/o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 3 2 1 0 1 2 3 Axial Position/µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='04 Mean Relative Error SP: MRE3D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='0121 CZT: MRE3D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='0088 VS: MRE3D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='0117 (b) 60 40 20 0 20 40 60 uaxial/o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 3 2 1 0 1 2 3 Axial Position/µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='3 Mean Relative Error GL: MRE3D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='2089 ScalarPSF: MRE3D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='2065 VectorPSF: MRE3D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='0035 (c) 60 40 20 0 20 40 60 uaxial/o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 3 2 1 0 1 2 3 Axial Position/µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='052 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='054 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='058 Mean Relative Error PSFGen_Good: MRE3D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='0561 PSFGen_Best: MRE3D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='0561 (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 13: Mean Relative Error of (a) our RW methods and SR, (b) the SP, CZT and VS and the state-of-the-art (c) GL for the Gibson and Lanni model, ScalarPSF and VectorPSF and (d) vector PSF from PSFGenerator for good and best accuracy with the GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 60 40 20 0 20 40 60 uaxial/o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 3 2 1 0 1 2 3 Axial Position/µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='99 NCC RW1: NCC3D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='99986 RW2: NCC3D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='9959 SR: NCC3D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='99963 (a) 60 40 20 0 20 40 60 uaxial/o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 3 2 1 0 1 2 3 Axial Position/µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='9996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='9997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='9998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='9999 NCC SP: NCC3D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='99965 CZT: NCC3D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='99965 VS: NCC3D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='99965 (b) 60 40 20 0 20 40 60 uaxial/o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 3 2 1 0 1 2 3 Axial Position/µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='995 NCC GL: NCC3D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='98863 ScalarPSF: NCC3D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='99158 VectorPSF: NCC3D = 1 (c) 60 40 20 0 20 40 60 uaxial/o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 3 2 1 0 1 2 3 Axial Position/µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='9988 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='9992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='9994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='9996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='9998 NCC PSFGen_Good: NCC3D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='99872 PSFGen_Best: NCC3D = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='99872 (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 14: Normalized cross correlation between the theoretical RW GS with the (a) RW1, RW2 and SR methods, (b) the SP, CZT and SR and the state-of-the-art (c) GL for the Gibson and Lanni model, ScalarPSF and VectorPSF and (d) vector PSF from PSFGenerator for good and best accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Calculating Point Spread Functions 17 Radial frequency Axial frequency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='55 Missing cone (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='448 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='678 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='227 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='169 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='875 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='875 GS RW1RW2SP CZT VS SR GL ScalarPSF VectorPSF PSFGen_Good PSFGen_Best Methods 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='5 2 Std of the Relative Integrated Intensity/arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' unit 10-2 (b) 50 0 50 uaxial/o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 2 1 0 1 2 Axial Position/µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='99 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='01 Integrated Intensity /arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' unit GS RW1 RW2 SR (c) 50 0 50 uaxial/o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 2 1 0 1 2 Axial Position/µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='998 1 Integrated Intensity /arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' unit GS SP CZT VS (d) 50 0 50 uaxial/o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 2 1 0 1 2 Axial Position/µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='995 1 Integrated Intensity /arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' unit GS GL ScalarPSF VectorPSF (e) 50 0 50 uaxial/o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 2 1 0 1 2 Axial Position/µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='98 1 Integrated Intensity /arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' unit GS PSFGen_Good PSFGen_Best (f) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 15: The violation of the missing cone is observed by calculating the integrated intensity of 90 % of the total surface xy−plane for each axial position z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (a) The OTF calculated from the GS and displayed at gamma of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The missing cone is presented in yellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (b) The standard deviation (Std) of the relative integrated intensity calculated within a z−range equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='3 µm near the focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The profiles of the integrated intensity along z from (c) our RW methods and SR, (d) the SP, CZT and VS and the state-of-the-art (e) GL for the Gibson and Lanni model Li, XUE, and BLU 8, ScalarPSF and VectorPSF9 and (f) vector PSF from PSFGenerator for good and best accuracy10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The variable uaxial denotes the axial optical unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Wrap-around effect In addition to the missing cone problem, the wrap-around effect due to FFT-based convolutions can contribute to the ac- curacy of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' To investigate this effect, we define a ref- erence window grid Wref where this wrap-around effect should be minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The size of the window Wref is defined such that the width of the PSF at the highest depth could still fit into it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' It is calculated to be: Wref|x = 2×2× � � zmax kx|max kz|max +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='3dres sx � �;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (12) Calculating Point Spread Functions 18 where the first applied factor 2 is to double the half window;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' the second factor 2 is to sample the frequency space two times finer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' zmax is the maximum depth expressed in nm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' kx|max and kz|max are the maximal radial and axial frequency respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' dres is the resolution limit of the optical system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' the factor 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='3 is an heuristic factor and, sx is the sampling size along x, expressed in nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The same formulation applies along y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' To quantify the wrap-around effect, we generate different PSFs from each model at different windows denoted by W and denote the PSF h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' We denote h0 the reference PSF which is calculated with the reference window Wref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' If W is smaller than Wref, the wrap-around effect in h relative to h0 is calcu- lated within the window W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' If W is bigger than Wref, no wrap- around effect is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' However, the energy is spread over a larger grid so the difference between h and h0 is not expected to be zero but is expected to converge to a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' In this case, we calculate the difference between h and h0 within the window Wref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The three-dimensional MRE and mean square error (MSE) between h and h0 for the given window for each model are calculated and plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' In this computation, Wref is calculated and rounded to be 300 × 300 pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The lateral window is square and the rest of the parameters for the comparison are the same as those described for calculating the model accuracy (see Section V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' As mentioned earlier, the scalar PSF based on Gibson and Lanni denoted by GL in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 16 uses Bessel series as a fast approach to calculate the PSF8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' As described in9, a numer- ical integration based on Simpson’s rule is used to compute ScalarPSF and the VectorPSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Thus, those models do not pos- sess any FFT wrap-around effect in their computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' They can therefore be used as reference to measure the wrap-around effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' A modified CZT PSF model, denoted by mCZT, is in- troduced in this particular PSFs comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This model works similarly as the CZT PSF model except the fact that if the given window W as described above is smaller than Wref, mCZT chooses Wref as window grid for the computa- tion, crops the generated PSF to get the input size W and scale the PSF by the integrated intensity in focus with a window size W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The errors between the PSF h at a window W and the reference PSF h0 in this case are small enough but are not zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' As it is also observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 16a and 16b, errors calculated from the Fourier-based models decrease generally as the win- dow grid becomes bigger except for the case of RW2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' How- ever, as it has been said computation of a PSF at a bigger win- dow grid can lead to an expensive computation and requires a larger computer memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The wrap-around effect can also be interpreted as under- sampling the phase in Fourier-space, which leads to aliasing effects in real space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Large propagation distances thus lead to more aliasing, particularly for small pupils in Fourier space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' small pixel-sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Computational time To investigate the computation time of each model, we use MATLAB R2018a under the same operating system stated in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' We run each model with the regular sampling 80 nm × 80 nm × 100 nm, the same microscope parameters in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' We choose two different window size 256 × 256 × 32 and 256 × 256 × 256 pixel and generate the computation time per voxel of each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The results are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' As mentioned previously, eight pixels are added on both borders of the z-range (32 and 256) to account for the “don’t care region” in the VS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The addition of 8 pixels on both sides of the z-range 32 pixels in the VS slows the computation down by a few milliseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The first run of the VS requires more computation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' It can go up to 20× or even slower than the second run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The computation is faster in the second run as the interpolators in the models are stored and used for the next computation if the PSF and imaging parameters re- main the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The computation time per voxel of a PSF using the CZT technique is considerably higher compared to the other tech- niques apart from the PSF generated from the PSFGenerator at the best accuracy10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This computation cost is explained from the fact that the CZT technique expands the lateral size of the PSF to avoid any FFT-based wrap-around problems if the z- range is large (See Section IV B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' As the z-range increases, the computation time per voxel of the CZT is growing because a large lateral window is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' A similar cost must be paid when one requires higher sampling to generate a more accu- rate model and to reduce as much as possible the missing cone error near the focus and any wrap-around effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Discussion In summary all the vector models agree closely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' VectorPSF has a 3D NCC value equal to 1 while CZT, VS, SP and SR differ deviate at the 4th and PSFGen_Good and PSFGen_Best at the 3rd digit from the perfect NCC of one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The scalar PSFs have the smallest 3D NCC values of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='9916 for ScalarPSF and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='9886 for GL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Similarly, the 3D mean relative error between the Vec- torPSF is the closest to the GS with a value of MRE ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='0035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This is followed by the CZT model with MRE ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='0088, VS and SP models with MRE ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This high similar- ity and accuracy is achieved at a high computation time cost (in average 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='432 µs·voxel −1 for the CZT if the window size 256×256×256 voxels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Slightly less accuracy can be achieved for a less expensive computation time with the SP and VS techniques and espe- cially if a repetitive computation of the PSF with the same imaging parameters is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' However, one needs to ex- pand the z-range to consider the “don’t care region”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' A higher z-range is therefore needed accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Another disadvan- tage of the VS is that the computation of a single slice PSF is not possible with this technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Similarly, as discussed in Section IV B 1, SP suffers from a wrap around problem at a Calculating Point Spread Functions 19 0 100 200 300 400 500 600 window size 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='45 MRE / arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' unit RW1 RW2 SP CZT mCZT VS SR GL ScalarPSF VectorPSF PSFGen_Good 50 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='0001 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='0002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='028 Zoom (a) 0 100 200 300 400 500 600 window size 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='6 MSE / arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' unit 10-7 RW1 RW2 SP CZT mCZT VS SR GL ScalarPSF VectorPSF PSFGen_Good 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='995 50 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='005 4 5 6 7 8 9 10 11 12 13 10-10 Zoom (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 16: Quantification of the wrap-around effect in each of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (a) Three-Dimensional Mean Relative error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (b) Three-Dimensional Mean Square Error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The right hand side windows entitled Zoom of (a) and (b) are the zoomed-view of the red dashed rectangular in the left hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' higher depth and with a smaller window grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Nevertheless, SP, CZT and VS are less prone, compared to the other method, to violate the missing cone properties of PSF in wide-field mi- croscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' On the other hand, the SR method can beat SP, VS and even CZT in terms of wrap-around effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The still observed wrap- around effect quantified and displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 16 are due to the jinc aperture added to the pupil aperture stop to limit the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' SR has a very high similarity to the GS especially near the focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Its missing cone problem is also considerably small but not as small as in SP, VS and CZT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' SR is faster to compute than CZT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' PSFGen_Good and PSFGen_Best have relatively higher MRE and lower NCC with the GS compared to the PSF models developed under this project and the state-of-the-art VectorPSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' PSFGen_Good yields to a lower MRE than PS- FGen_Best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' We speculate that the reason for this could be that the PSFGen_Best is overestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The probability to violate the missing cone is higher in the computation of this model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The model is also very expensive in time compared to the other models discussed in this manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Compared to the rest of the models, the scalar model (GL and ScalarPSF) have the lowest 3D NCC and highest 3D MRE (after RW2) with the GS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' These values quantify the difference Calculating Point Spread Functions 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='226 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='213 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='526 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='94 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='578 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='644 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='384 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='432 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='421 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='278 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='665 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='107 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='106 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='206 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='686 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='392 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='897 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='348 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='01 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='512 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='023 RW1 RW2 SP CZT VS SR GL ScalarPSF VectorPSF PSFGen_Good PSFGen_Best 0 10 20 30 40 Computation time/µs voxel-1 256 256 32 256 256 256 window size: FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 17: Computational time of each technique per pixel in µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' RW1 and RW2 are two ways for computing the Richards and Wolf models7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' SP: slice propagation technique (Section IV B),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' CZT: SP technique using the chirp Z transform (Section IV B 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' VS: Fourier shell interpolation technique (Section IV D),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' SR: sinc-shell method (Section IV C),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' GL: scalar PSF based on the work of Gibson and Lanni 6 and developed by Li,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' XUE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' and BLU 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' ScalarPSF and VectorPSF: scalar and vector PSF from9 and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' PSFGen_Good and PSFGen_Best: vector PSFs calculated with the Richards and Wolf 3D optical model with good and best accuracy respectively from the PSFGenerator toolbox in reference10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' between a vector PSF and scalar PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' These scalar PSFs are however the fastest models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Their precision and accuracy can be sufficient for some application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Although RW2 is among the models which have higher ability to satisfy the missing cone properties, its accuracy tends to be close to the scalar PSF models, the least accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Experimental validation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Experimental PSF data To validate the model experimentally, a PSF measurement is performed by averaging the images of beads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' For this, we use Tetraspeak beads of 100 nm in diameter (Invitrogen by Thermo Fisher Scientific, REF T7279).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The beads fluoresce with an emission wavelength of 580 nm at an excitation light of 561 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The beads are first diluted with distilled water to get 1/400 concentration and vortexed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' We drop some droplets of the solution on a coverslip and let it dry for over 4 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Our imaging system is composed of a 63× Zeiss C-Apochromat oil immersion (n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='518) objective lens of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='4 NA (DIC M27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The choice of the oil immersion for the microscope is very important as this can contribute to any observed aber- ration in the recorded image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Any contributions from back- ground light is reduced and eliminated during the measure- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The image of background is recorded before any mea- surement and several measurements are averaged to reduce the noise in the offset estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The exposure time of the detector camera is set to be 100 ms and the output laser power illuminating the sample is measured at 25 mW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' To obtain the experimental PSF data, an offset image is sub- tracted from the raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The PSF is afterwards constructed by averaging over 64 beads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This averaged PSF data is of size 40 × 40 × 159 pixel and voxel size 80 nm × 80 nm × 50 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Let hexp denote the experimentally averaged PSF data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' To compare our theoretical models with hexp , two differ- ent sets of comparison are conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The first set consists of retrieving the phase aberration contained in hexp and add it in the corresponding pupil plane of our theoretical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This comparison is only achieved with the PSFs in our toolbox (RW1, RW2, SP, CZT, VS, SR) since we do not have access to the amplitudes fields of GL, ScalarPSF, VectorPSF, PSF- Gen_Good and PSFGen_Best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The phase retrieval technique developed by Hanser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='37 is conducted to retrieve the pupil function hence phase aberration of hexp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' By testing on a the- oretically computed known phase aberration and pupil func- tion with the same parameters as the imaging system here, the phase retrieval algorithm is able to retrieve the phase with a MRE of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='0084×10−4 and the retrieved PSF has a 3D MRE of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='786×10−8 compared to the preprocessed measured PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Only the z−range within the indicated blue dashed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 18a are used for this set of comparison in order to avoid a compar- ison with only noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The size of hexp within this region is 40×40×80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The second set of comparison consists of comparing a block region near the focus with higher signal and discard those at higher depth and compare the region with non-aberrant theo- retical PSFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Theoretical PSF data The required window to avoid any wrap-around is calcu- lated using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Each theoretical model is computed with a lateral size equal to the result from this computation (242×242 pixel) and with a depth equal to 80 pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The com- puted PSFs are cropped to get the same size 40 × 40 × 80 as hexp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The xz−profile can be observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' First set of comparison: aberrated PSFs To retrieve the phase aberration Ψ contained in the aver- aged experimental data, we take each second axial position of hexp within the blue dashed rectangle indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 18a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The retrieved phase is displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 19a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' To generate the theoretical aberrant PSFs, the complex-valued amplitude PSFs are 2D Fourier transformed to obtain their pupil func- tions and the retrieved phase is added by multiplying the pupil by expiΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The aberrant pupil is inverse Fourier transformed back to real space and we get the aberrant PSF by taking the absolute square of the aberrant complex-valued amplitude PSFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The results are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The similarity and difference between the theoretical aber- rant PSFs and hexp are quantified by computing the NCC slice by slice and the MRE respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The 2D NCC results are plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 20a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Calculating Point Spread Functions 21 (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 18: Meridional sections at gamma 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='05 of the PSFs of a wide-field microscope with an oil objective of a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='4 NA, wavelength emission of 580 nm, window size 40×40×80 and voxel size 80 nm × 80 nm × 50 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (a) Experimental PSF with the blue dashed rectangle indicating the selected region to be the ground truth for the first set PSF comparison and the red dashed rectangle delimiting the ground truth for the second set of PSF comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (b) The Richards and Wolf model RW1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (c) RW2 using the RW model but computed in different way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (d) SP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (e) CZT model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (f) Fourier-shell interpolation model (VS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (g) Sinc-shell method (SR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (h) Gibson and Lanni model by Li, XUE, and BLU 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (i,j) Scalar and vector models respectively from9 (k) Vector model using the RW technique computed at the corresponding best accuracy from10 3 2 1 0 1 2 3 (a) (b) (c) (d) (e) (f) (g) (h) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 19: Comparison of aberrant PSFs with the experimentally averaged PSF data: (a) Retrieved phase aberration from the experimental data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (b) Experimental averaged PSFdata;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (c) aberrant RW1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (d) aberrant RW2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (e) aberrant SP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (f) aberrant CZT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (g) aberrant VS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (h) aberrant SR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' PSFs are displayed at gamma equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Calculating Point Spread Functions 22 40 30 20 10 0 10 20 30 uaxial 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='5 Axial position/µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='9 NCC between aberrated PSF and experimental PSF data RW1: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='95411 RW2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='9497 SP: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='98363 CZT: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='98365 VS: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='98363 SR: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='98356 NCC3D (a) 40 30 20 10 0 10 20 30 uaxial 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='5 Axial position/µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='55 MRE between aberrated PSF and experimental PSF data RW1: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='56228 RW2: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='59195 SP: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='55861 CZT: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='55866 VS: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='55857 SR: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='55933 MRE3D (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 20: Evaluation of the similarity and difference between the experimental PSF data and the theoretical PSFs which are aberrant by the retrieved phase from the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (a) Similarity quantified by the NCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' (b) Difference quantified by the MRE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Second set of comparison: non-aberrant PSFs The errors considered in the first set of comparison include error due to phase retrieval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' To not consider that error and to include the state-of-art PSFs into the comparison, we compare non-aberrant theoretical PSFs with the experimental data in a region of interest where the aberration and noise in the exper- imental data are minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Firstly, to check the region where aberrations are minimal, we conduct a slice by slice compari- son between the computed theoretical aberrant PSF described in Section V B 3 and the corresponding non-aberrant PSFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The NCC results are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Secondly, as we would want to compare our non-aberrant models with an ex- perimental data having as little noise as possible, the standard deviation (Std) of the experimental data is calculated at each axial position (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' We fit this Std distribution into a Gaussian function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The Std distribution is not originally cen- tered to z = 0 so we shift it to zero accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The same shift in z is applied to the experimental measured PSF data before the region of interest is extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' To construct a comparison metric, we thus chose a common region where the slicewise NCC (after z-alignment) is high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This corresponds to posi- tions near the focus where the Std is above half of its maxi- mum as estimated by a Gaussian fit (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This region of interest is indicated by the red dashed rectangle in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 18a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The NCC results between the experimental data and the non-aberrant theoretical data within the region of interest are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Discussion By observing Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 20, on one side we can conclude that there is a high similarity between the aberrant theoretical PSFs with the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' On the other side, the MRE results are not symmetric about the focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This asymmetry is ex- plained by the fact that the shift of focus in the experimental measured PSF was not perfectly retrieved through the phase retrieval technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This induces the error detected in the MRE since the computation of the MRE is conducted point 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='5 Axial position/µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='8 1 Normalized std of the experimental PSF data Experimental data Gaussian fit std 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='5 Axial position/µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='9 NCC between aberrated PSF and non-aberrated PSF RW1 RW2 SP CZT VS SR NCC FWHM FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 21: Selection of the region of interest (ROI) for comparing the non-aberrant PSFs with experimental PSF data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The ROI is the volume delimited by the full-width half maximum (FWHM) of the Gaussian fit of the standard deviation (Std) of the experimental data along the axial axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The Std distribution is normalized by its maximum for display.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' by point and slice by slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' However, we can conclude from the graph that SP, CZT and VS are the models which have higher similarity and less error compared to the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' By considering the region of interest where the signal-to- noise ratio is higher and the aberration is small, the quantifi- cation of the NCC between the non-aberrant PSFs and the ex- perimental data allows us to short the models in order of accu- racy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The VectorPSF described in reference9 leads with a 3D NCC equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='98124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This lead is followed closely by PSF- Gen_Good and PSFGen_Best in reference Kirshner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 10, RW1, VS, CZT, SP, SR, ScalarPSF Aguet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 9, RW2 and GL in reference Li, XUE, and BLU 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The scalar PSFs are the least accurate especially near the focus in this particular com- parison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The difference in NCC between the models and with the experimental data is however very close for the vector PSF models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' Calculating Point Spread Functions 23 8 6 4 2 0 2 4 6 8 uaxial 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='4 Axial position/µm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='98 NCC between non-aberrated PSF and experimental PSF data RW1 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='97888 RW2 : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='97673 SP : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='97873 CZT : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='97874 VS : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='97876 SR : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='97869 GL : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='96156 ScalarPSF : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='96725 VectorPSF : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='98124 PSFGen_Good : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='97933 PSFGen_Best : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content='97933 NCC3D FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' 22: NCC between experimental PSF and non-aberrant theoretical PSFs within a particular region of interest near the focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' CONCLUSION In this work, we provided a general approach for calculat- ing the 3D PSF of a system satisfying the Abbe sine condi- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' We focused on Fourier based techniques and compared the results of a variety of PSF calculations schemes with a gold standard from a Richard and Wolf model computed at higher sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' We explained the algorithmic details of each technique and potential advantages and pitfalls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The Fourier models agree with high precision with the state-of-the-art and are validated experimentally to have good accuracy around the focus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' We also showed in this work that vector PSFs are more accurate than scalar models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The study of the PSFs at higher depth of focus as well as the inclusion of refractive index mis- matching in the theoretical model is not covered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This constitutes the next step for PSFs comparisons in addi- tion to the test of each model in image reconstruction (de- convolution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The Fourier based 3-D PSF models are already fast enough given the fact that there is no radial symmetry in- cluded in them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The models discussed in this manuscript are under the condition that all the planes in the optical system are perpendicular to the optical axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The ability of our models to accommodate radial asymmetry is advantageous compared to the state-of-the-art because our models can accommodate any non-circular aberration and tilted planes caused in the system such as a tilt of a coverslip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' We plan to combine some of the models such that the computation is still faster without com- promising the accuracy of the models at any axial depth z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' The models can be adjusted for confocal microscopy, STED (Stim- ulated emission depletion microscopy) and PSF engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' FUNDING This work was funded by the DAAD through the African Institute for Mathematical Sciences and Stellenbosch Univer- sity, and Friedrich Schiller University Jena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} +page_content=' This work was also supported by the German Research Foundation (DFG) through the Collaborative Research Center PolyTarget 1278, project number 316213987, subproject C04 and the Council for Scientific and Industrial Research (CSIR), project number 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtFRT4oBgHgl3EQfOzfW/content/2301.13515v1.pdf'} diff --git a/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf b/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..6654e95671380618e27a44f9a15277a90fecde45 --- /dev/null +++ b/ZNE0T4oBgHgl3EQfnAEV/content/2301.02505v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:eb69a66033dce871b56c759ec07fb519e216cffa1dd885fa1682ab782cfab40e +size 14478692 diff --git a/_NFAT4oBgHgl3EQfqh2x/content/tmp_files/2301.08648v1.pdf.txt b/_NFAT4oBgHgl3EQfqh2x/content/tmp_files/2301.08648v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2e51702b5af150f546c97200b4ac9a3f9545ac1e --- /dev/null +++ b/_NFAT4oBgHgl3EQfqh2x/content/tmp_files/2301.08648v1.pdf.txt @@ -0,0 +1,1510 @@ +STORM-GAN: Spatio-Temporal Meta-GAN for +Cross-City Estimation of Human Mobility +Responses to COVID-19 +Han Bao, Xun Zhou* +University of Iowa +{han-bao, xun-zhou}@uiowa.edu +Yiqun Xie +University of Maryland +xie@umd.edu +Yanhua Li +Worcester Polytechnic Institute +yli15@wpi.edu +Xiaowei Jia +University of Pittsburgh +xiaowei@pitt.edu +Abstract—Human mobility estimation is crucial during the +COVID-19 pandemic due to its significant guidance for pol- +icymakers to make non-pharmaceutical interventions. While +deep learning approaches outperform conventional estimation +techniques on tasks with abundant training data, the continuously +evolving pandemic poses a significant challenge to solving this +problem due to data nonstationarity, limited observations, and +complex social contexts. Prior works on mobility estimation +either focus on a single city or lack the ability to model the +spatio-temporal dependencies across cities and time periods. +To address these issues, we make the first attempt to tackle +the cross-city human mobility estimation problem through a +deep meta-generative framework. We propose a Spatio-Temporal +Meta-Generative Adversarial Network (STORM-GAN) model +that estimates dynamic human mobility responses under a set +of social and policy conditions related to COVID-19. Facilitated +by a novel spatio-temporal task-based graph (STTG) embedding, +STORM-GAN is capable of learning shared knowledge from +a spatio-temporal distribution of estimation tasks and quickly +adapting to new cities and time periods with limited training +samples. The STTG embedding component is designed to capture +the similarities among cities to mitigate cross-task heterogeneity. +Experimental results on real-world data show that the proposed +approach can greatly improve estimation performance and out- +perform baselines. +Index Terms—Meta-Learning, Generative Adversarial Net- +works, Spatio-Temporal, Graph Embedding, COVID-19 +I. INTRODUCTION +Evolving developments (e.g., spread, mutation, vaccination) +around the COVID-19 pandemic have continued to pressure +policymakers to come up with effective and changing policies +that can protect public health while avoiding breakdowns of +economics, and maintain the support of essential needs in daily +lives. To mitigate this dilemma, staged reopening to avoid +infections caused by eased social distancing policies have +been implemented. As a result, estimation on dynamic human +mobility responses to the pandemic condition and policies +remains a crucial task in policymaking. +Due to the asynchronous spread of the disease, it is par- +ticularly hard for cities in the early stage of an outbreak or +wave (e.g., the Omicron variant) to estimate future human +mobility responses under unprecedented severity levels or +unseen policies as there is very limited historical data. To +* Corresponding author +address this issue, it is crucial for such cities to be able to +leverage other cities’ past experiences and knowledge for its +own estimation. To this end, mobility response estimation +methods that can leverage cross-city knowledge to achieve +promising results are urgently needed. +In this paper, we make the first attempt to solve the cross- +city human mobility responses estimation problem: Given a set +of inputs on contextual (e.g., population, point-of-interest POI +counts), epidemic (e.g., COVID-19 cases), policy (e.g., stay- +at-home orders) conditions and corresponding human mobility +responses measures (e.g., POI visit counts, home dwell time) +from multiple cities, we aim to learn a model, which can +quickly adapt to previously unseen cities and time periods, +and estimate human mobility response dynamics under any +projected conditions. +Challenges. The cross-city human mobility response estima- +tion problem has three major challenges. First, human mobility +responses depend on many complex social-physical factors +which are unknown or uncertain. For example, responses +can be affected by people’s willingness in cooperating with +policies, decisions from service providers (e.g, whether a +restaurant will open or allow dine-in options), changes in +public transportation, supply, and many more [1]. Second, +human mobility responses often have spatio-temporal non- +stationarity. For example, the contribution of different factors +in mobility tends to vary from region to region due to +cultural and economic differences, and can quickly evolve +over time. Such spatial and temporal nonstationarity greatly +limits the availability of training data for each estimation +task, making it difficult to leverage the approximation power +of data-driven approaches. Third, there also exist complex +spatial and temporal dependencies across different estimation +tasks, i.e., cities and time periods, which need to be explicitly +considered for robust parameter sharing. For example, cities +may share similar mobility dynamic patterns based on their +spatial adjacency (e.g., distances, travel connections such as +airlines), and their stages in the pandemic. +Related Work. Many recent efforts have attempted to use ma- +chine learning methods for spatio-temporal estimation tasks. +For example, [2] uses a conditional generative adversarial +network (cGAN) to estimate traffic volume. Similarly, a re- +cent work [3] proposed a COVID-GAN for human mobility +arXiv:2301.08648v1 [cs.LG] 20 Jan 2023 + +estimation, where policies (e.g., school closure) are used as +constraints to help improve estimation results. However, both +of them only consider estimation in a single city, without +modeling the spatial and temporal dependencies across cities +or stages. Therefore, they lack the ability to quickly adapt to +unseen cities. In addition, COVID-GAN is a purely spatial +model, which does not explicitly model the dynamics of hu- +man mobility responses over time. A POI embedding transfer +learning approach is proposed [4] to predict urban traffic +from one city to another city. This approach adapts model +parameters without using meta-learning method. In recent +years, many meta-learning approaches have been proposed +to solve the few-shot learning problem [5], [6], where the +training samples are limited for new tasks. However, few +of them are designed for spatio-temporal tasks. Among the +exceptions, [7] uses a model-based meta-learning approach +with a variational autoencoder structure to generate traffic +volume. Another traffic prediction work [6] combines the +attention mechanism and CNNs, and uses functionality zones +to group cities into tasks before applying the model-agnostic +meta-learning (MAML [8]) framework. However, this model +is designed with no time-based tasks, which is insufficient to +model the continued and dynamic changes in our problem. +Moreover, it does not model and utilize the spatio-temporal +dependencies and correlations across different tasks. +Proposed Work. To address the limitations of prior works, +we formulate the problem as a meta-learning-based conditional +data generation problem, where each task of the meta-learning +framework is to estimate a time-series of human mobility +response maps for a specific city during a specific time +period under designated conditions. As a solution, we design a +Spatio-TempORal (conditional) Meta-Generative Adversarial +Network (STORM-GAN). Building on top of a conditional +GAN (cGAN) model [9], the STORM-GAN model learns to +generate spatio-temporal mobility dynamics in different cities +under a set of geographic, epidemic, social and other factors. +It utilizes a meta-learning paradigm to learn a general model +initialization from a distribution of tasks (i.e., mobility estima- +tion for each city over a time period) for fast adaption to new +spatio-tempral tasks (e.g., new cities, future projection). To +explicitly model the spatio-temporal relationships across tasks, +we propose a Spatio-Temporal Task-based Graph (STTG) +embedding method for better model generalization and adap- +tation, which further improves STORM-GAN’s performance. +Overall, the contributions of this paper are as follows: +• We formulate the cross-city human mobility response es- +timation problem as a spatio-temporal meta-learning-based +data generation problem. To the best of our knowledge, this +is the first attempt to estimate human mobility through a +deep meta-generative framework. The proposed novel meta- +generative framework models the uncertainty, spatial and +temporal patterns simultaneously. +• Specifically, we propose a COVID-19 spatio-temporal task- +based graph, which is embedded into the framework to +explicitly model spatio-temporal dependency among dif- +ferent tasks, further improving the learning of the shared- +knowledge. +• We perform various experiments on real-world datasets to +evaluate the performance of the proposed approach under +different scenarios, and the results show that STORM-GAN +can greatly improve mobility response estimation compared +to other candidate approaches. We have released our code +and the sample data in a temporary GitHub link.1 +II. PROBLEM STATEMENT +This section introduces a set of basic concepts about our +data modeling, and then provides a formal problem statement. +The overall solution framework is shown in Fig. 1. +A. Basic Concepts +Definition 1. Spatial grid S is a grid-discretization of a +spatial field (e.g., a city), where each grid cell si represents an +equally-sized squared area. Given S, the location of any POI +can be mapped into a grid cell. For simplicity, in this work +we choose the grid cells to be 1km × 1km. +Definition 2. Temporal period T is a temporal period (e.g., +a 7-day window) containing equal-length slots (e.g., a day), +denoted as T = {t1, t2, ..., tn}, where each slot t represents +the finest temporal resolution of the data. +Definition 3. Mobility related conditions: All conditions that +will influence human mobility responses are mobility related +conditions including contextual conditions (e.g., population, +household income), epidemic conditions (e.g., COVID-19 con- +firmed cases and deaths), and policy conditions (e.g., strict +stay-at-home or shelter-in-place orders). We denote a list of k +conditions as F = {f1, f2, ..., fk}. For a grid cell s, we denote +f s,t as all the conditions of s in time slot t. +Definition 4. Human mobility responses: The human mobility +responses M is a two-dimensional tensor, representing the total +number of visits to POIs (e.g., grocery stores, hardware stores, +restaurants, gas stations) in each grid cell s for time slot t. +Note here we use POI visit counts simply as an example to +demonstrate the solution framework. Other mobility measures +(e.g., median home dwell time) can also be used with our +model. The choice of the measure is not the focus of this +paper. +Definition 5. Generator G: A deep neural network model that +is used to generate a series of human mobility response maps +M′ +G given a set of conditions. +Definition 6. Discriminator D: A deep neural network model +that outputs a probability preal at which a map of human +mobility responses is classified as from real-world rather than +from a generator G. +Definition 7. Spatio-temporal mobility estimation tasks: A +task Ti consists of a series of pairs (Mt, Ft) for a few +consecutive time periods T (e.g., 5 weeks) in a partitioned +1https://github.com/BaoHan88/STROM-GAN.git + +area S (e.g., 10 × 10 grids of a city). Each sample is a 4D +tensor with size l × l × k × |T|, where l × l is the size of the +spatial window, and k is the number of conditions. Each Ti is +divided into a training set Dtrain +i +and a testing set Dtest +i +. +B. Problem Definition +We construct the tasks (Def. 7) by a spatio-temporal parti- +tion of all conditions F and mobility responses M. Each spatio- +temporal task Ti contains data from the grid S of a single +city for w consecutive time periods {T1, ..., Tw}, and tasks +are mutually exclusively (i.e., no overlap along the temporal +dimension). Each data sample in a task contains a time-series +of length |T| with any start time (but the time-span of a sample +must be completely within the span of a task). +Inputs: +• A time-series of conditions {Ft−|T |+1, ..., Ft} for each data +sample in training tasks; +• Mobility response Mt for each sample in training tasks; +Outputs: +• A generator G to generate/estimate mobility responses; +• A meta-initialization θ for G for fast adaptation to training +and new testing tasks. +Objective: +• Minimize average generation error on new testing tasks. +In this work, given a series of tasks sampled from multiple +cities (e.g., Boston and NYC), we train a meta-generative +model. When a new city (e.g., Houston) comes in with a small +training set, we quickly fine-tune the meta-model parameters +to obtain a tailored model for the new city to generate its +mobility responses. +III. METHODOLOGY +In this section, we present the details of STORM-GAN. We +first present the architecture of the proposed spatio-temporal +meta-generative model. Then, we show the details of meta- +parameter updates in STROM-GAN. The model architecture +is shown in Fig. 2. +A. STORM-GAN with Spatio-Temporal Task-based graph- +embedding. +We now provide details for the three components in our pro- +posed spatio-temporal meta-generative model: spatio-temporal +generator, discriminator and spatio-temporal task-based graph +embedding. +1) Spatio-Temporal Generator: The spatio-temporal gener- +ator aims to generate human mobility responses while cap- +turing the spatial patterns and temporal dependencies. The +utilization of GAN structure allows known factors be learned +as conditions and unknown factors be represented by latent +noise which help the model to express these uncertainties +(i.e., mobility response estimations may have some degree +of variations). As shown in Fig. 2, the generator uses a +stack of CNN and LSTM elements where CNN captures local +spatial patterns and maintains the spatial representation (e.g., +neighbor relationships), and LSTM is able to capture temporal +trends in a given sequence. The generator takes a condition +tensor F ∈ Rl×l×k×|T | (we skip the batch dimension here for +simplicity) and a latent code tensor U ∈ Rl×l×u×|T |, where k +is the number of conditions (e.g., policy, COVID statistics and +contextual conditions), u is the dimension of the noise vector +for modeling the uncertainties, and |T| is the length of a time +period. +In the generator, denote the CNN output as ˆr ∈ Rd×|T |, +where d is the number of output features. Next, to capture +temporal patterns and trends, ˆr is fed into a LSTM layer, where +the memory vector is concatenated to ˆr. Then, the output from +the last timestamp of the LSTM layer ˆrh will be concatenated +with the graph embedding, and further passes through a fully +connected layer to generate the final output. This ˆrh is not +yet the estimated mobility response M′ +G. For more robust +estimation, the spatio-temporal generator additionally uses +a proposed spatio-temporal task-based graph embedding to +characterize task-level spatio-temporal features and potential +dependency across multiple cities and their mobility patterns, +as discussed in the next section. +2) Spatio-Temporal Task-based Graph (STTG) Embedding: +In real-world scenarios, spatial meta-learning tasks may have +a very diverse distribution. For example, in our problem, tasks +sampled from multiple cities can have significantly different +human mobility patterns due to different urban contexts. +Meanwhile, there may also exist underlying dependencies +among cities due to traffic connections, geo-socio similarities, +etc. Such spatial distribution of tasks, if properly utilized, +would greatly enhance the performance of the learned meta- +learning model. +To better model heterogeneity and dependency across +spatio-temporal tasks, we propose a novel spatio-temporal +task-based graph (STTG) to incorporate such information +and facilitate the learning of transferable knowledge from +related tasks. In the following part, we will first introduce +the construction rules of STTG, and then discuss STTG-based +embedding learning. +The STTG in our proposed STORM-GAN framework is +a directed weighted graph G (V, E), where nodes represent +the spatial locations of tasks (e.g., cities) and edges (and +weights) represent the relevance among spatial locations. The +graph is attributed, meaning that the nodes are associated with +attributes f(νi) to describe the characteristics of each spatial +location in the task space. +STTG can be defined in various ways depending on the +underlying analysis goal and the network data used. In our +particular application, we define each node νi as a major +metropolitan area in the U.S, which contains features f(νi) of +the city such as the current stage of the pandemic. Each edge +eij connecting cities νi and νj indicates that there is geo- +socio similarity between νi and νj in the pandemic, where +the edge weight represents the strengths of such similarity. +Depending on how “similarity” is measured, we can define the +edge and weights differently. Examples of such measures may +include the infection spreading between cities [10], correlation +between cities’ mobility patterns, etc. + +Fig. 1: Overall framework. +Fig. 2: STORM-GAN architecture. +In this paper, we present two examples of STTG con- +struction cases, although other definitions can be used with +our method as well. In the first case, we define the edges +and their weights based on physical reachability, i.e., the +number of direct flights and driving distance between cities, +with the assumptions that the COVID spreading is tightly +related to traveling and that cities with stronger transportation +connections tend to have more relevance in COVID situation. +In the second case, we define the edges based on the similarity +of historical mobility pattern distribution measured by the +Kullback–Leibler (KL) divergence [11] between cities. We +provide details on the STTG construction in Sec. IV-D, and +show effectiveness in Sec. IV-E. +Next, we use the built STTG in the meta-training phase +to help learn more useful knowledge across tasks. As Fig. +2 shows, during the training on generator, we first sample a +task-specific 1-hop subgraph H for the corresponding node (a +city) on the STTG. Then, we obtain a sub-graph embedding +using Variational Graph Autoencoder (VGAE) which consists +of Graph Convolution Neural network (GCNs) [12] by solving: +f(XL, A) = α( ˆD− 1 +2 ˆA ˆD− 1 +2 XL−1W L−1), +(1) +where A is the adjacency matrix, ˆA = A + I, I is the identity +matrix, ˆD is the diagonal node degree matrix of ˆA, α(·) is +a activation function (e.g., ReLU), X is the feature matrix +of each node from the graph, and W L−1 is a weight matrix +for the L − 1th layer. The encoder takes A and X as inputs +and generates the latent variable Z as output. The decoder +reconstructs a adjacency matrix defined by the inner product +between latent variable Z. +The graph feature representation Z is concatenated with the +output ˆrh (Fig. 2), and flows through a final fully connected +layer in the spatio-temporal generator to achieve M′ +G. The new +STTG and GCN-based embedding, being part of the genera- +tion process, will also help the meta-learner to incorporate the +similarity and dependency among tasks. +3) Spatio-Temporal Discriminator: Fig. 2 shows the struc- +ture of the discriminator, which takes a tensor of size +Rl×l×(k+1)×|T |, where k is the number of conditions (same + +Mobility related conditions +Time-Period T +City 1 +A Task +City 2 +& +A Sample +Graph +Mobility Responses +Training +Estimated Mobility +One Batch +New City +STTG +Responses +City 1 +City 2 +City 3 +Dtrain +new city +Dtrain +Dtrain +fe3(G, D): +Initial Meta- +fe1(G, D) +fe2(G, D) +国用 +Parameters +由围 +Fine-tuning +Adapt +Adapt +Adapt +new city +Spatio-Temporal +fe(G,D) +Dtest +Dfest +Graph Representation +Adapt +fenew city(G) +Update fe(G, D)CNN +CNN +CNN +LSTM +rh +GCN +FC +STTGA +Embed +concat. +CNN +CNN +CNN +LSTM +Reshape + FC + Sigmoidas that for generator) and the added one dimension is for the +mobility response layer. +To create training data for “fake” or “real” labels, the +input tensors are created in three ways: (1) generated mobility +M′ +G concatenated with conditions; (2) real mobility Mreal +concatenated with corresponding conditions; (3) conditions +concatenated with mismatched real mobility Mmismatch. Only +samples from the second combination are labeled “real”. Using +these inputs, the discriminator learns to determine whether an +input is “real” or “fake”. +STORM-GAN training on a single city is performed through +adversarial configuration between the generator and discrimi- +nator. A min-max objective function is used to train G and D +jointly by solving: +LG,D = EM∼Pdata[log D(M, F)] ++ EU∼PU [log(1 − D(G(F, U, STTG), M))] +(2) +where LG,D is the binary cross-entropy loss. +B. STORM-GAN Training and Testing +1) MAML-based Outer Loop Updates: As defined in Sec. +I, our goal is to learn the shared knowledge or initialization +across tasks drawn from multiple cities. To transfer the struc- +tural knowledge from graph and spatio-temporal knowledge +from mobility data in multiple cities, we adopt the model- +agnostic meta-learning (MAML) framework to learn the meta +parameter θD and θG, specifically in our case for all spatio- +temporal tasks. The learned initialization is expected to contain +common knowledge that can be fast-adapted to new tasks. +With MAML, we sample a batch of tasks in each step, where +each task Ti consists of (F, M) and their corresponding one- +hop subgraph in STTG. The general optimization formulation +is as follows. Given a set of tasks {T1, T2, ...} drawn from a +task distribution p (T ), where each task Ti ∼ p(T ) consists +of a training and a test set {Dtrain +i +, Dtest +i +}, we optimize the G +and D with parameters θG and θD to minimize the expected +empirical loss across all tasks during meta-training. The meta- +update rules are given by: +θD = θD − β∇θDLG,D(fθ′ +D) +(3) +θG = θG − β∇θGLG,D(fθ′ +G) +(4) +where β is the learning rate for meta-update, and θ′ +G and +θ′ +D represent temporary task-specific parameters. Following +the recommendation in [8], we use the first-order MAML for +meta-weight update. +2) STORM-GAN Inner Loop Updates: The detailed meta- +training procedure is shown in Alg. 1. The training of dis- +criminator uses the three types of combinations: (M′ +G, F), +(Mreal, F) and (Mmismatch, F). Denote α as the learning rate +of discriminator, θ′ +D as the parameters of discriminator, the +loss function and the update rule of D are shown in Eq. (5) +and Eq. (6), respectively. +fD = − 1 +m +m +� +i=1 +� +log(1 − D((M′ +G)i, Fi)) + log(D(Mi +real, Fi)) ++ log(1 − D(Mi +mismatch, Fi)) +� +(5) +Fig. 3: POI visit counts data pre-processing. +θ′ +D = θ′ +D − α∇fD(θ′ +D) +(6) +where m is the total number of samples in a batch, and index +i refers to the ith sample. Denote θ′ +G as the parameters in G, +we have the loss function and update rule of G as: +fG = 1 +m +m +� +i=1 +� +log(1 − D((M′ +G)i, Fi)) +� += 1 +m +m +� +i=1 +� +log(1 − D(G(Fi, Ui, STTG), Fi)) +� +(7) +θ′ +G = θ′ +G + α∇fG(θ′ +G) +(8) +3) STORM-GAN Adaptation on New Tasks: During the +model adaptation phase (e.g., updating the optimal initializa- +tion for a new task from a new city), we first copy θG and +θD from the meta-training phase as the initialization for fast- +adaptation, and then use training samples from the new task +to perform STORM-GAN for updating the meta-parameter +θD and θG. Finally, the updated model outputs the estimated +mobility using testing samples. The tasks used for meta-testing +adaptation are held out from meta-training. +IV. EVALUATION +A. Dataset Description +Data Sources. We elaborate four types of data as described in +Def. 3 & Def. 4 (pandemic, contextual, policy, and mobility). +They are collected from Centers for Disease Control and +Prevention [13], Census Bureau [14], the date of disease +prevention policies were collected from the corresponding city +government website news, and SafeGraph [15], respectively. +SafeGraph provides free access to data for academic purposes +with upon request and all the other data are publicly available. +Data Granularity. The original POI dataset from SafeGraph +is obtained by collecting the location from cell phone records +with latitude and longitude information. Then, the location +information is used to determine the visits to POIs [15]. The +POI visit counts data is in point data format. Fig. 3 illustrates +the discretization of one city, where we sum the total POI visit +counts that fall into each grid cell, and use this aggregated +visit counts value to represents the human mobility responses +of each grid cell. +Data Pre-processing. To construct the list of conditions for +our input, for each grid cell, we preprocess data collected from +different sources with different geographic units. We first adopt +a commonly used space-partitioning method to segment each +spatial domain into grid cells of size of 1km × 1km, and +segment all mobility related conditions using the same grid +cells. Then, each spatial region (or unit spatial window) we +used to create a data sample is a 10 × 10 spatial window + +. +Walam +Bosibn +MeitdnAlgorithm 1: STORM-GAN Training and Testing +Require: +• Set of training cities Ttrain; set of testing cities Ttest +• Conditions F, mobility Mreal, a STTG G +• Inner learning rate α; outer learning rate β; number of +epochs epoch +Ensure: θG, θG, estimated mobility M ′ +G for Ttest +{# Meta-learning on training cities} +1: G = initG(); D = initD() +2: Randomly initialize meta θG, θG +3: for e = 1 to epoch do +4: +Sample a batch of T from Ttrain +5: +Sample the subgragh H of T from G +6: +for Ti in {F, Mreal,H} do +7: +Sample a set of disjoint Dtrain +i +, Dtest +i +8: +Generate graph embedding E of H +9: +M′ +G = G(F, E, rand(PU)) +10: +Update D using Dtrain +i +by Eqs. (5) and (6) +11: +Update G using Dtrain +i +by Eqs. (7) and (8) +12: +Evaluate estimation loss using Dtest +i +by Eq. (2) +13: +end for +14: +Update θD and θG by Eq. (3) and Eq. (4) +15: end for +16: Return θG, θD +{# Fast-adaptation on testing cities} +17: Sample batch of testing tasks T from Ttest +18: for Ti in {F, Mreal,H} do +19: +Sample a disjoint Dtrain +i +, Dtest +i +from Ttest +20: +Generate graph embedding E of H +21: +Copy θG, θD +22: +Evaluate performance by Eq. (2) using Dtrain +i +23: +Update G through Eqs. (7) and (8) +24: +Estimate M′ +G using updated G and Dtest +i +25: end for +on the grid. For each grid cell, the value of human mobility +response is the total number of POI visit counts in a day. +Note that some conditions are re-scaled during this process. +For example, population and median household income data +are collected at the census tract level, and we linearly re-scaled +the data using the corresponding area ratios between the area +of the original census tract polygon and the proposed 10 × 10 +grid cells. Similarly, COVID-19 statistics and policy data are +collected at the county level. We assign each grid cell with +the corresponding data on which county it belongs. +Training Data Description. We collect the mobility related +datasets from six cities. The dataset spans over six cities in +different states located from the west coast, midwest to the east +coast (i.e., Boston, Chicago, Houston, Iowa City, Los Angeles, +and NYC). The list of cities also covers regions from large +metropolitan areas to less populous places. Detailed statistics +of these datasets (e.g., number of POIs, number of cells +covered for each city) are listed in Table I and Table II. The +duration of data is from 02/24/2020 to 10/25/2020, covering 35 +weeks in total. As discussed in Methodology section, the data +is segmented into a spatio-temporal distribution of tasks, where +each task contains one single city for five consecutive weeks +(no mutual overlaps among tasks). The candidate methods +are trained on five cities (meta-training) with one left out +as the new city for meta-testing. Adaptation on test cities is +performed with data samples from the most recent two weeks +(out of 35 weeks in total). Overall, we have 35 spatio-temporal +estimation tasks in total. For methods with meta-learning, 80% +of data in each task is used for meta-training, and the rest for +testing (Def. 7). +TABLE I: Detailed data statistics. +City +Time Span (M/D/Y) +POIs +Size +Boston +02/24/2020 +-10/25/2020 +35 Weeks +26054 +37 × 48 +NYC +133520 +58 × 72 +LA +86721 +52 × 64 +Chicago +47356 +50 × 40 +Houston +37315 +50 × 60 +Iowa City +1401 +20 × 32 +TABLE II: Average number of POI visit counts per grid cell +City +Boston +NYC +LA +Chicago +Houston +Iowa City +POI Counts +28 +64 +52 +30 +24 +6 +B. Evaluation Metrics +We evaluate the performance of STORM-GAN by using the +following measures: mean absolute error (MAE) and rooted +mean square error (RMSE). +MAE = 1 +n +n +� +i=0 +���MG − +ˆ +MG +��� +(9) +RMSE = +� +� +� +� 1 +n +n +� +i=1 +� +MG − +ˆ +MG +�2 +(10) +where MG is the real mobility response and +ˆ +MG is the +generated mobility response values by candidate approach. +Since model generates the spatial unit windows multiple times +for each grid cell during estimation, the outputs of generator +are averaged before comparing with the ground truth. +To evaluate the model performance of learning the data +distribution, we calculate the KL divergence to indicate the +similarity between the learned human mobility responses dis- +tribution ˆP and real human mobility responses distribution P +on different bin sizes. The KL divergence is defined as follow: +DKL(P|| ˆP) = +N +� +i=1 +P(M ′ +G)log( P(M ′ +G) +ˆP(Mreal) +) +(11) +C. Baseline Methods +We compare our proposed method with the following base- +line methods, and fine-tune each method using Houston and +Iowa City as testing cities respectively. + +• HA: Historical Average. The average of human mobility +responses calculated using observed values from the same +location in the past two weeks (same weekday). +• Spatial smoothing with neighborhood regions [16]. This +method uses the mobility response values in a local 3 × 3 +window to compute a mean as the estimated value. The +values for smoothing are from the same weekday in the +most recent week. +• Ridge [17]. We use ridge regression with the same input +features and mobility responses. +• cGAN [2]. A conditional GAN where the generator and +discriminator use three fully-connected layers (no layer +structure to learn spatial or temporal patterns). +• COVID-GAN [3]: COVID-GAN has the same structure as +the above cGAN, and it adds a correction layer, which is +used to add constraints based on policy to refine the results. +• MAML-DAWSON +[5]: +An +optimization-based +meta- +learning approach using MAML. As DAWSON originally +works on music generation tasks, we modify its inner +structure with a regression-focused conditional GAN. +• MetaST [6]: MetaST fuses CNN, LSTM and attention +mechanism to predict urban traffic volume through MAML +framework. +D. STTG Construction Examples +In this section, we provide two different STTG construction +scenarios to evaluate the effectiveness of graph embedding in +human mobility estimation. +Scenario 1 (S1). We assume that cities of similar sizes, +socio-economic environment, and land-use design may share +similar human mobility pattern that could help the estimation +of new cities. To build the graph Gs1 (V, E), we enumerate +major metropolitan cities from every region in U.S., and define +each city as a node νi. Next, we extract human mobility +maps for all the cities from a same date, and calculate the +pairwise distribution similarity score between cities using KL- +divergence. The KL-divergence indicates the strength of hu- +man mobility correlation. Each edge is added if the correlation +is ≦ 0.5, and is weighted by the correlation. Gs1 contains 55 +nodes and 682 edges. Node attribute store the outbreak stage +of COVID-19. Each stage value is in {1, 2, 3}, where a smaller +value means earlier in COVID-19 outbreak. The stage value +is assigned based on the month when an exponential growth +is first appeared. +Scenario 2 (S2). Intuitively, urban environment increases the +chance of infection as people move around and interact with +others and the environment. As a hub for migration and travel, +urban areas may quickly spread infections to nearby places +through short-distance travel, and to major cities through +connection flights. +To construct S2, similar to S1, we enumerate major +metropolitan areas in U.S, and define a graph Gs2 (V, E) to +represent the relationships of these cities. We divide the nodes +V into two categories: the hub nodes Vh are major cities with +more than 100 airlines; the second-tier nodes Vs are cities +with more than 35 but less than 100 airlines. Moreover, each +Fig. 4: Subgraph of a node in STTG S2. +directed edge νi → νj ∈ E is added if its two nodes are: +(1) both major cities that have direct flights; or (2) within a +spatial proximity threshold (500 km in this paper). Our graph +contains 69 nodes and 776 edges. +The graph is then weighted by spatio-temporal attributes +associated with nodes and edges. Edge attributes contain +the number of directed flights between the cities and their +geographic distance. Node attributes store the sum of flights +from connected edges as well as the outbreak stage of COVID- +19 which is the same as scenario 1. Fig. 4 shows an illustration +example of our S2 which is a 1-hop subgraph for Kansas City, +a second-tier city by the above-mentioned classification. Major +cities that have direct flights to Kansas City (e.g., Denver, +Atlanta, Minneapolis) and second-tier cities (e.g., Oklahoma, +Omaha) within the spatial proximity threshold are shown on +the subgraph. +We use both of the two STTG definitions with our STORM- +GAN (namely, STORM-GAN(S1) and STORM-GAN(S2)) to +evaluate its performance in the next subsection. +E. Estimation Quality Evaluation +The benefit of meta-learning is that the model can quickly +update the model parameters, and generate good results on +a new task by seeing a small fraction of new data. During +the adaptation phase, we evaluate the performance of the +candidate methods on the two test cities (i.e., Houston and +Iowa City) using their last two weeks (Monday to Sunday) of +data. The length of a time period we use is 7-day since human +mobility pattern is influenced by strong weekly periodicity. +For each testing city and week, we use 2 consecutive weeks +of data ahead of the week for adaptation, and then use the +parameters to generate the next 7-day (one week) human +mobility responses. +Performance comparison of proposed STORM-GAN +and other candidate methods. Tables III and IV show +the results of the candidate methods obtained using Houston +and Iowa City as the testing city, respectively. We apply +S1 and S2 graph construction scenarios on STROM-GAN, +respectively. The evaluation results show that both STORM- +GAN scenarios overall achieve the lowest RMSE and MAE +for each day in the week, with major improvements from +91.7% to 17.6%. It is interesting to observe that historical +average and spatial smoothing methods perform better than +the basic cGAN, which to some degree shows the spatio- +temporal auto-correlation effects. However, these methods + +Minneapolis +Chicago +Portland +Tulsa +Oklahoma +Detroit +LOS +Kansas ( +it. +Omaha +Angeles +Hub +Second-tier +Denver +Atlanta +Edge +DallasTABLE III: Human mobility responses estimation by candidate methods for Houston +RMSE +MAE +Model +Mon +Tues +Wed +Thu +Fri +Sat +Sun +Mon +Tues +Wed +Thu +Fri +Sat +Sun +HA +194.8 +193.2 +193.5 +196.3 +193.6 +194.5 +195.1 +81.2 +80.9 +80.4 +80.8 +79.6 +81.3 +81.2 +Smoothing +150.9 +168.1 +169.2 +177.1 +187.3 +202.4 +162.1 +82.3 +90.2 +90.4 +94.6 +100.1 +99.8 +104.2 +cGAN +278.2 +283.4 +284.6 +279.5 +286.3 +286.1 +286.2 +118.7 +122.4 +125.3 +120.6 +128.3 +130.6 +129.3 +Ridge +189.4 +192.1 +182.3 +181.5 +187.3 +188.6 +195.7 +95.8 +99.5 +101.9 +95.3 +99.4 +98.1 +95.2 +COVID-GAN +171.5 +175.6 +174.3 +172.2 +176.4 +171.8 +170.6 +75.1 +80.1 +73.2 +77.1 +80.5 +81.7 +82.8 +MAML-DAWSON +169.5 +169.7 +166.1 +168.9 +164.6 +166.7 +168.1 +68.3 +67.4 +70.3 +69.4 +68.2 +75.5 +78.2 +MetaST +170.2 +171.4 +173.8 +170.6 +169.5 +170.3 +169.4 +72.8 +76.1 +71.5 +74.2 +72.9 +80.2 +81.3 +STROM-GAN (S1) +151.2 +150.5 +149.3 +162.9 +163.9 +164.2 +167.2 +66.7 +66.4 +64.7 +63.1 +70.4 +71.8 +72.2 +STORM-GAN (S2) +145.1 +142.6 +141.9 +141.6 +152.5 +156.7 +160.2 +61.7 +60.4 +59.3 +53.8 +58.4 +64.2 +67.2 +TABLE IV: Human mobility responses estimation by candidate methods for Iowa City +RMSE +MAE +Model +Mon +Tues +Wed +Thu +Fri +Sat +Sun +Mon +Tues +Wed +Thu +Fri +Sat +Sun +HA +22.2 +23.1 +21.3 +24.2 +22.3 +25.2 +26.5 +13.4 +13.1 +15.2 +12.6 +14.2 +16.1 +13.2 +Smoothing +18.4 +16.3 +18.6 +21.7 +18.3 +19.2 +19.5 +11.2 +10.3 +10.6 +9.1 +11.6 +9.2 +11.6 +cGAN +34.3 +33.7 +34.8 +36.5 +32.8 +31.2 +34.4 +21.1 +19.5 +21.3 +18.7 +17.8 +19.9 +20.3 +Ridge +20.6 +22.3 +21.1 +20.5 +19.8 +19.2 +20.1 +12.4 +11.6 +13.3 +13.6 +12.2 +13.5 +12.2 +COVID-GAN +17.1 +17.3 +16.6 +16.3 +15.2 +14.3 +15.6 +11.3 +10.7 +13.1 +12.3 +14.7 +13.8 +13.3 +MAML-DAWSON +16.5 +17.6 +17.7 +15.2 +15.3 +13.6 +13.4 +10.3 +11.6 +10.1 +9.5 +8.1 +10.9 +9.2 +MetaST +17.1 +17.4 +17.8 +17.6 +16.5 +16.3 +16.1 +10.8 +10.6 +10.4 +10.5 +9.9 +10.1 +9.7 +STROM-GAN (S1) +15.8 +16.2 +16.1 +15.8 +15.9 +15.4 +14.3 +8.9 +9.6 +9.9 +10.2 +9.1 +9.3 +9.3 +STORM-GAN (S2) +14.1 +15.6 +14.9 +14.4 +14.2 +13.6 +13.3 +8.2 +7.4 +9.1 +8.3 +7.8 +9.1 +8.5 +can mainly estimate a rough base but are limited in captur- +ing complex spatio-temporal relationships between features +and mobility responses. Comparing to COVID-GAN and +MAML-DAWSON, our model outperforms COVID-GAN by +20.5% (RMSE) and 23.5% (MAE) on average, and MAML- +DAWSON by 15.1% (RMSE) and 23.4% (MSE). The results +show that the design of spatio-temporal architecture (i.e., +CNN and LSTM substructures and the STTG graph) and +meta-learning adaptation can significantly improve the solution +quality. Furthermore, our model achieves 14.7% (RMSE) and +15.2% (MSE) better than MetaST, demonstrating that task- +based graph embedding can contribute to model performance +by learning the inter-task similarities. We evaluate the model +performance on less populous areas using Iowa City as testing +city, the POI numbers and city size of Iowa City are sig- +nificantly smaller than large metropolitan areas according to +Tables I and II. As Table IV shows, the improvements are +relatively smaller due to smaller number of POI visit counts +in less populated city. However, both STORM-GAN scenarios +still achieve the lowest errors in all of the testing days. +We calculate the KL-divergence using results from Houston +and Iowa City (Fig. 5). The X-axis represents the number +of equal-size bins used to discretize the value needed for +the computation, and the Y-axis shows the KL divergence +values. A lower KL divergence value means the result better +matches the real distribution. As shown in Fig. 5, STORM- +GAN achieves lowest KL-divergence compare to the baseline +methods consistently for all numbers of bins. +Impact of STTG choice: Our results show that both +of the two STTG constructed can significantly improve the +performance of STORM-GAN in Houston and Iowa City. This +proves that the spatio-temporal task-graph embedding design is +effective and robust, rather than tailored for a specific STTG +Fig. 5: Kullback-Leibler divergence. +definition. Between the two choices, S2 generally achieves +better performance as it uses more information that are directly +related to the spreading of COVID-19. +Ablation Study. We study the effect of different compo- +nents proposed in our method using Houston as the testing +city on one day (Monday). +• Base: Baseline conditional GAN. +• Base + Spatial (S): Equivalent to COVID-GAN, which has +a correction layer to add policy constraints, but purely a +spatial model. +• Base + Spatio-Temporal (ST) + Meta: Proposed STORM- +GAN with spatio-temporal meta-learning, but without the +STTG graph. +• Base + ST + Meta + Graph(S2): Complete STORM-GAN. +Table V shows the estimation performance of STORM- +GAN and its variants. First, the base + spatial (S) achieves +a lower RMSE and MAE (a reduction of 15.3% and 15.2%, +respectively) compared to cGAN, showing the effectiveness +of the correction layer from COVID-GAN. Next, we can + +Houston +Iowa City +0.6 +0.6 + Divergence +0.5 +0.5 +0.4 +0.4 +0.3 +TY +0.3 +0.2 +0.2 +0.1 +0.1 +0 +30 +60 +90 +120 +30 +60 +90 +120 +HA +cGAN +Smoothing +Ridge +-MAML-DAWSON +COVID-GAN +MetaST +OSTORM-GAN(S1) +→STORM-GAN(S2) +Number of binsFig. 6: Mobility estimation results of the Houston study area. +Fig. 7: Mobility estimation results of the Iowa City study area. +see that the addition of spatio-temporal meta-learning further +reduces RMSE and MAE by 12.7% and 10%, respectively. +This result demonstrates that CNN, LSTM and meta-learning +can better capture the complex spatio-temporal relationships +across multiple cities. Finally, the complete STROM-GAN +achieves the lowest RMSE and MAE with the sptaio-temporal +task-based graph. +Visualization. We compare the solution quality of seven +candidate approaches through map visualization. Fig. 6 and +Fig. 7 (a) to (f) show the results of baseline methods, and (g), +(h) show the STORM-GAN (S2) and ground truth. The results +TABLE V: Comparison among STORM-GAN variations +Method +RMSE +MAE +Base +202.2 +80.6 +Base + S +171.5 +75.1 +Base + ST + Meta +149.8 +67.6 +Base + ST + Meta + Graph (S2) +145.1 +61.7 +show the full Houston and Iowa City study areas for a day +in the data. Here STORM-GAN generates fine-scale mobility +values that are closer to the ground truth. As we can see, the +mobility pattern generated by the STORM-GAN can capture +the spatial pattern of human mobility responses better than +other baselines. The reason may be that similar functionality +zones at different cities may have similar mobility patterns. +The meta-learning framework successfully learns this shared +knowledge from training tasks. Moreover, the utilization of +CNN and LSTM helps capture the spatio-temporal correlation +from region to region. +V. OTHER RELATED WORK +Deep Learning for Spatio-Temporal Prediction. There +have been many deep learning based techniques developed for +spatiotemporal data. For example, LSTM were widely used in +traffic accident prediction [18] due to its capability in capturing +spatio-temporal correlation and thus provide good prediction +results. Geospatial object mapping [19], taxi driver behavior +imitation [20], taxi demand [21], travel time estimation [22], +etc, they all combine the deep learning model with spatio- +temporal perspective in their model design and obtain good +performance. Most of these techniques typically are stationary +predictors (i.e., same result from two runs on same data) rather +than generative models, and they do not consider the unknown +factors in prediction, and their performance relies on large data +sets. Besides, they do not leverage domain knowledge based +constraints to assist learning (e.g., cGAN [2], [3]). +Meta-Learning. Meta-learning learns new tasks quickly +and effectively with a few examples. Existing optimization- +based meta-learning algorithms such as MAML [8] and Rep- +tile [23] rely on optimization through gradient descent, and +both are compatible with any model. Recently, the idea of +optimization-based meta-learning has been applied to many +domains including classification and reinforcement learning. +However, there only are a few work address the spatial +and temporal problems simultaneously. In traffic prediction, +a recent work [7] focuses on knowledge transfer in a single +city, which only deals with temporal tasks with no spatial- +based tasks. [24] proposes a transfer learning framework for +traffic prediction through learning region matching function. +Another work [6] which is based on multiple cities does not +consider temporal patterns and dynamic scenarios. +Mobility Estimation. There have been many studies [1], +[25], [26] exploring the interplay between human mobility +responses, social distancing policies, and transmission dynam- +ics in response to the COVID-19 pandemic. A US mobility +change map was created in [25] to increase risk awareness +of the public and to visualize dynamic changes in mobility +as COVID-19 situation and policy evolves. These studies are +timely in showing the important role played by mobility in the +spread of COVID-19, but they do not address the challenges +in real-world mobility estimation/simulation (e.g., effects of +unknown, uncertain, and random factors), and they analyze +the mobility changes in city or country scale. A study [27] +simulated the human mobility which allows policymakers to + +inspect mobility changes under different policies. But this +approach utilizes a traditional epidemiological model, and +does not transfer the simulation from city to city by shared +knowledge. However, these studies have not explored the +potential use of deep learning based generative models and +meta-learning to assist the estimation. +VI. CONCLUSIONS +We made the first attempt to tackle the human mobility +estimation problem through a spatio-temporal meta-generative +framework. Specifically, we proposed a STORM-GAN model +to capture complex spatio-temporal patterns using a set of +social and policy conditions related to COVID-19. We also +proposed a novel spatio-temporal task-based graph (STTG) to +represent the spatio-temporal relationships among cities, with a +graph convolution network to learn embeddings of its subgraph +for cross-task learning enhancements. Finally, STORM-GAN +utilized the meta-learning paradigm to learn shared-knowledge +from a spatio-temporal distribution of estimation tasks and can +quickly adapt to new tasks (e.g., new cities). The experiment +results showed that our proposed approach can significantly +improve the estimation performance compared to baselines. +The model can assist policymakers to better understand the +dynamic mobility pattern changes under different social and +policy conditions, and can potentially be leveraged to inform +decisions in resource allocation and provisioning, event plan- +ning, response management, etc. +ACKNOWLEDGMENT +This paper is funded in part by Safety Research using +Simulation University Transportation Center (SAFER-SIM). +SAFER-SIM is funded by a grant from the U.S. Department +of Transportation’s University Transportation Centers Program +(69A3551747131). However, the U.S. Government assumes +no liability for the contents or use thereof. Yiqun Xie is +supported in part by NSF grants 2105133, 2126474, 2147195, +Google’s AI for Social Good Impact Scholars program, and +the DRI award at the University of Maryland; and Xiaowei +Jia is supported in part by NSF award 2147195, USGS award +G21AC10207, Pitt Momentum Funds award, and CRC at the +University of Pittsburgh. Yanhua Li was supported in part by +NSF grants IIS-1942680 (CAREER), CNS-1952085, CMMI- +1831140, and DGE-2021871. +REFERENCES +[1] M. U. Kraemer, C.-H. Yang, B. Gutierrez, C.-H. Wu, B. Klein, D. M. +Pigott, L. Du Plessis, N. R. Faria, R. Li, W. P. Hanage et al., “The effect +of human mobility and control measures on the covid-19 epidemic in +china,” Science, vol. 368, no. 6490, pp. 493–497, 2020. +[2] Y. Zhang, Y. Li, X. Zhou, X. Kong, and J. Luo, “Trafficgan: Off- +deployment traffic estimation with traffic generative adversarial net- +works,” in 2019 IEEE International Conference on Data Mining +(ICDM). +IEEE, 2019, pp. 1474–1479. +[3] H. Bao, X. Zhou, Y. 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Grusky et al., +“Supporting covid-19 policy response with large-scale mobility-based +modeling,” in Proceedings of the 27th ACM SIGKDD Conference on +Knowledge Discovery & Data Mining, 2021, pp. 2632–2642. + diff --git a/_NFAT4oBgHgl3EQfqh2x/content/tmp_files/load_file.txt b/_NFAT4oBgHgl3EQfqh2x/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..eb8a6333883d725c5f9cdc604b739b7433be106b --- /dev/null +++ b/_NFAT4oBgHgl3EQfqh2x/content/tmp_files/load_file.txt @@ -0,0 +1,979 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf,len=978 +page_content='STORM-GAN: Spatio-Temporal Meta-GAN for Cross-City Estimation of Human Mobility Responses to COVID-19 Han Bao, Xun Zhou* University of Iowa {han-bao, xun-zhou}@uiowa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='edu Yiqun Xie University of Maryland xie@umd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='edu Yanhua Li Worcester Polytechnic Institute yli15@wpi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='edu Xiaowei Jia University of Pittsburgh xiaowei@pitt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='edu Abstract—Human mobility estimation is crucial during the COVID-19 pandemic due to its significant guidance for pol- icymakers to make non-pharmaceutical interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' While deep learning approaches outperform conventional estimation techniques on tasks with abundant training data, the continuously evolving pandemic poses a significant challenge to solving this problem due to data nonstationarity, limited observations, and complex social contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Prior works on mobility estimation either focus on a single city or lack the ability to model the spatio-temporal dependencies across cities and time periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' To address these issues, we make the first attempt to tackle the cross-city human mobility estimation problem through a deep meta-generative framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' We propose a Spatio-Temporal Meta-Generative Adversarial Network (STORM-GAN) model that estimates dynamic human mobility responses under a set of social and policy conditions related to COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Facilitated by a novel spatio-temporal task-based graph (STTG) embedding, STORM-GAN is capable of learning shared knowledge from a spatio-temporal distribution of estimation tasks and quickly adapting to new cities and time periods with limited training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The STTG embedding component is designed to capture the similarities among cities to mitigate cross-task heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Experimental results on real-world data show that the proposed approach can greatly improve estimation performance and out- perform baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Index Terms—Meta-Learning, Generative Adversarial Net- works, Spatio-Temporal, Graph Embedding, COVID-19 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' INTRODUCTION Evolving developments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', spread, mutation, vaccination) around the COVID-19 pandemic have continued to pressure policymakers to come up with effective and changing policies that can protect public health while avoiding breakdowns of economics, and maintain the support of essential needs in daily lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' To mitigate this dilemma, staged reopening to avoid infections caused by eased social distancing policies have been implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' As a result, estimation on dynamic human mobility responses to the pandemic condition and policies remains a crucial task in policymaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Due to the asynchronous spread of the disease, it is par- ticularly hard for cities in the early stage of an outbreak or wave (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', the Omicron variant) to estimate future human mobility responses under unprecedented severity levels or unseen policies as there is very limited historical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' To Corresponding author address this issue, it is crucial for such cities to be able to leverage other cities’ past experiences and knowledge for its own estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' To this end, mobility response estimation methods that can leverage cross-city knowledge to achieve promising results are urgently needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' In this paper, we make the first attempt to solve the cross- city human mobility responses estimation problem: Given a set of inputs on contextual (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', population, point-of-interest POI counts), epidemic (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', COVID-19 cases), policy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', stay- at-home orders) conditions and corresponding human mobility responses measures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', POI visit counts, home dwell time) from multiple cities, we aim to learn a model, which can quickly adapt to previously unseen cities and time periods, and estimate human mobility response dynamics under any projected conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The cross-city human mobility response estima- tion problem has three major challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' First, human mobility responses depend on many complex social-physical factors which are unknown or uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' For example, responses can be affected by people’s willingness in cooperating with policies, decisions from service providers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='g, whether a restaurant will open or allow dine-in options), changes in public transportation, supply, and many more [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Second, human mobility responses often have spatio-temporal non- stationarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' For example, the contribution of different factors in mobility tends to vary from region to region due to cultural and economic differences, and can quickly evolve over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Such spatial and temporal nonstationarity greatly limits the availability of training data for each estimation task, making it difficult to leverage the approximation power of data-driven approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Third, there also exist complex spatial and temporal dependencies across different estimation tasks, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', cities and time periods, which need to be explicitly considered for robust parameter sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' For example, cities may share similar mobility dynamic patterns based on their spatial adjacency (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', distances, travel connections such as airlines), and their stages in the pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Related Work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Many recent efforts have attempted to use ma- chine learning methods for spatio-temporal estimation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' For example, [2] uses a conditional generative adversarial network (cGAN) to estimate traffic volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Similarly, a re- cent work [3] proposed a COVID-GAN for human mobility arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='08648v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='LG] 20 Jan 2023 estimation, where policies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', school closure) are used as constraints to help improve estimation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' However, both of them only consider estimation in a single city, without modeling the spatial and temporal dependencies across cities or stages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Therefore, they lack the ability to quickly adapt to unseen cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' In addition, COVID-GAN is a purely spatial model, which does not explicitly model the dynamics of hu- man mobility responses over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' A POI embedding transfer learning approach is proposed [4] to predict urban traffic from one city to another city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' This approach adapts model parameters without using meta-learning method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' In recent years, many meta-learning approaches have been proposed to solve the few-shot learning problem [5], [6], where the training samples are limited for new tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' However, few of them are designed for spatio-temporal tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Among the exceptions, [7] uses a model-based meta-learning approach with a variational autoencoder structure to generate traffic volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Another traffic prediction work [6] combines the attention mechanism and CNNs, and uses functionality zones to group cities into tasks before applying the model-agnostic meta-learning (MAML [8]) framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' However, this model is designed with no time-based tasks, which is insufficient to model the continued and dynamic changes in our problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Moreover, it does not model and utilize the spatio-temporal dependencies and correlations across different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Proposed Work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' To address the limitations of prior works, we formulate the problem as a meta-learning-based conditional data generation problem, where each task of the meta-learning framework is to estimate a time-series of human mobility response maps for a specific city during a specific time period under designated conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' As a solution, we design a Spatio-TempORal (conditional) Meta-Generative Adversarial Network (STORM-GAN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Building on top of a conditional GAN (cGAN) model [9], the STORM-GAN model learns to generate spatio-temporal mobility dynamics in different cities under a set of geographic, epidemic, social and other factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' It utilizes a meta-learning paradigm to learn a general model initialization from a distribution of tasks (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', mobility estima- tion for each city over a time period) for fast adaption to new spatio-tempral tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', new cities, future projection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' To explicitly model the spatio-temporal relationships across tasks, we propose a Spatio-Temporal Task-based Graph (STTG) embedding method for better model generalization and adap- tation, which further improves STORM-GAN’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Overall, the contributions of this paper are as follows: We formulate the cross-city human mobility response es- timation problem as a spatio-temporal meta-learning-based data generation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' To the best of our knowledge, this is the first attempt to estimate human mobility through a deep meta-generative framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The proposed novel meta- generative framework models the uncertainty, spatial and temporal patterns simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Specifically, we propose a COVID-19 spatio-temporal task- based graph, which is embedded into the framework to explicitly model spatio-temporal dependency among dif- ferent tasks, further improving the learning of the shared- knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' We perform various experiments on real-world datasets to evaluate the performance of the proposed approach under different scenarios, and the results show that STORM-GAN can greatly improve mobility response estimation compared to other candidate approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' We have released our code and the sample data in a temporary GitHub link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='1 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' PROBLEM STATEMENT This section introduces a set of basic concepts about our data modeling, and then provides a formal problem statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The overall solution framework is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Basic Concepts Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Spatial grid S is a grid-discretization of a spatial field (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', a city), where each grid cell si represents an equally-sized squared area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Given S, the location of any POI can be mapped into a grid cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' For simplicity, in this work we choose the grid cells to be 1km × 1km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Temporal period T is a temporal period (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', a 7-day window) containing equal-length slots (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', a day), denoted as T = {t1, t2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', tn}, where each slot t represents the finest temporal resolution of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Mobility related conditions: All conditions that will influence human mobility responses are mobility related conditions including contextual conditions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', population, household income), epidemic conditions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', COVID-19 con- firmed cases and deaths), and policy conditions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', strict stay-at-home or shelter-in-place orders).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' We denote a list of k conditions as F = {f1, f2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', fk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' For a grid cell s, we denote f s,t as all the conditions of s in time slot t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Human mobility responses: The human mobility responses M is a two-dimensional tensor, representing the total number of visits to POIs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', grocery stores, hardware stores, restaurants, gas stations) in each grid cell s for time slot t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Note here we use POI visit counts simply as an example to demonstrate the solution framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Other mobility measures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', median home dwell time) can also be used with our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The choice of the measure is not the focus of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Generator G: A deep neural network model that is used to generate a series of human mobility response maps M′ G given a set of conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Discriminator D: A deep neural network model that outputs a probability preal at which a map of human mobility responses is classified as from real-world rather than from a generator G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Spatio-temporal mobility estimation tasks: A task Ti consists of a series of pairs (Mt, Ft) for a few consecutive time periods T (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', 5 weeks) in a partitioned 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='com/BaoHan88/STROM-GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='git area S (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', 10 × 10 grids of a city).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Each sample is a 4D tensor with size l × l × k × |T|, where l × l is the size of the spatial window, and k is the number of conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Each Ti is divided into a training set Dtrain i and a testing set Dtest i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Problem Definition We construct the tasks (Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' 7) by a spatio-temporal parti- tion of all conditions F and mobility responses M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Each spatio- temporal task Ti contains data from the grid S of a single city for w consecutive time periods {T1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', Tw}, and tasks are mutually exclusively (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', no overlap along the temporal dimension).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Each data sample in a task contains a time-series of length |T| with any start time (but the time-span of a sample must be completely within the span of a task).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Inputs: A time-series of conditions {Ft−|T |+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', Ft} for each data sample in training tasks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Mobility response Mt for each sample in training tasks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Outputs: A generator G to generate/estimate mobility responses;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' A meta-initialization θ for G for fast adaptation to training and new testing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Objective: Minimize average generation error on new testing tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' In this work, given a series of tasks sampled from multiple cities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', Boston and NYC), we train a meta-generative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' When a new city (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', Houston) comes in with a small training set, we quickly fine-tune the meta-model parameters to obtain a tailored model for the new city to generate its mobility responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' METHODOLOGY In this section, we present the details of STORM-GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' We first present the architecture of the proposed spatio-temporal meta-generative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Then, we show the details of meta- parameter updates in STROM-GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The model architecture is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' STORM-GAN with Spatio-Temporal Task-based graph- embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' We now provide details for the three components in our pro- posed spatio-temporal meta-generative model: spatio-temporal generator, discriminator and spatio-temporal task-based graph embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' 1) Spatio-Temporal Generator: The spatio-temporal gener- ator aims to generate human mobility responses while cap- turing the spatial patterns and temporal dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The utilization of GAN structure allows known factors be learned as conditions and unknown factors be represented by latent noise which help the model to express these uncertainties (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', mobility response estimations may have some degree of variations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' 2, the generator uses a stack of CNN and LSTM elements where CNN captures local spatial patterns and maintains the spatial representation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', neighbor relationships), and LSTM is able to capture temporal trends in a given sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The generator takes a condition tensor F ∈ Rl×l×k×|T | (we skip the batch dimension here for simplicity) and a latent code tensor U ∈ Rl×l×u×|T |, where k is the number of conditions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', policy, COVID statistics and contextual conditions), u is the dimension of the noise vector for modeling the uncertainties, and |T| is the length of a time period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' In the generator, denote the CNN output as ˆr ∈ Rd×|T |, where d is the number of output features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Next, to capture temporal patterns and trends, ˆr is fed into a LSTM layer, where the memory vector is concatenated to ˆr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Then, the output from the last timestamp of the LSTM layer ˆrh will be concatenated with the graph embedding, and further passes through a fully connected layer to generate the final output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' This ˆrh is not yet the estimated mobility response M′ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' For more robust estimation, the spatio-temporal generator additionally uses a proposed spatio-temporal task-based graph embedding to characterize task-level spatio-temporal features and potential dependency across multiple cities and their mobility patterns, as discussed in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' 2) Spatio-Temporal Task-based Graph (STTG) Embedding: In real-world scenarios, spatial meta-learning tasks may have a very diverse distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' For example, in our problem, tasks sampled from multiple cities can have significantly different human mobility patterns due to different urban contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Meanwhile, there may also exist underlying dependencies among cities due to traffic connections, geo-socio similarities, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Such spatial distribution of tasks, if properly utilized, would greatly enhance the performance of the learned meta- learning model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' To better model heterogeneity and dependency across spatio-temporal tasks, we propose a novel spatio-temporal task-based graph (STTG) to incorporate such information and facilitate the learning of transferable knowledge from related tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' In the following part, we will first introduce the construction rules of STTG, and then discuss STTG-based embedding learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The STTG in our proposed STORM-GAN framework is a directed weighted graph G (V, E), where nodes represent the spatial locations of tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', cities) and edges (and weights) represent the relevance among spatial locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The graph is attributed, meaning that the nodes are associated with attributes f(νi) to describe the characteristics of each spatial location in the task space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' STTG can be defined in various ways depending on the underlying analysis goal and the network data used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' In our particular application, we define each node νi as a major metropolitan area in the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='S, which contains features f(νi) of the city such as the current stage of the pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Each edge eij connecting cities νi and νj indicates that there is geo- socio similarity between νi and νj in the pandemic, where the edge weight represents the strengths of such similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Depending on how “similarity” is measured, we can define the edge and weights differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Examples of such measures may include the infection spreading between cities [10], correlation between cities’ mobility patterns, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' 1: Overall framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' 2: STORM-GAN architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' In this paper, we present two examples of STTG con- struction cases, although other definitions can be used with our method as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' In the first case, we define the edges and their weights based on physical reachability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', the number of direct flights and driving distance between cities, with the assumptions that the COVID spreading is tightly related to traveling and that cities with stronger transportation connections tend to have more relevance in COVID situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' In the second case, we define the edges based on the similarity of historical mobility pattern distribution measured by the Kullback–Leibler (KL) divergence [11] between cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' We provide details on the STTG construction in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' IV-D, and show effectiveness in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' IV-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Next, we use the built STTG in the meta-training phase to help learn more useful knowledge across tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' As Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' 2 shows, during the training on generator, we first sample a task-specific 1-hop subgraph H for the corresponding node (a city) on the STTG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Then, we obtain a sub-graph embedding using Variational Graph Autoencoder (VGAE) which consists of Graph Convolution Neural network (GCNs) [12] by solving: f(XL, A) = α( ˆD− 1 2 ˆA ˆD− 1 2 XL−1W L−1), (1) where A is the adjacency matrix, ˆA = A + I, I is the identity matrix, ˆD is the diagonal node degree matrix of ˆA, α(·) is a activation function (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', ReLU), X is the feature matrix of each node from the graph, and W L−1 is a weight matrix for the L − 1th layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The encoder takes A and X as inputs and generates the latent variable Z as output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The decoder reconstructs a adjacency matrix defined by the inner product between latent variable Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The graph feature representation Z is concatenated with the output ˆrh (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' 2), and flows through a final fully connected layer in the spatio-temporal generator to achieve M′ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The new STTG and GCN-based embedding, being part of the genera- tion process, will also help the meta-learner to incorporate the similarity and dependency among tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' 3) Spatio-Temporal Discriminator: Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' 2 shows the struc- ture of the discriminator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' which takes a tensor of size Rl×l×(k+1)×|T |,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' where k is the number of conditions (same Mobility related conditions Time-Period T City 1 A Task City 2 & A Sample Graph Mobility Responses Training Estimated Mobility One Batch New City STTG Responses City 1 City 2 City 3 Dtrain new city Dtrain Dtrain fe3(G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' D): Initial Meta- fe1(G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' D) fe2(G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' D) 国用 Parameters 由围 Fine-tuning Adapt Adapt Adapt new city Spatio-Temporal fe(G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='D) Dtest Dfest Graph Representation Adapt fenew city(G) Update fe(G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' D)CNN CNN CNN LSTM rh GCN FC STTGA Embed concat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' CNN CNN CNN LSTM Reshape + FC + Sigmoidas that for generator) and the added one dimension is for the mobility response layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' To create training data for “fake” or “real” labels, the input tensors are created in three ways: (1) generated mobility M′ G concatenated with conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' (2) real mobility Mreal concatenated with corresponding conditions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' (3) conditions concatenated with mismatched real mobility Mmismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Only samples from the second combination are labeled “real”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Using these inputs, the discriminator learns to determine whether an input is “real” or “fake”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' STORM-GAN training on a single city is performed through adversarial configuration between the generator and discrimi- nator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' A min-max objective function is used to train G and D jointly by solving: LG,D = EM∼Pdata[log D(M, F)] + EU∼PU [log(1 − D(G(F, U, STTG), M))] (2) where LG,D is the binary cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' STORM-GAN Training and Testing 1) MAML-based Outer Loop Updates: As defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' I, our goal is to learn the shared knowledge or initialization across tasks drawn from multiple cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' To transfer the struc- tural knowledge from graph and spatio-temporal knowledge from mobility data in multiple cities, we adopt the model- agnostic meta-learning (MAML) framework to learn the meta parameter θD and θG, specifically in our case for all spatio- temporal tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The learned initialization is expected to contain common knowledge that can be fast-adapted to new tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' With MAML, we sample a batch of tasks in each step, where each task Ti consists of (F, M) and their corresponding one- hop subgraph in STTG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The general optimization formulation is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Given a set of tasks {T1, T2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='} drawn from a task distribution p (T ), where each task Ti ∼ p(T ) consists of a training and a test set {Dtrain i , Dtest i }, we optimize the G and D with parameters θG and θD to minimize the expected empirical loss across all tasks during meta-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The meta- update rules are given by: θD = θD − β∇θDLG,D(fθ′ D) (3) θG = θG − β∇θGLG,D(fθ′ G) (4) where β is the learning rate for meta-update, and θ′ G and θ′ D represent temporary task-specific parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Following the recommendation in [8], we use the first-order MAML for meta-weight update.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' 2) STORM-GAN Inner Loop Updates: The detailed meta- training procedure is shown in Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The training of dis- criminator uses the three types of combinations: (M′ G, F), (Mreal, F) and (Mmismatch, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Denote α as the learning rate of discriminator, θ′ D as the parameters of discriminator, the loss function and the update rule of D are shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' (5) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' (6), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' fD = − 1 m m � i=1 � log(1 − D((M′ G)i, Fi)) + log(D(Mi real, Fi)) + log(1 − D(Mi mismatch, Fi)) � (5) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' 3: POI visit counts data pre-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' θ′ D = θ′ D − α∇fD(θ′ D) (6) where m is the total number of samples in a batch, and index i refers to the ith sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Denote θ′ G as the parameters in G, we have the loss function and update rule of G as: fG = 1 m m � i=1 � log(1 − D((M′ G)i, Fi)) � = 1 m m � i=1 � log(1 − D(G(Fi, Ui, STTG), Fi)) � (7) θ′ G = θ′ G + α∇fG(θ′ G) (8) 3) STORM-GAN Adaptation on New Tasks: During the model adaptation phase (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', updating the optimal initializa- tion for a new task from a new city), we first copy θG and θD from the meta-training phase as the initialization for fast- adaptation, and then use training samples from the new task to perform STORM-GAN for updating the meta-parameter θD and θG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Finally, the updated model outputs the estimated mobility using testing samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The tasks used for meta-testing adaptation are held out from meta-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' EVALUATION A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Dataset Description Data Sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' We elaborate four types of data as described in Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' 3 & Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' 4 (pandemic, contextual, policy, and mobility).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' They are collected from Centers for Disease Control and Prevention [13], Census Bureau [14], the date of disease prevention policies were collected from the corresponding city government website news, and SafeGraph [15], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' SafeGraph provides free access to data for academic purposes with upon request and all the other data are publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Data Granularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The original POI dataset from SafeGraph is obtained by collecting the location from cell phone records with latitude and longitude information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Then, the location information is used to determine the visits to POIs [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The POI visit counts data is in point data format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' 3 illustrates the discretization of one city, where we sum the total POI visit counts that fall into each grid cell, and use this aggregated visit counts value to represents the human mobility responses of each grid cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Data Pre-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' To construct the list of conditions for our input, for each grid cell, we preprocess data collected from different sources with different geographic units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' We first adopt a commonly used space-partitioning method to segment each spatial domain into grid cells of size of 1km × 1km, and segment all mobility related conditions using the same grid cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Then, each spatial region (or unit spatial window) we used to create a data sample is a 10 × 10 spatial window .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Walam Bosibn MeitdnAlgorithm 1: STORM-GAN Training and Testing Require: Set of training cities Ttrain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' set of testing cities Ttest Conditions F, mobility Mreal, a STTG G Inner learning rate α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' outer learning rate β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' number of epochs epoch Ensure: θG, θG, estimated mobility M ′ G for Ttest {# Meta-learning on training cities} 1: G = initG();' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' D = initD() 2: Randomly initialize meta θG, θG 3: for e = 1 to epoch do 4: Sample a batch of T from Ttrain 5: Sample the subgragh H of T from G 6: for Ti in {F, Mreal,H} do 7: Sample a set of disjoint Dtrain i , Dtest i 8: Generate graph embedding E of H 9: M′ G = G(F, E, rand(PU)) 10: Update D using Dtrain i by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' (5) and (6) 11: Update G using Dtrain i by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' (7) and (8) 12: Evaluate estimation loss using Dtest i by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' (2) 13: end for 14: Update θD and θG by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' (3) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' (4) 15: end for 16: Return θG, θD {# Fast-adaptation on testing cities} 17: Sample batch of testing tasks T from Ttest 18: for Ti in {F, Mreal,H} do 19: Sample a disjoint Dtrain i , Dtest i from Ttest 20: Generate graph embedding E of H 21: Copy θG, θD 22: Evaluate performance by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' (2) using Dtrain i 23: Update G through Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' (7) and (8) 24: Estimate M′ G using updated G and Dtest i 25: end for on the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' For each grid cell, the value of human mobility response is the total number of POI visit counts in a day.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Note that some conditions are re-scaled during this process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' For example, population and median household income data are collected at the census tract level, and we linearly re-scaled the data using the corresponding area ratios between the area of the original census tract polygon and the proposed 10 × 10 grid cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Similarly, COVID-19 statistics and policy data are collected at the county level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' We assign each grid cell with the corresponding data on which county it belongs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Training Data Description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' We collect the mobility related datasets from six cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The dataset spans over six cities in different states located from the west coast, midwest to the east coast (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', Boston, Chicago, Houston, Iowa City, Los Angeles, and NYC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The list of cities also covers regions from large metropolitan areas to less populous places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Detailed statistics of these datasets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', number of POIs, number of cells covered for each city) are listed in Table I and Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The duration of data is from 02/24/2020 to 10/25/2020, covering 35 weeks in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' As discussed in Methodology section, the data is segmented into a spatio-temporal distribution of tasks, where each task contains one single city for five consecutive weeks (no mutual overlaps among tasks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The candidate methods are trained on five cities (meta-training) with one left out as the new city for meta-testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Adaptation on test cities is performed with data samples from the most recent two weeks (out of 35 weeks in total).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Overall, we have 35 spatio-temporal estimation tasks in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' For methods with meta-learning, 80% of data in each task is used for meta-training, and the rest for testing (Def.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' TABLE I: Detailed data statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' City Time Span (M/D/Y) POIs Size Boston 02/24/2020 10/25/2020 35 Weeks 26054 37 × 48 NYC 133520 58 × 72 LA 86721 52 × 64 Chicago 47356 50 × 40 Houston 37315 50 × 60 Iowa City 1401 20 × 32 TABLE II: Average number of POI visit counts per grid cell City Boston NYC LA Chicago Houston Iowa City POI Counts 28 64 52 30 24 6 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Evaluation Metrics We evaluate the performance of STORM-GAN by using the following measures: mean absolute error (MAE) and rooted mean square error (RMSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' MAE = 1 n n � i=0 ���MG − ˆ MG ��� (9) RMSE = � � � � 1 n n � i=1 � MG − ˆ MG �2 (10) where MG is the real mobility response and ˆ MG is the generated mobility response values by candidate approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Since model generates the spatial unit windows multiple times for each grid cell during estimation, the outputs of generator are averaged before comparing with the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' To evaluate the model performance of learning the data distribution, we calculate the KL divergence to indicate the similarity between the learned human mobility responses dis- tribution ˆP and real human mobility responses distribution P on different bin sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The KL divergence is defined as follow: DKL(P|| ˆP) = N � i=1 P(M ′ G)log( P(M ′ G) ˆP(Mreal) ) (11) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Baseline Methods We compare our proposed method with the following base- line methods, and fine-tune each method using Houston and Iowa City as testing cities respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' HA: Historical Average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The average of human mobility responses calculated using observed values from the same location in the past two weeks (same weekday).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Spatial smoothing with neighborhood regions [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' This method uses the mobility response values in a local 3 × 3 window to compute a mean as the estimated value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The values for smoothing are from the same weekday in the most recent week.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Ridge [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' We use ridge regression with the same input features and mobility responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' cGAN [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' A conditional GAN where the generator and discriminator use three fully-connected layers (no layer structure to learn spatial or temporal patterns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' COVID-GAN [3]: COVID-GAN has the same structure as the above cGAN, and it adds a correction layer, which is used to add constraints based on policy to refine the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' MAML-DAWSON [5]: An optimization-based meta- learning approach using MAML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' As DAWSON originally works on music generation tasks, we modify its inner structure with a regression-focused conditional GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' MetaST [6]: MetaST fuses CNN, LSTM and attention mechanism to predict urban traffic volume through MAML framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' STTG Construction Examples In this section, we provide two different STTG construction scenarios to evaluate the effectiveness of graph embedding in human mobility estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Scenario 1 (S1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' We assume that cities of similar sizes, socio-economic environment, and land-use design may share similar human mobility pattern that could help the estimation of new cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' To build the graph Gs1 (V, E), we enumerate major metropolitan cities from every region in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', and define each city as a node νi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Next, we extract human mobility maps for all the cities from a same date, and calculate the pairwise distribution similarity score between cities using KL- divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The KL-divergence indicates the strength of hu- man mobility correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Each edge is added if the correlation is ≦ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='5, and is weighted by the correlation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Gs1 contains 55 nodes and 682 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Node attribute store the outbreak stage of COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Each stage value is in {1, 2, 3}, where a smaller value means earlier in COVID-19 outbreak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The stage value is assigned based on the month when an exponential growth is first appeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Scenario 2 (S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Intuitively, urban environment increases the chance of infection as people move around and interact with others and the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' As a hub for migration and travel, urban areas may quickly spread infections to nearby places through short-distance travel, and to major cities through connection flights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' To construct S2, similar to S1, we enumerate major metropolitan areas in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='S, and define a graph Gs2 (V, E) to represent the relationships of these cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' We divide the nodes V into two categories: the hub nodes Vh are major cities with more than 100 airlines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' the second-tier nodes Vs are cities with more than 35 but less than 100 airlines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Moreover, each Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' 4: Subgraph of a node in STTG S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' directed edge νi → νj ∈ E is added if its two nodes are: (1) both major cities that have direct flights;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' or (2) within a spatial proximity threshold (500 km in this paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Our graph contains 69 nodes and 776 edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The graph is then weighted by spatio-temporal attributes associated with nodes and edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Edge attributes contain the number of directed flights between the cities and their geographic distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Node attributes store the sum of flights from connected edges as well as the outbreak stage of COVID- 19 which is the same as scenario 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' 4 shows an illustration example of our S2 which is a 1-hop subgraph for Kansas City, a second-tier city by the above-mentioned classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Major cities that have direct flights to Kansas City (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', Denver, Atlanta, Minneapolis) and second-tier cities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', Oklahoma, Omaha) within the spatial proximity threshold are shown on the subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' We use both of the two STTG definitions with our STORM- GAN (namely, STORM-GAN(S1) and STORM-GAN(S2)) to evaluate its performance in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Estimation Quality Evaluation The benefit of meta-learning is that the model can quickly update the model parameters, and generate good results on a new task by seeing a small fraction of new data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' During the adaptation phase, we evaluate the performance of the candidate methods on the two test cities (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', Houston and Iowa City) using their last two weeks (Monday to Sunday) of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The length of a time period we use is 7-day since human mobility pattern is influenced by strong weekly periodicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' For each testing city and week, we use 2 consecutive weeks of data ahead of the week for adaptation, and then use the parameters to generate the next 7-day (one week) human mobility responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Performance comparison of proposed STORM-GAN and other candidate methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Tables III and IV show the results of the candidate methods obtained using Houston and Iowa City as the testing city, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' We apply S1 and S2 graph construction scenarios on STROM-GAN, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The evaluation results show that both STORM- GAN scenarios overall achieve the lowest RMSE and MAE for each day in the week, with major improvements from 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='7% to 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' It is interesting to observe that historical average and spatial smoothing methods perform better than the basic cGAN, which to some degree shows the spatio- temporal auto-correlation effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' However, these methods Minneapolis Chicago Portland Tulsa Oklahoma Detroit LOS Kansas ( it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Omaha Angeles Hub Second-tier Denver Atlanta Edge DallasTABLE III: Human mobility responses estimation by candidate methods for Houston RMSE MAE Model Mon Tues Wed Thu Fri Sat Sun Mon Tues Wed Thu Fri Sat Sun HA 194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='8 193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='2 193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='5 196.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='5 can mainly estimate a rough base but are limited in captur- ing complex spatio-temporal relationships between features and mobility responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Comparing to COVID-GAN and MAML-DAWSON, our model outperforms COVID-GAN by 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='5% (RMSE) and 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='5% (MAE) on average, and MAML- DAWSON by 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='1% (RMSE) and 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='4% (MSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The results show that the design of spatio-temporal architecture (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', CNN and LSTM substructures and the STTG graph) and meta-learning adaptation can significantly improve the solution quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Furthermore, our model achieves 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='7% (RMSE) and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='2% (MSE) better than MetaST, demonstrating that task- based graph embedding can contribute to model performance by learning the inter-task similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' We evaluate the model performance on less populous areas using Iowa City as testing city, the POI numbers and city size of Iowa City are sig- nificantly smaller than large metropolitan areas according to Tables I and II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' As Table IV shows, the improvements are relatively smaller due to smaller number of POI visit counts in less populated city.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' However, both STORM-GAN scenarios still achieve the lowest errors in all of the testing days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' We calculate the KL-divergence using results from Houston and Iowa City (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The X-axis represents the number of equal-size bins used to discretize the value needed for the computation, and the Y-axis shows the KL divergence values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' A lower KL divergence value means the result better matches the real distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' 5, STORM- GAN achieves lowest KL-divergence compare to the baseline methods consistently for all numbers of bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Impact of STTG choice: Our results show that both of the two STTG constructed can significantly improve the performance of STORM-GAN in Houston and Iowa City.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' This proves that the spatio-temporal task-graph embedding design is effective and robust, rather than tailored for a specific STTG Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' 5: Kullback-Leibler divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Between the two choices, S2 generally achieves better performance as it uses more information that are directly related to the spreading of COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Ablation Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' We study the effect of different compo- nents proposed in our method using Houston as the testing city on one day (Monday).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Base: Baseline conditional GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Base + Spatial (S): Equivalent to COVID-GAN, which has a correction layer to add policy constraints, but purely a spatial model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Base + Spatio-Temporal (ST) + Meta: Proposed STORM- GAN with spatio-temporal meta-learning, but without the STTG graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Base + ST + Meta + Graph(S2): Complete STORM-GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Table V shows the estimation performance of STORM- GAN and its variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' First, the base + spatial (S) achieves a lower RMSE and MAE (a reduction of 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='3% and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='2%, respectively) compared to cGAN, showing the effectiveness of the correction layer from COVID-GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Next, we can Houston Iowa City 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='6 Divergence 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='3 TY 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='1 0 30 60 90 120 30 60 90 120 HA cGAN Smoothing Ridge MAML-DAWSON COVID-GAN MetaST OSTORM-GAN(S1) →STORM-GAN(S2) Number of binsFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' 6: Mobility estimation results of the Houston study area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' 7: Mobility estimation results of the Iowa City study area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' see that the addition of spatio-temporal meta-learning further reduces RMSE and MAE by 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='7% and 10%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' This result demonstrates that CNN, LSTM and meta-learning can better capture the complex spatio-temporal relationships across multiple cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Finally, the complete STROM-GAN achieves the lowest RMSE and MAE with the sptaio-temporal task-based graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' We compare the solution quality of seven candidate approaches through map visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' 6 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' 7 (a) to (f) show the results of baseline methods, and (g), (h) show the STORM-GAN (S2) and ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The results TABLE V: Comparison among STORM-GAN variations Method RMSE MAE Base 202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='2 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='6 Base + S 171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='5 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='1 Base + ST + Meta 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='8 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='6 Base + ST + Meta + Graph (S2) 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='1 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='7 show the full Houston and Iowa City study areas for a day in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Here STORM-GAN generates fine-scale mobility values that are closer to the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' As we can see, the mobility pattern generated by the STORM-GAN can capture the spatial pattern of human mobility responses better than other baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The reason may be that similar functionality zones at different cities may have similar mobility patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The meta-learning framework successfully learns this shared knowledge from training tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Moreover, the utilization of CNN and LSTM helps capture the spatio-temporal correlation from region to region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' OTHER RELATED WORK Deep Learning for Spatio-Temporal Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' There have been many deep learning based techniques developed for spatiotemporal data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' For example, LSTM were widely used in traffic accident prediction [18] due to its capability in capturing spatio-temporal correlation and thus provide good prediction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Geospatial object mapping [19], taxi driver behavior imitation [20], taxi demand [21], travel time estimation [22], etc, they all combine the deep learning model with spatio- temporal perspective in their model design and obtain good performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Most of these techniques typically are stationary predictors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', same result from two runs on same data) rather than generative models, and they do not consider the unknown factors in prediction, and their performance relies on large data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Besides, they do not leverage domain knowledge based constraints to assist learning (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', cGAN [2], [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Meta-Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Meta-learning learns new tasks quickly and effectively with a few examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Existing optimization- based meta-learning algorithms such as MAML [8] and Rep- tile [23] rely on optimization through gradient descent, and both are compatible with any model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Recently, the idea of optimization-based meta-learning has been applied to many domains including classification and reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' However, there only are a few work address the spatial and temporal problems simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' In traffic prediction, a recent work [7] focuses on knowledge transfer in a single city, which only deals with temporal tasks with no spatial- based tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' [24] proposes a transfer learning framework for traffic prediction through learning region matching function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Another work [6] which is based on multiple cities does not consider temporal patterns and dynamic scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Mobility Estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' There have been many studies [1], [25], [26] exploring the interplay between human mobility responses, social distancing policies, and transmission dynam- ics in response to the COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' A US mobility change map was created in [25] to increase risk awareness of the public and to visualize dynamic changes in mobility as COVID-19 situation and policy evolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' These studies are timely in showing the important role played by mobility in the spread of COVID-19, but they do not address the challenges in real-world mobility estimation/simulation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', effects of unknown, uncertain, and random factors), and they analyze the mobility changes in city or country scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' A study [27] simulated the human mobility which allows policymakers to inspect mobility changes under different policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' But this approach utilizes a traditional epidemiological model, and does not transfer the simulation from city to city by shared knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' However, these studies have not explored the potential use of deep learning based generative models and meta-learning to assist the estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' CONCLUSIONS We made the first attempt to tackle the human mobility estimation problem through a spatio-temporal meta-generative framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Specifically, we proposed a STORM-GAN model to capture complex spatio-temporal patterns using a set of social and policy conditions related to COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' We also proposed a novel spatio-temporal task-based graph (STTG) to represent the spatio-temporal relationships among cities, with a graph convolution network to learn embeddings of its subgraph for cross-task learning enhancements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Finally, STORM-GAN utilized the meta-learning paradigm to learn shared-knowledge from a spatio-temporal distribution of estimation tasks and can quickly adapt to new tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=', new cities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The experiment results showed that our proposed approach can significantly improve the estimation performance compared to baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' The model can assist policymakers to better understand the dynamic mobility pattern changes under different social and policy conditions, and can potentially be leveraged to inform decisions in resource allocation and provisioning, event plan- ning, response management, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' ACKNOWLEDGMENT This paper is funded in part by Safety Research using Simulation University Transportation Center (SAFER-SIM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' SAFER-SIM is funded by a grant from the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Department of Transportation’s University Transportation Centers Program (69A3551747131).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' However, the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NFAT4oBgHgl3EQfqh2x/content/2301.08648v1.pdf'} +page_content=' Government assumes no liability for the contents or use 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b/aNAzT4oBgHgl3EQfK_vP/content/tmp_files/2301.01108v1.pdf.txt @@ -0,0 +1,1721 @@ +arXiv:2301.01108v1 [quant-ph] 3 Jan 2023 +Towards optimization under uncertainty for fundamental models in energy +markets using quantum computers +M.C. Braun, T. Decker, N. Hegemann, S.F. Kerstan, and F. Lorenz +JoS QUANTUM GmbH, Frankfurt am Main, Germany ∗ +January 4, 2023 +Abstract +We present a method to formulate the unit commitment problem in energy production as quadratic unconstrained +binary optimization (QUBO) problem, which can be solved by classical algorithms and quantum computers. +We +suggest a first approach to consider uncertainties in the renewable energy supply, power demand and machine failures. +We show how to find cost-saving solutions of the UCP under these uncertainties on quantum computers. We also +conduct a study with different problem sizes and we compare results of simulated annealing with results from quantum +annealing machines. +1 +Introduction +Electrical energy production and supply is a foundation of developed economies and essential for the stability of +societies. Energy markets provide producers, consumers and grid operators an efficient way for pricing electricity, +depending on forecasted and actual demand and supply. As the grid frequency needs to be stable (50 Hz in Europe) +and as big amounts of electrical energy cannot be stored efficiently, transmission operators need to balance the output +of power generation units and the consumption in real-time. +Energy production based on solar and wind power depends on weather conditions and these can change rather +quickly, introducing extreme volatility in production levels and prices. The European Union plans to continue the +substitution of power plants powered by fossil fuels, i.e. coal, oil and gas, with renewable sources of energy [1]. This +introduces an even higher volatility while forecasting the feed-in to the energy grid is limited. Therefore, smart ways +to steer the production as well as the consumption need to be introduced to produce energy with the lowest cost while +satisfying demand. Finding a power plant dispatch schedule whose expected supply matches the demand at any given +point in time and comes with the lowest possible costs is a well-known optimization problem in energy markets. +Energy prices are usually derived from the power plant with the lowest marginal costs that match the energy +demand. This is called merit order [2]. Beside this, a number of technical constraints on the power generating units +must be observed. Conventional power plants cannot freely be switched on and off and have to obey limits in their power +generation range. The problem of finding a suitable power unit schedule is known as unit commitment problem (UCP) +and it is NP-hard to solve. Several approaches exist to solve such types of problems including dynamic programming [3], +Lagrangian relaxation [4], Benders decomposition [5], mixed integer programming [6] and reinforcement learning [7]. +The feed-in from renewable power plants like solar parks and wind farms depend on factors like sunshine hours and +wind speed. The accuracy of predictions of the weather is limited and this leads to an unknown residual future power +demand that conventional power plants have to compensate. Other uncertainties come in the form of machine failure +events or unpredictable demand fluctuations on small time scales. Considering such uncertainties in the UCP leads to +the stochastic unit commitment problem (SUCP). There exist many approaches to the SUCP and an overview is given +in [8]. Since it can be shown that the UCP is NP-hard [9] (and so is the SUCP), there exists no efficient method to +solve this problem exactly. Instead, approximate solutions and solutions to relaxed problems are used in practice. +Quantum computing may open new ways of solving the UCP and the SUCP efficiently. Recent theoretical work +has shown that with quantum algorithms we can achieve advantages over classical methods for several problems +in theory [10, 11]. +One way is to use annealing methods that are carried out by Ising machines. +One example +of commercially available quantum annealers are the machines from D-Wave. +These machines leverage quantum +mechanical effects like quantum tunneling to find optimal solutions to annealing problems quickly. This opens the +path to provide solutions with higher quality in less time. +Although the technology still has to overcome technical limitations before reliable large-scale quantum computers +are available, the fast progress in the field promises a true alternative to classical solvers in the future. +In this +∗For contact email: niklas.hegemann@jos-quantum.de. Authors are listed in alphabetical order. +1 + +paper, we explore the possibility of formulating the UCP in terms of a binary quadratic unconstrained optimization +(QUBO) problem that can be solved by many different solvers, e.g. by quantum inspired algorithms [12], quantum +annealers [13, 14] and variational quantum algorithms like the variational quantum eigensolver [15] or the quantum +approximate optimization algorithm [16]. +We use simulated annealing [17] as well as quantum annealing to find +solutions for our QUBO formulations. +A similar approach for the static UCP was published recently in [18]. Here, we go beyond this method by advancing +the QUBO formulation to solve a relaxed version of the UCP. In this version, we assume that the overall supply needs +to match the overall demand, but at each timestep supply and demand can deviate and a small over- or undersupply +can be compensated with storage devices. +The results for several examples were obtained by classical solver algorithms and by D-Wave annealing hardware. +This work might help hardware companies to better understand the hardware requirements for real-world business +applications. +2 +Elements of the unit commitment problem +In this work, we restrict ourselves to the single bus approximation where the geographical structure of the electricity +grid is ignored. The demand is considered to be satisfied whenever the total production equals the total demand. +The UCP refers to the optimization task of finding the cost optimized production plan of N conventional power +generators whose total energy supply adds up to demand(t) at every timestep t ∈ {1, 2, . . ., T}. Each power generator +k ∈ {1, 2, . . . , N} can generate a power output genk with variable costs varcostk, i.e. the costs per produced unit in a +timestep. Startup costs startcostk are introduced whenever a unit is turned on. +We extend the problem by introducing R renewable units that run without variable costs and that have a fixed +power-time profile supplyRE +r (t) which is determined by external factors, like the weather or tides. These renewable +units effectively reduce the required amount of energy and give a residual demand that has to be compensated by +conventional power plants. We call this an effective demand d(t) and we require a solution of the UCP to provide +d(t) = demand(t) − +R +� +r=1 +supplyRE +r +(t) = +N +� +k=1 +genk(t) for all t ∈ {1, 2, . . ., T} . +More complexity arises when restrictions that originate from power generators are introduced. The power generating +units need to obey the following constraints: The power supply of each unit has a lower bound mingenk and an upper +bound maxgenk, i.e., genk(t) ∈ [mingenk, maxgenk] for all t ∈ {1, 2, . . ., T}. Furthermore, power generators exhibit +technical minimum run times minupk and minimum shut down times mindownk. This means that once power unit +k is turned on it can not be turned off after running for at least minupk timesteps. +A similar condition ensures +compliance of downtimes. To summarize, a specific problem instance of the UCP is fully determined by the following +set of parameters: +• demand(t) for t ∈ {1, . . . , T} +• supplyRE +r (t) for r ∈ {1, . . ., R} and t ∈ {1, . . ., T} +• mingenk, maxgenk, varcostk, minupk, mindownk for k ∈ {1, . . ., N} +3 +QUBO formulation with penalty terms +3.1 +Formulation of QUBO problems +In this section, we discuss techniques that can be used to formulate combinatorial optimization problems as Quadratic +Unconstrained Binary Optimization (QUBO) problems. It is a mathematical construct equivalent to Ising spin glasses. +Once formulated, these QUBO problems can be solved by different techniques, including quantum annealers, quantum +computers and other types of Ising-machines [19]. +Such optimization problems can be formulated as follows when we set B = {0, 1}. Let fQ : Bn → R be the quadratic +polynomial +fQ(x1, . . . , xn) = +n +� +i=1 +n +� +j=1 +qijxixj +(1) +with qij ∈ R, which can be understood as real valued entries of a matrix Q. The function fQ is called the objective +function or objective of the corresponding QUBO matrix Q. For properly chosen coefficients qij, the global extrema +2 + +of the objective function correspond to a solution of an optimization problem, where the solution itself is encoded in +the binary-valued vector x = (x1, . . . , xn) ∈ Bn that fulfills1 +min(fQ) = +n +� +i=1 +n +� +j=1 +qijxixj +or +max(fQ) = +n +� +i=1 +n +� +j=1 +qijxixj . +A more elegant way to write equation (1) is using Dirac’s bracket notation: +fQ(x) = ⟨x | Qx⟩ +Here, ⟨x|y⟩ is the scalar product of two vectors |x⟩ and |y⟩. The matrix Q acts as an operator on the corresponding +vector space and ⟨x| is the adjoint of |x⟩. Subsequently, this notation is used in some cases for simplifying the notation. +In analogy to the quantum mechanical origin of this notation, the solution to a minimization problem corresponds to +a global minimum in the energy landscape. +An efficient QUBO solver algorithm is a machinery that takes Q as an input and outputs the vector |x⟩ of the global +minima in a reasonable amount of time, i.e. such an algorithm would provide us with a tool to solve optimization +problems of this kind. To the best of our knowledge, there exists no reliable and fast algorithm to solve universal +problems of this type as it can be shown that QUBO problems are NP-hard [20]. However, several kinds of algorithms +exist that can be used to find approximate good solutions. Here, we compare simulated annealing to the results of +hardware-based quantum annealing. A quantum annealer represents a special type of quantum computer that performs +approximate adiabatic quantum annealing and is able to provide a QUBO solver, at least on a theoretical level. +3.2 +Building the QUBO for the UCP +The strategy for defining the QUBO elements qij is to use quadratic terms that give higher objective values for expensive +and forbidden states and lower objective values for cheaper and allowed states, respectively. Forbidden and allowed in +this context refers to the adherence to constraints. As introduced in section 2, we want the following conditions and +technical constraints to be fulfilled in the UCP: +• minimizing costs +• matching power demand +• obey minimum and maximum power generation constraints +• obey minup and mindown times +We start the QUBO formulation of the UCP by introducing a resolution of the power supply in the form of a binary +representation of numbers. Since we work in a discrete model, it is not possible for genk(t) to take continuous values +between mingenk and maxgenk. However, we can approximate the continuous case arbitrarily well by introducing a +discretization of the interval [mingenk, maxgenk] and we write +maxgenk − mingenk = +B +� +b=1 +pk,b with pk,b = 2b−1 · (maxgenk − mingenk) +2B − 1 +. +(2) +Here, the number of bits used for the discretization is B. Furthermore, we introduce the variables xp +k,b(t) for b ∈ +{1, 2, . . ., B} for each timestep t. The variable xp +k,b(t) describes how much the power supply of unit k contributes to +the sum (2) depending on level b. We also introduce the binary variables x1/0 +k +(t), which describe whether unit k is on +or off in timestep t. Then we can write +genk(t) = mingenk · x1/0 +k +(t) + +B +� +b=1 +pk,b · xp +k,b(t) = mingenk · x1/0 +k +(t) + ⟨pk|xp +k(t)⟩ +(3) +with pk = (pk,1, . . . , pk,B) and xp +k(t) = (xp +k,1(t), . . . , xp +k,B(t)), which suits the QUBO formalism. Increasing the resolution +B increases the number R(B) = 2B of possible power supply realizations and it improves the fineness +F(B) = maxgenk − mingenk +2B − 1 +of the power range as can be seen in table 1. The introduction of an additional variable xs +k(t) per unit and timestep +is required to describe the start of the power production of a unit. If unit k is off in timestep t − 1 and if it is turned +on in timestep t, then we set the start-variable xs +k(t) = 1 and xs +k(t) = 0 else. +1In the following, we write vectors (x1, . . . , xn)T as row vectors (x1, . . . , xn) to simplify notation and to avoid ambiguities with the number +T of timesteps. +3 + +Table 1: Discretization of the interval [x, y]. Number R(B) of power supply realizations and the relative fineness +F(B)/(y − x) as functions of the resolution parameter B. +B +R(B) +F(B)/(y − x) +2 +4 +0.3333... +3 +8 +0.1429... +4 +16 +0.0607... +6 +64 +0.0159... +10 +1024 +0.001... +20 +1048576 +≈ 1 · 10−6 +For timestep t, we choose the corresponding part of the full solution vector to be arranged like +x(t) = (xp(t), x1/0(t), xs(t)), where xp(t) = (xp +1(t), xp +2(t), . . . , xp +N(t)) +encodes the power supply stages of all units. The vectors +x1/0(t) = (x1/0 +1 +(t), x1/0 +2 +(t), . . . , x1/0 +N (t)) +and +xs(t) = (xs +1(t), xs +2(t), . . . , xs +N(t)) +contain the on/off and start information, respectively. Each element xp +k(t) has the structure +xp +k(t) = (xp +k,1(t), xp +k,2(t), . . . , xp +k,B(t)). +The introduction of x1/0(t) enables us to implement the minimum power constraint, which comes at the cost of +increasing the problem size. The full solution vector for T timesteps is given by +x = (x(1), x(2), . . . , x(T )). +(4) +The length of this vector is given by dim(x) = T N(B + 2), i.e. it grows linearly with the number of timesteps T , units +N and resolution B. The next step is to build the QUBO matrix Q by translating the UCP into matrix elements. +We do this step by step and implement the individual problem specification via defining corresponding penalty terms, +which are added together at the end. +The minimal costs for the optimal power unit allocation include the variable costs, i.e. the running cost per unit +time, as well as start costs. A suitable penalty term is given by +P cost · +T +� +t=1 +N +� +k=1 +varcostk · +� +mingenk · x1/0 +k +(t) + ⟨pk | xp +k(t)⟩ +� ++ startcostk · xs +k(t), +(5) +where P cost is a penalty parameter that has to be adjusted. Note that this form is equivalent to equation (1), since +for binary variables we can use the equality x = x2 to substitute linear terms with quadratic ones. It is also possible +to directly introduce a quadratic cost term by squaring (5). We obtain +P cost · +� T +� +t=1 +N +� +k=1 +varcostk · +� +mingenk · x1/0 +k +(t) + ⟨pk | xp +k(t)⟩ +� ++ startcostk · xs +k(t) +�2 +and this enlarges the differences of the objective function for different states by increasing the range of the energy +spectrum. We quantify this by defining +κk(x(t)) = varcostk · +� +mingenk · x1/0 +k +(t) + ⟨pk | xp +k(t)⟩ +� ++ startcostk · xs +k(t) +and observe that κk(x(t)) ∈ [0, κmax +k +] where κmax +k += varcostk · maxgenk + startcostk is the upper bound. It is now +obvious that +P cost · +T +� +t=1 +N +� +k=1 +κk(x(t)) < P cost +� T +� +t=1 +N +� +k=1 +κk(x(t)) +�2 +if �T +t=1 +�N +k=1 κk(x(t)) > 1. Since the penalty term can only take non-negative values, we can rescale the costs to +ensure that this is always true. Hence, the energy spectrum in the case of the quadratic term has a larger range. Also, +if we additionally define κmax = max{κmax +1 +, κmax +2 +, . . . , κmax +K +}, the following inequalities are true: +P cost · +T +� +t=1 +N +� +k=1 +κk(x(t)) < P costT Nκmax +P cost · +� T +� +t=1 +N +� +k=1 +κk(x(t)) +�2 +< P costT 2N 2(κmax)2 +4 + +This shows that the energy range is potentially larger in the case of a quadratic cost term. From now on, we choose +to continue with the quadratic cost term. +The next step is to match the power demand in every timestep. We require the supply resulting from an optimal +power unit schedule to match the forecasted energy amount for all timesteps. As described in section 2, we use an +effective demand d(t) in every timestep and find that a suitable penalty term is given by +P demand · +T +� +t=1 +� N +� +k=1 +mingenk · x1/0 +k +(t) + ⟨pk | xp +k(t)⟩ − d(t) +�2 +. +(7) +Again, P demand is a penalty strength which has to be chosen properly. Note that this gives multiple terms of the form +d(t)2, which do not include x and therefore cannot take the form of equation (1). However, we can subtract these +constant terms from (7) and this leads to a constant energy shift for all states. Consequently, minimizing this new +function also minimizes (7). +We now add a penalty to respect the technical restriction of minimum run times. We have specified the minimum +run time of unit k via minupk. When unit k is turned on in timestep t, the earliest shutdown of this unit is possible +after timestep t + minupk, i.e. the unit has to run for minupk timesteps. With the penalty strength P minup, we use +the penalty term +P minup · +T +� +t=1 +N +� +k=1 +xs +k(t) +� +minupk · xs +k(t) − +t+minupk−1 +� +τ=t +x1/0 +k +(τ) +� +. +If the minimum run time condition is fulfilled, the penalty vanishes. Note that in the summation over τ we only +consider elements up to at most τ = T . +The penalty terms for minimum down times work in a similar way. When unit k is turned off in timestep t, the +earliest possible start of this unit is after timestep t + mindownk, i.e., the unit has to be shut down for mindownk +timesteps. Introducing the penalty strength P mindown, we choose +P mindown · +T +� +t=1 +N +� +k=1 +� +x1/0 +k +(t − 1) − x1/0 +k +(t) + xs +k(t) +� +· +t+mindownk−1 +� +τ=t +x1/0 +k +(τ) +as the corresponding term in the QUBO. As in the case of minimum run time, fulfilling the minimum down time +condition gives a zero penalty. Note that we do not sum over elements for values of t − 1 and τ that are outside of +{1, . . . , T}. +The minimum and maximum power supply of each power generator are automatically met due to equation (3), if +we can ensure that whenever at least one of the xp +k,b(t) is non-zero, the variable x1/0 +k +(t) is also set to one. This requires +a penalty term of the form +P inter1 · +T +� +t=1 +N +� +k=1 +B +� +b=1 +xp +k,b(t) +� +1 − x1/0 +k +(t) +� +, +where P inter1 denotes the penalty strength. This term interrelates the variables xp +k,b(t) and x1/0 +k +(t). +Finally, setting the start variable is required to ensure that we correctly connect the start variable xs +k(t) to the +on/off-variable x1/0 +k +(t) of unit k such that we have +xs +k(t) = +� +1 +for x1/0 +k +(t − 1) = 0 and x1/0 +k +(t) = 1 +0 +else +A suitable penalty term is +P inter2 · +T +� +t=1 +N +� +k=1 +� � +x1/0 +k +(t + 1) − xs +k(t + 1) +�2 ++ x1/0 +k +(t) · +� +xs +k(t + 1) − x1/0 +k +(t + 1) +� � +with the corresponding penalty strength P inter2. Here, we do not sum over elements for values of t + 1 and τ that are +outside of {1, . . . , T}. +We can now write the full QUBO as the sum of the penalty terms above: +5 + +Q(x) =P cost +� T +� +t=1 +N +� +k=1 +varcostk · +� +mingenk · x1/0 +k +(t) + ⟨pk | xp +k(t)⟩ +� ++ startcostk · xs +k(t) +�2 ++P demand · +T +� +t=1 +� N +� +k=1 +mingenk · x1/0 +k +(t) + ⟨pk | xp +k(t)⟩ − d(t) +�2 ++P minup · +T +� +t=1 +N +� +k=1 +xs +k(t) +� +minupk · xs +k(t) − +t+minupk−1 +� +τ=t +x1/0 +k +(τ) +� ++P mindown · +T +� +t=1 +N +� +k=1 +� +x1/0 +k +(t − 1) − x1/0 +k +(t) + xs +k(t) +� +· +t+mindownk−1 +� +τ=t +x1/0 +k +(τ) ++P inter1 · +T +� +t=1 +N +� +k=1 +B +� +b=1 +xp +k,b(t) +� +1 − x1/0 +k +(t) +� ++P inter2 · +T +� +t=1 +N +� +k=1 +� � +x1/0 +k +(t + 1) − xs +k(t + 1) +�2 ++ x1/0 +k +(t) · +� +xs +k(t + 1) − x1/0 +k +(t + 1) +� � +(8) +3.3 +Tuning penalty parameters +In this section we derive relations between the different penalty strength parameters that appear in equation (8) to +ensure that the model correctly prioritizes different problem constraints and relates good solutions of the UCP to low +energy states. +The first requirement is that we want to avoid demand mismatches in favor of cost savings. Therefore, consider +a solution x that matches the demand for all timesteps. Further, a second solution is given which equals the first +one except that in time t′ there is a supply deficit of ∆gen(t′) = gen(t′) − d(t′) < 0 which comes at the cost saving +∆κ(t′) < 0. Using equation (8) and the Kronecker delta notation δt,t′, the corresponding condition for the difference +of these two solutions is +P cost · +� T +� +t=1 +N +� +k=1 +κk(x(t)) +�2 +< P cost · +� T +� +t=1 +� N +� +k=1 +κk(x(t)) + ∆κ(t′)δt,t′ +��2 ++ P demand∆gen(t′)2 +We now exploit the following inequality: With ǫ < 0 < |ǫ| < z it follows that (z + ǫ)2 < (z + ǫ)(z − ǫ) = z2 − ǫ2. We +use this to estimate the right hand side and with a multiplication with N 2 and we obtain: +P cost · +� T +� +t=1 +N +� +k=1 +κk(x(t)) +�2 +

+N 2∆�κ2 +mink,b (pk,b)2 P cost . +To fullfill this inequality in all cases, we choose +P demand = 2 · +N 2∆�κ2 +mink,b (pk,b)2 P cost . +Since the penalty parameter P cost defines the energy scale of the objective function, we can set P cost = 1 without loss +of generality. +6 + +The other penalty parameters represent hard constraints of the problem and have to be prioritized over the cost +and demand matching, i.e., we want to penalize violations of these technical constraints more than demand mismatches +or high costs. Empirically, we find that suitable values for these parameters are given by +P minup = P mindown = 102 · N 2∆�κ2 , +P inter1 = 104 · N 2∆�κ2 and +P inter2 = 106 · N 2∆�κ2 . +3.4 +Example +We now present results of the method applied to a small-sized problem, which consists of N = 2 power supply units +and T = 3 timesteps with a resolution of B = 10. We also consider two renewable supply units, which contribute +fixed, but time dependent amounts of energy. As a result, the two power units have to match the residual demand. +The individual parameters of each unit are given in table 2. +unit k +varcostk +startcostk +mingenk +maxgenk +minupk +mindownk +1 +65 +200 +34 +505 +2 +1 +2 +25 +500 +250 +900 +2 +2 +Table 2: Parameters of the power generating units for the example. +Table 3 shows the power demand as well as the supply from renewables for three timesteps and the resulting +effective demand. +t +demand(t) +supplyRE +1 (t) +supplyRE +2 (t) +d(t) +1 +618 +50 +100 +468 +2 +1145 +50 +150 +945 +3 +710 +25 +125 +560 +Table 3: Power demand, renewable supply and effective demand for the individual timesteps. +These parameters fix the elements of the resulting (72 × 72)-dimensional QUBO matrix, which is illustrated in +figure 1 as a heatmap. +0 +3 +6 +9 +12 +15 +18 +21 +24 +27 +30 +33 +36 +39 +42 +45 +48 +51 +54 +57 +60 +63 +66 +69 +0 +4 +8 +12 +16 +20 +24 +28 +32 +36 +40 +44 +48 +52 +56 +60 +64 +68 +10 +4 +10 +6 +10 +8 +10 +10 +10 +12 +10 +14 +10 +16 +Figure 1: Heatmap of the (72×72)-dimensional QUBO matrix for the UCP instance specified by the parameters from +table 2 and table 3. +In this case, it is easy to find the optimal solution, which is shown in figure 2c. The minimum costs are given by +cmin = 139775. +7 + +We now compare the results from a simulated annealing algorithm 2 with results from real quantum annealing +hardware. We performed quantum annealing on the D-Wave Advantage system 5.2 containing 5760 qubits [21]. After +one run of simulated or quantum annealing, we subsequently performed a gradient descent method to find the nearest +local minimum. +Figure 2d compares the resulting costs from simulated annealing and quantum annealing to the +optimal solution. It should be emphasized that the RE supply units that appear in table 3 are not directly part of +the optimization process, but contribute with a fixed amount of energy in each timestep which is previously set in the +problem parameters. +0.0 +0.5 +1.0 +1.5 +2.0 +time [h] +0 +200 +400 +600 +800 +1000 +1200 +1400 +power [MWh] +demand +RE supply 1 +RE supply 2 +unit 1 +unit 2 +(a) Best solution obtained using simulated annealing with 1000 shots. +0.0 +0.5 +1.0 +1.5 +2.0 +time [h] +0 +200 +400 +600 +800 +1000 +1200 +1400 +power [MWh] +demand +RE supply 1 +RE supply 2 +unit 1 +unit 2 +(b) Best solution obtained using quantum annealing on D-Wave’s Ad- +vantage 5.2 with 1000 shots +0.0 +0.5 +1.0 +1.5 +2.0 +time [h] +0 +200 +400 +600 +800 +1000 +1200 +1400 +power [MWh] +demand +RE supply 1 +RE supply 2 +unit 1 +unit 2 +(c) Optimal power unit schedule for the UCP specified in table 2 and +table 3. +140000 +140500 +141000 +141500 +142000 +142500 +143000 +143500 +absolute costs +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +0.030 +relative error +simulated annealing +quantum annealing +optimal solution +(d) Absolute costs and relative costs error of the solutions from sim- +ulated annealing and quantum annealing compared to costs of +optimal solution. +Figure 2: Comparison of results from the two different solvers to the optimal solution. 2a - 2c show the calculated +power unit schedules and costs are compared in 2d. +It can be seen that all three solutions provide the same power unit schedule, but they assign slightly different power +levels to the units which explains the small difference in the costs. Nevertheless, in this case, simulated annealing as +well as quantum annealing find solutions of high quality, which both are approximately equal to the optimal power +unit schedule. +3.5 +Benchmarking the formulation and solvers +3.5.1 +Evaluating results +In this section, we introduce an evaluation function that can be used as a measure for the quality of the model with +respect to the solutions it produces. It enables us to compare the results from different solving methods. We introduce +2D-Wave’s simulated annealing sampler available in the neal-package. +8 + +the probability ν for a solution to violate hard constraints. +This values is a priori unknown. +We estimate ν by +performing 10 runs of one solver and divide the number of results with violated constraints by the total number of +runs. We denote this quantity ¯ν and use it as an estimator for ν. Furthermore, we define the critical probability νcrit +of constraint violations that is tolerated. If we denote the minimal costs of the optimal solution as cmin and the costs +obtained from one run of the algorithm as c, then we can define +f(ν, c) = + + + +� +e− +ν +νcrit − νe−ν−1 +crit +� +· +� +1 − |c−cmin| +cmin +� +for νcrit ∈ (0, 1] +Θ(−ν) · +� +1 − |c−cmin| +cmin +� +for νcrit = 0 +to be our evaluation function, where +Θ(x) = +� +0 for x < 0 +1 for x ≥ 0 +is the Heaviside step-function. +Note that in most cases cmin will be unknown. +However, here we only consider +cases where we know the exact optimal solution. The critical value νcrit should be chosen such that problem specific +requirements are met. In our case, we choose a zero-tolerance policy and we set νcrit = 0, i.e. in the following we use +f(¯ν, c) = Θ(−¯ν) · +� +1 − |c − cmin| +cmin +� +for evaluating our results. When we calculate the value of f(¯ν, c) several times with different experiments, then we +denote the average value of it by f(¯ν, c). +3.5.2 +Simulated annealing vs. quantum annealing +In the following, we present the results of comparing simulated annealing with quantum annealing for several examples. +We introduce the following examples with different problem sizes: +Table 4: We define the number N of conventional units and the number T of timesteps for different example set +sizes. The length dim (x) of the corresponding solution vector equals the number of rows and columns of +the QUBO matrix for a resolution of B = 10. +example set +XXS +XS +S +M +L +XL +XXL +[N, T] +[2,1] +[2,3] +[2,5] +[5, 24] +[50, 24] +[500, 24] +[5000, 24] +dim (x) +24 +72 +120 +1440 +14400 +144000 +1440000 +Modeling very large power plant grids like the European electricity network, which contain thousands of power +plants would lead to problems of similar size as the XXL example. Due to the current size of the quantum annealers +from D-Wave (Advantage system 5.2, Europe) the comparison is only feasible for our example sets XXS, XS and S +with B = 10, i.e. a resolution of 10 bits. +20 +40 +60 +80 +100 +120 +dim(x) +0.88 +0.90 +0.92 +0.94 +0.96 +0.98 +1.00 +f(¯ν, c) +XXS +XS +S +Benchmark UCP, B=10, shots=1000 +Simulated annealing +QPU D-Wave Advantage system 5.2 +20 +40 +60 +80 +100 +120 +dim(x) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +time to solve QUBO [s] +XXS +XS +S +Benchmark UCP, B=10, shots=1000 +Simulated annealing +QPU D-Wave Advantage system 5.2 +Figure 3: Solution quality f(ν, c) (left) and time so solution in seconds (right) over the QUBO size dim(x). The +different example set sizes are marked. +9 + +For the small problem set XXS with a QUBO size of 24 × 24, both simulated annealing and the quantum annealer +find the global solution. For bigger problem sets, the quality of the solutions is comparable but the runtimes differ +vastly. For the problem set S, simulated annealing takes 3.6 seconds 3 while the quantum annealer takes only 0.25 +seconds. For simulated annealing, the pro-cessing time was measured using D-Wave’s neal package. For quantum +annealing, the "qpu_access_time" 4 was taken, which does not consider pre-, post-processing, embedding time and +queuing time. This gives us a hint that also for big problem sizes the time to an approximate solution increases only +moderately. +4 +Relaxed QUBO with uncertainty +4.1 +Elements with uncertainty +In reality, it is not possible to predict the exact amount of future electricity demand or supply. Rather, the forecasting +is subject to different kinds of uncertainty, especially in the exact energy consumption and the power supply from +renewable supply units. However, we can account for these uncertainties by considering probability distributions for +individual future scenarios. The task is then to find an optimal unit allocation strategy that exploits the stochastic +character of the problem. Consequently, before continuing, we need to establish a stochastic framework. +Let us denote by R a set of renewable energy supply units with R elements. +In timestep t, for each renewable +unit r ∈ R we introduce a discrete random variable SRE +r +(t) whose realizations sir +r (t) with ir ∈ {1, 2, . . ., nr(t)} rep- +resent possible power supply stages of that unit. Moreover, a second discrete random variable for the demand D(t), +which can take values δj(t) for j ∈ {1, 2, . . ., nD(t)} is used to describe the uncertainty in demand at time t. Having +these variables, we can define the probability +pir +r (t) = Pr[SRE +r +(t) = sir +r (t)] +for every power supply realization of unit r ∈ R and and the probability +pj +D(t) = Pr[D(t) = δj(t)] +for a specific demand at time t. Now, a specific effective demand value is given by +d(j,i1,...,iR)(t) = δj(t) − +� +r∈R +sir +r (t). +(10) +Since the effective demand is a function of two independent random variables SRE +r +and D, it also represents a random +variable with the realizations of equation (10) to which the probabilities +p(j,i1,...,iR) +d +(t) = Pr[d(j,i1,...,iR)(t)] = pj +D(t) · +� +r∈R +pir +r (t) +can be assigned. A scenario for the UCP is then defined by the collection of the realizations d(j,i1,...,iR)(t) for t ∈ +{1, . . . , T}. The probability for a specific scenario is then given by +T +� +t=1 +� +r∈R +pj +D(t) · pir +r (t). +We will also need the expectation value for the effective demand: +⟨d⟩ (t) = +� +r∈R +nr(t) +� +ir=1 +nD(t) +� +j=1 +p(j,i1,...,iR) +d +(t) · d(j,i1,...,iR)(t). +Having defined these quantities, our aim is now to use them in a formalism to find an optimal power unit allocation +such that we exploit the uncertainty contained in the probability distributions. We could simply look for the most +probable scenario. However, this strategy fails in the case of a uniform probability distribution. A similar approach +that incorporates the stochastic aspects of the problem more profoundly is to work with the expectation values for +the effective demand in every timestep and to optimize the expected scenario. However, this also does not exploit the +stochastics and ignores the additional degrees of freedom that enter the problem through the underlying uncertainties. +3We used a laptop with a 10 core CPU and the D-Wave package dwave-neal 0.5.9. +4D-Wave Operation and Timing: https://docs.dwavesys.com/docs/latest/c_qpu_timing.html +10 + +We propose a method that exploits the variety of possible effective demands in every timestep by introducing a +relaxation of the optimization problem. Besides searching an optimal power unit allocation, we include the demand +and renewable power supply in the optimization. We do this by allowing these quantities to take unique values SRE +r +(t) +and D(t) in all timesteps. The selection of these values is controlled via a minimization of +� T +� +t=1 +� +d(t) − ⟨d⟩ (t) +� +�2 +(11) +in addition to the cost minimization explained in section 3.2. Note that we use d(t) here not as a random variable, but +as a placeholder for the optimal realization that has to be found. The term (11) plays a similar role as a measure for +the variance with respect to the expected scenario. This ensures that the power demand to be matched is close to the +expectation values, but this formulation allows greater deviations from it, in favor of lower costs. We now show how +to adapt the QUBO model such that this strategy can be implemented. +4.2 +Building multi-scenario QUBO’s +To take the minimization of the value (11) into account, the term +P var · +� T +� +t=1 +� +d(t) − ⟨d⟩ (t) +� +�2 +(12) +is added to the QUBO of equation (8). Here, P var is the penalty strength of the new term. Since we want the model +to choose between different values for demand and renewable supply, we introduce more auxiliary variables to our +solution vector. These contain information about whether a specific power stage is chosen or not. For simplicity, we +choose a constant amount nR of possible power stages for all renewable units and nD demand values for every timestep. +This leads to an overhead of T · (R · nR + nD) variables. If we denote the new solution vector to this UCP including +uncertainty as xrelaxed, then we can attach the auxiliary variables to the end of the solution vector of equation (4) and +we get +xrelaxed = (x, xRE(1), xRE(2), . . . , xRE(T ), eD(1), eD(2), eD(T )) . +(13) +This new solution vector is of length +dim (xrelaxed) = T · (N(B + 2) + R · nR + nD) +and xRE(t) = (eRE1(t), eRE2(t), . . . , eRER(t)) is a vector containing R·nR elements for the power stages of the renewable +energy suppliers. The unit length vectors eD(t) encode the chosen demand in a timestep and the unit length vectors +eREr(t) with r ∈ {1, . . . , R} encode the power stage of RE-unit r in timestep t. Consequently, with the vectors +eREr;i = (0, . . . , +1 +���� +ith +, . . . , 0) +and sr(t), which holds the elements s1 +r(t), . . . , snR +r (t), the choice of power stage i for the RE-unit r in timestep t is +equivalent to evaluating ⟨eREr;i|sr(t)⟩. A specific unique effective demand is calculated via +d(t) = ⟨eD(t) | D(t)⟩ − +� +r∈R +⟨eREr(t) | sr(t)⟩ , +(14) +where D(t) contains the elements δ1(t), . . . , δnD(t). +We need to add terms that ensure that we choose unique values for demand and the power stages of the renewables. +These are +P setD · +T +� +t=1 +� +|eD(t)| − 1 +�2 +and +(15) +P setRE · +T +� +t=1 +� +r∈R +� +|eREr(t)| − 1 +�2 +(16) +with the individual penalty strengths P setD and P setRE. Here, |eD(t)| and |eREr(t)| are the sums of the binary entries +of the vectors. +11 + +The new QUBO is given by +Qrelaxed(x) = Q(x) + P var +� T +� +t=1 +� +d(t) − ⟨d⟩ (t) +� +�2 ++ P setD · +T +� +t=1 +� +|eD(t)| − 1 +�2 ++ P setRE · +T +� +t=1 +� +r∈R +� +|eREr(t)| − 1 +�2 +Due to the structure of the solution vector xrelaxed from equation (13), the new QUBO matrix has the following +structure: +Qrelaxed = +�Q +Qcouple +0 +Qnew +� +The submatrix Qnew contains the corresponding matrix elements of (12), (15) and (16). The submatrix Qcouple ensures +that we can choose between all possible scenarios and it originates from replacing the d(t) value in equation (8) with +d(t) from equation (14). As before, Q is responsible for finding the best power unit schedule and contains the same +information as (8). Finally, we need to fix the numerical values of P setD, P setRE and P var. Since we highly prioritize +the uniqueness conditions, but we do not require a dominance of the P var term, we empirically find the suitable values +P setD = P setRE = 108 · P demand +and +P var = 10−2 · P demand. +4.3 +Example +We now come back to the example set XS, which was introduced in section 3.4. We extend the example by considering +uncertainty in the demand and in the production of the renewable units, as it can be seen in table 5. +Table 5: Probability distribution for power demand and renewable supply for the individual timesteps. This is an +extension of the example set XS. +t +D(t) +Pr [D(t)] +SRE +1 +(t) +Pr [SRE +1 +(t)] +SRE +2 +(t) +Pr [SRE +2 +(t)] +1 +{518, 618, 718} +{0.1, 0.8, 0.1} +{40, 50, 60} +{0.1, 0.8, 0.1} +{80, 100, 120} +{0.1, 0.8, 0.1} +2 +{1145, 1145, 1745} +{0.15, 0.7, 0.15} +{30, 50, 70} +{0.15, 0.7, 0.15} +{120, 150, 180} +{0.15, 0.7, 0.15} +3 +{310, 710, 1110} +{0.2, 0.6, 0.2} +{5, 25, 45} +{0.2, 0.6, 0.2} +{80, 125, 170} +{0.2, 0.6, 0.2} +A heatmap of the QUBO matrix that is built from the parameters from table 2 and the probability distributions from +table 5 is shown in figure 4. +0 +4 +8 +12 +16 +20 +24 +28 +32 +36 +40 +44 +48 +52 +56 +60 +64 +68 +72 +76 +80 +84 +88 +92 +96 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +55 +60 +65 +70 +75 +80 +85 +90 +95 +104 +106 +108 +1010 +1012 +1014 +1016 +1018 +Figure 4: Heatmap of the (99 × 99)-dimensional QUBO matrix for the UCP including uncertainty for the example set +XS with resolution B = 10. The parameters are specified in table 2 and table 5. +12 + +Again, we compare the results from simulated annealing (figure 5a) and quantum annealing (figure 5b) to the optimal +solution, shown in figure 5c. +−0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +time [h] +0 +200 +400 +600 +800 +1000 +1200 +1400 +power [MWh] +demand +expected demand +RE suppl 1 +RE suppl 2 +unit 1 +unit 2 +(a) Best solution obtained from one run of simulated annealing +(cost=137447). +−0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +time [h] +0 +200 +400 +600 +800 +1000 +1200 +1400 +power [MWh] +demand +expected demand +RE suppl 1 +RE suppl 2 +unit 1 +unit 2 +(b) Best solution obtained from one quantum annealing run on D- +Wave Advantage 5.2 (cost=151944). +−0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +time [h] +0 +200 +400 +600 +800 +1000 +1200 +1400 +power [MWh] +demand +expected demand +RE suppl 1 +RE suppl 2 +unit 1 +unit 2 +(c) Optimal power unit schedule for the UCP specified in table 2 +and table 3 (cost=106950). +110000 +120000 +130000 +140000 +150000 +absolute costs +0.0 +0.1 +0.2 +0.3 +0.4 +relative error +simulated annealing +quantum annealing +opt. sol. relaxed +opt. sol. non-relaxed +(d) Absolute costs and relative error of the solutions from simulated +annealing and quantum annealing, compared to costs of optimal +solution of the UCP and the relaxed UCP from section 3.4. +Figure 5: Comparison of results from the two different solvers to the optimal solution. Figures 5a - 5c show the +calculated power unit schedules and costs are compared in figure 5d. +−0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +time [h] +0 +200 +400 +600 +800 +1000 +1200 +1400 +power [MWh] +demand +expected demand +RE suppl 1 +RE suppl 2 +unit 1 +unit 2 +(a) Best solution obtained from one run of simulated annealing (rel- +ative cost error=2.29 %). +−0.5 +0.0 +0.5 +1.0 +1.5 +2.0 +time [h] +0 +200 +400 +600 +800 +1000 +1200 +1400 +power [MWh] +demand +expected demand +RE suppl 1 +RE suppl 2 +unit 1 +unit 2 +(b) Best solution obtained from one run of quantum annealing (rel- +ative cost error=2.27 %). +Figure 6: Comparison of results from the two different solvers for a solution of B = 4. Both solvers find the +optimal power unit schedule. +13 + +The difference in the quality of the solutions of simulated annealing and quantum annealing is bigger than in the +example without uncertainty due to the increased size of the QUBO. However, it can be seen that simulated and +quantum annealing find comparable solutions close but not exact to the optimal solution. As a further example for +the results of an optimization, figures 6a and 6b show the solutions for B = 4. +It can be seen that in this case quantum annealing is even able to beat simulated annealing. However, a smaller +value for the resolution B comes in general with a decreased resolution in demand matching. +4.4 +Benchmarking +We now compare the results of simulated and quantum annealing for the UCP with uncertainty. We introduce several +examples with different problem sizes as shown in table 6. The complexity of the formulation scales with the size +T · (n · (B + 2) + R · nr + nD) +of the solution vector. +Table 6: We consider N units and R renewable units in different example set sizes. Further, we have the number +nR of power stages of the RE units, the number nD of power stages of the demand and T timesteps. +The resulting length of the solution vector equals the number of rows and columns of the corresponding +QUBO matrix for a resolution with B = 10. +example set +XXS +XS +S +M +[N,R,nR,nD,T] +[2,2,2,2,1] +[2,2,2,2,3] +[2,2,3,3,5] +[5,3,5,5,24] +dim(x) +33 +99 +165 +1920 +example set +L +XL +XXL +[N,R,nR,nD,T] +[50,50,5,5,24] +[500,100,20,20,24] +[5000,1000,10,10,24] +dim(x) +9000 +168240 +1680240 +The comparison of simulated and quantum annealing using the D-Wave advantage 5.2 system is shown in figure 7. +20 +40 +60 +80 +100 +120 +dim(x) +0.4 +0.5 +0.6 +0.7 +0.8 +f(¯ν, c) +XXS +XS +S +Benchmark relaxed UCP, B=6, shots=1000 +Simulated annealing +QPU D-Wave Advantage system 5.2 +20 +40 +60 +80 +100 +120 +dim(x) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +time to solve QUBO [s] +XXS +XS +S +Benchmark relaxed UCP, B=6, shots=1000 +Simulated annealing +QPU D-Wave Advantage system 5.2 +Figure 7: Solution quality f(ν, c) (left) and time to solution in seconds (right) over the QUBO size dim(x). The +different example set sizes are marked. +It can be seen that the quantum annealer achieves a slightly better solution quality for bigger problem sets, e.g. we +have a 3% points improvement with example set S. In terms of runtime, the difference is much larger. For example set S, +the quantum annealer only requires 0.16 seconds, which is more than 14 times faster than the simulated annealing +method5. For simulated annealing, the processing time was measured using D-Wave’s neal package. For quantum +annealing, the ""qpu_access_time" was taken. +5We used a laptop with a 10 core CPU and the D-Wave package dwave-neal 0.5.9. +14 + +5 +Discussion and outlook +The hardness of the UCP suggests to reformulate the problem in such a way that advanced non-classical methods can +provide shorter optimization times for finding solutions with a high quality. We showed that such a reformulation can +be achieved by using QUBO’s, which can be solved by Ising machines leveraging quantum computing techniques. +The complexity of the problem correlates with the size of the corresponding QUBO matrix. +Including more +technical constraints may lead to a larger overhead, i.e., introducing additional auxiliary variables like xs or x1/0 +becomes necessary. Regardless of the size of the QUBO, it is possible to construct a correct mathematical model in +which low energy states represent feasible solutions. +The problem comes with the sampling of states corresponding to the desired solution. Algorithms like simulated +annealing fail because the search space is too big. The large number of states make it unlikely for an algorithm that +runs for a limited amount of time to find the global minimum or solutions with a sufficiently low energy. Here, quantum +annealing provides a possible solution, at least at the theoretical level. In the optimal case, the system would always +end up in the global minimum. However, as it can be seen from our results, current state-of-the-art machines are far +away from providing a real advantage over simulated annealing. While good solutions are found for small problem +sizes, the hardware is still several orders of magnitude too small for real-world problems. +For example, a model of the German electricity market would require to include roughly 2000 conventional units +alone [22]. +The number of renewable power supply stations increases steadily. +Therefore, a model including all +power stations individually seems to be unrealistic for several reasons, including a lack of computational resources and +sufficiently large quantum annealing machines. However, a model of local networks with a smaller number of energy +supply units might be possible. +Besides the shortcomings of the current Ising machines, difficulties arise when the QUBO matrix is passed to the +hardware, which needs to map the problem to the QPU topology. This mapping is called minor embedding and requires +time, which has to be added to the actual solving time of the machine. An increasing problem size also leads to longer +embedding times, which, at some point, start to exceed the actual solving time. A pre-defined embedding routine +might fix this issue, but this would strongly depend on the yet unknown topology of bigger QPU’s. +We conclude that our quantum approach to the UCP is a promising alternative to classical solutions under the +assumptions that future annealing machines will eventually reach appropriate levels of size and accuracy. The ongoing +ambitious developments in the field give hope for a future generation of quantum annealers, which may actually be +able to solve such hard problems like the UCP efficiently. +Besides this technical point of view, we also would like to discuss the underlying principle that is used in our +construction. Searching for a solution that lies in the neighborhood of a mean value scenario is a first approach to treat +uncertainties. Since we choose to match only the accumulated mean demand over time, it happens that sometimes +the solution underestimates the power demand in one timestep and overestimates it in the next, as it is the case for +the example in section 4.3. Such a property of the solution would require elements in the power net that compensate +such demand dips and tips. Energy storage systems provide such elements and can act as a damping element for +fluctuations in the net. A more advanced modeling should account for these storage systems and integrate them into +the UCP. +Another issue concerns the principle of the approach we presented here. If one deals with probability distributions +that have multiple maxima, which belong to quite separated scenarios, then this approach leads to insufficient solutions. +For example, consider a solar park that is located in a geographical position such that a passing cloud front in an +otherwise clear sky may or may not block the sun from shining on the solar panels over a period of time. In the most +extreme case, one can then assign a 50% chance of getting no power and a 50% of getting the full supply, since the +exact path that the cloud takes is hard to predict. Using our approach, we would find a solution near a scenario where +50 percent of the full solar power is supplied. Therefore, the situation that actually occurs could mean that we would +have to change the calculated power unit schedule to compensate for a rather big demand deficit or surplus. This +means that we have costs that were not considered in the original computation of a solution. +To avoid this kind of problem, we suggest to modify our approach in such a way that we consider weighted costs +for different scenarios, where the weight is determined by the costs it would take to change the power unit schedule in +a worst case scenario. We then try to find the scenario that balances low operating costs and low schedule changing +costs. +As a final note, this work can also be seen as the foundation to benchmark further quantum algorithms (e.g. VQE +and QAOA) that can also solve QUBO problems. +6 +Acknowledgments +This work is supported by the Federal Ministry for Economics and Climate Action through the project ‘EnerQuant’ +(Project-ID 03EI1025B). +15 + +References +[1] European Commission, “REPowerEU Plan,” 2022. https://energy.ec.europa.eu/system/files/2022-05/COM_2022_230_1_EN_ACT_part1_v5.pdf. +[2] F. Sensfuß, M. Ragwitz, and M. Genoese, “The merit-order effect: A detailed analysis of the price effect of +renewable electricity generation on spot market prices in germany,” Energy Policy vol. 36, pp. 3086–3094, 2008. +[3] W. L. Snyder, H. D. Powell, and J. C. Rayburn, “Dynamic programming approach to unit commitment,” IEEE +Transactions on Power Systems, vol. 2, no. 2, pp. 339–348, 1987. +[4] D. Murata and S. Yamashiro, “Unit commitment scheduling by lagrange relaxation method taking into account +transmission losses,” Electrical Engineering in Japan, vol. 152, no. 4, pp. 27–33, 2005. +[5] M. Paredes, L. Martins, S. Soares, and H. Ye, “Benders’ decomposition of the unit commitment problem with +semidefinite relaxation of ac power flow constraints,” Electric Power Systems Research, vol. 192, p. 106965, 2021. +[6] B. Knueven, J. Ostrowski, and J.-P. Watson, “On mixed-integer programming formulations for the unit commit- +ment problem,” INFORMS Journal on Computing 32(4), pp. 857–876, 2020. +[7] G. Dalal and S. Mannor, “Reinforcement learning for the unit commitment problem,” jun 2015. +[8] M. Håberg, “Fundamentals and recent developments in stochastic unit commitment,” International Journal of +Electrical Power and Energy Systems, vol. 109, pp. 38–48, 2019. +[9] P. Bendotti, P. Fouilhoux, and C. Rottner, “On the complexity of the unit commitment problem,” Ann Oper Res +274, p. 119–130, 2019. +[10] A. Montanaro, “Quantum speedup of branch-and-bound algorithms,” Phys. Rev. Res., vol. 2, p. 013056, Jan 2020. +[11] S. Chakrabarti, P. Minssen, R. Yalovetzky, and M. Pistoia, “Universal quantum speedup for branch-and-bound, +branch-and-cut, and tree-search algorithms,” 2022. +[12] H. Oshiyama and M. Ohzeki, “Benchmark of quantum-inspired heuristic solvers for quadratic unconstrained binary +optimization,” Scientific Reports 12, 2146, 2022. +[13] T. Albash and D. A. Lidar, “Adiabatic quantum computation,” Rev. Mod. Phys., vol. 90, p. 015002, Jan 2018. +[14] P. Hauke, H. G. Katzgraber, W. Lechner, H. Nishimori, and W. D. Oliver, “Perspectives of quantum annealing: +methods and implementations,” Reports on Progress in Physics, vol. 83, p. 054401, may 2020. +[15] A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’Brien, +“A variational eigenvalue solver on a photonic quantum processor,” Nat Commun 5, 4213, 2014. +[16] E. Farhi and A. W. Harrow, “Quantum supremacy through the quantum approximate optimization algorithm,” +2016. +[17] B. Dimitris and J. Tsitsiklis, “Simulated annealing,” Statistical science 8.1, pp. 10–15, 1993. +[18] P. Halffmann, P. Holzer, K. Plociennik, and M. Trebing, “A quantum computing approach for the unit commitment +problem,” 2022. +[19] A. Lucas, “Ising formulations of many np problems,” Frontiers in Physics, vol. 2, 2014. +[20] F. Glover, G. Kochenberger, and Y. Du, “A tutorial on formulating and using qubo models,” 2018. +[21] C. McGeoch and P. Farré, “D-wave technical report series, the advantage system: Performance update,” 2021. +https://www.dwavesys.com/media/kjtlcemb/14-1054a-a_advantage_system_performance_update.pdf. +[22] Bundesnetzagentur, “Kraftwerkliste,” 2022. https://www.bundesnetzagentur.de/SharedDocs/Downloads/DE/Sachgebiete/Energie/Unternehmen_Institutionen/Versorgungssicherheit/Erzeugungskapazitaeten/Kraftwerksliste/Kraftwerksliste_2022.xlsx?__blob=publicationFile&v=11. +16 + diff --git a/aNAzT4oBgHgl3EQfK_vP/content/tmp_files/load_file.txt b/aNAzT4oBgHgl3EQfK_vP/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..16be7134d3ef57131e19b4fc42ce7362155764fb --- /dev/null +++ b/aNAzT4oBgHgl3EQfK_vP/content/tmp_files/load_file.txt @@ -0,0 +1,856 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf,len=855 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='01108v1 [quant-ph] 3 Jan 2023 Towards optimization under uncertainty for fundamental models in energy markets using quantum computers M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Braun, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Decker, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Hegemann, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Kerstan, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Lorenz JoS QUANTUM GmbH, Frankfurt am Main, Germany ∗ January 4, 2023 Abstract We present a method to formulate the unit commitment problem in energy production as quadratic unconstrained binary optimization (QUBO) problem, which can be solved by classical algorithms and quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' We suggest a first approach to consider uncertainties in the renewable energy supply, power demand and machine failures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' We show how to find cost-saving solutions of the UCP under these uncertainties on quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' We also conduct a study with different problem sizes and we compare results of simulated annealing with results from quantum annealing machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' 1 Introduction Electrical energy production and supply is a foundation of developed economies and essential for the stability of societies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Energy markets provide producers, consumers and grid operators an efficient way for pricing electricity, depending on forecasted and actual demand and supply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' As the grid frequency needs to be stable (50 Hz in Europe) and as big amounts of electrical energy cannot be stored efficiently, transmission operators need to balance the output of power generation units and the consumption in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Energy production based on solar and wind power depends on weather conditions and these can change rather quickly, introducing extreme volatility in production levels and prices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' The European Union plans to continue the substitution of power plants powered by fossil fuels, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' coal, oil and gas, with renewable sources of energy [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' This introduces an even higher volatility while forecasting the feed-in to the energy grid is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Therefore, smart ways to steer the production as well as the consumption need to be introduced to produce energy with the lowest cost while satisfying demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Finding a power plant dispatch schedule whose expected supply matches the demand at any given point in time and comes with the lowest possible costs is a well-known optimization problem in energy markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Energy prices are usually derived from the power plant with the lowest marginal costs that match the energy demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' This is called merit order [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Beside this, a number of technical constraints on the power generating units must be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Conventional power plants cannot freely be switched on and off and have to obey limits in their power generation range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' The problem of finding a suitable power unit schedule is known as unit commitment problem (UCP) and it is NP-hard to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Several approaches exist to solve such types of problems including dynamic programming [3], Lagrangian relaxation [4], Benders decomposition [5], mixed integer programming [6] and reinforcement learning [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' The feed-in from renewable power plants like solar parks and wind farms depend on factors like sunshine hours and wind speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' The accuracy of predictions of the weather is limited and this leads to an unknown residual future power demand that conventional power plants have to compensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Other uncertainties come in the form of machine failure events or unpredictable demand fluctuations on small time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Considering such uncertainties in the UCP leads to the stochastic unit commitment problem (SUCP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' There exist many approaches to the SUCP and an overview is given in [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Since it can be shown that the UCP is NP-hard [9] (and so is the SUCP), there exists no efficient method to solve this problem exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Instead, approximate solutions and solutions to relaxed problems are used in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Quantum computing may open new ways of solving the UCP and the SUCP efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Recent theoretical work has shown that with quantum algorithms we can achieve advantages over classical methods for several problems in theory [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' One way is to use annealing methods that are carried out by Ising machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' One example of commercially available quantum annealers are the machines from D-Wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' These machines leverage quantum mechanical effects like quantum tunneling to find optimal solutions to annealing problems quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' This opens the path to provide solutions with higher quality in less time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Although the technology still has to overcome technical limitations before reliable large-scale quantum computers are available, the fast progress in the field promises a true alternative to classical solvers in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' In this ∗For contact email: niklas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='hegemann@jos-quantum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='de.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Authors are listed in alphabetical order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' 1 paper, we explore the possibility of formulating the UCP in terms of a binary quadratic unconstrained optimization (QUBO) problem that can be solved by many different solvers, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' by quantum inspired algorithms [12], quantum annealers [13, 14] and variational quantum algorithms like the variational quantum eigensolver [15] or the quantum approximate optimization algorithm [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' We use simulated annealing [17] as well as quantum annealing to find solutions for our QUBO formulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' A similar approach for the static UCP was published recently in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Here, we go beyond this method by advancing the QUBO formulation to solve a relaxed version of the UCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' In this version, we assume that the overall supply needs to match the overall demand, but at each timestep supply and demand can deviate and a small over- or undersupply can be compensated with storage devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' The results for several examples were obtained by classical solver algorithms and by D-Wave annealing hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' This work might help hardware companies to better understand the hardware requirements for real-world business applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' 2 Elements of the unit commitment problem In this work, we restrict ourselves to the single bus approximation where the geographical structure of the electricity grid is ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' The demand is considered to be satisfied whenever the total production equals the total demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' The UCP refers to the optimization task of finding the cost optimized production plan of N conventional power generators whose total energy supply adds up to demand(t) at every timestep t ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=', T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Each power generator k ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' , N} can generate a power output genk with variable costs varcostk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' the costs per produced unit in a timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Startup costs startcostk are introduced whenever a unit is turned on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' We extend the problem by introducing R renewable units that run without variable costs and that have a fixed power-time profile supplyRE r (t) which is determined by external factors, like the weather or tides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' These renewable units effectively reduce the required amount of energy and give a residual demand that has to be compensated by conventional power plants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' We call this an effective demand d(t) and we require a solution of the UCP to provide d(t) = demand(t) − R � r=1 supplyRE r (t) = N � k=1 genk(t) for all t ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=', T} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' More complexity arises when restrictions that originate from power generators are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' The power generating units need to obey the following constraints: The power supply of each unit has a lower bound mingenk and an upper bound maxgenk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=', genk(t) ∈ [mingenk, maxgenk] for all t ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=', T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Furthermore, power generators exhibit technical minimum run times minupk and minimum shut down times mindownk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' This means that once power unit k is turned on it can not be turned off after running for at least minupk timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' A similar condition ensures compliance of downtimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' To summarize, a specific problem instance of the UCP is fully determined by the following set of parameters: demand(t) for t ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' , T} supplyRE r (t) for r ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=', R} and t ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=', T} mingenk, maxgenk, varcostk, minupk, mindownk for k ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=', N} 3 QUBO formulation with penalty terms 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='1 Formulation of QUBO problems In this section, we discuss techniques that can be used to formulate combinatorial optimization problems as Quadratic Unconstrained Binary Optimization (QUBO) problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' It is a mathematical construct equivalent to Ising spin glasses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Once formulated, these QUBO problems can be solved by different techniques, including quantum annealers, quantum computers and other types of Ising-machines [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Such optimization problems can be formulated as follows when we set B = {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Let fQ : Bn → R be the quadratic polynomial fQ(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' , xn) = n � i=1 n � j=1 qijxixj (1) with qij ∈ R, which can be understood as real valued entries of a matrix Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' The function fQ is called the objective function or objective of the corresponding QUBO matrix Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' For properly chosen coefficients qij, the global extrema 2 of the objective function correspond to a solution of an optimization problem, where the solution itself is encoded in the binary-valued vector x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' , xn) ∈ Bn that fulfills1 min(fQ) = n � i=1 n � j=1 qijxixj or max(fQ) = n � i=1 n � j=1 qijxixj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' A more elegant way to write equation (1) is using Dirac’s bracket notation: fQ(x) = ⟨x | Qx⟩ Here, ⟨x|y⟩ is the scalar product of two vectors |x⟩ and |y⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' The matrix Q acts as an operator on the corresponding vector space and ⟨x| is the adjoint of |x⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Subsequently, this notation is used in some cases for simplifying the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' In analogy to the quantum mechanical origin of this notation, the solution to a minimization problem corresponds to a global minimum in the energy landscape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' An efficient QUBO solver algorithm is a machinery that takes Q as an input and outputs the vector |x⟩ of the global minima in a reasonable amount of time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' such an algorithm would provide us with a tool to solve optimization problems of this kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' To the best of our knowledge, there exists no reliable and fast algorithm to solve universal problems of this type as it can be shown that QUBO problems are NP-hard [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' However, several kinds of algorithms exist that can be used to find approximate good solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Here, we compare simulated annealing to the results of hardware-based quantum annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' A quantum annealer represents a special type of quantum computer that performs approximate adiabatic quantum annealing and is able to provide a QUBO solver, at least on a theoretical level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='2 Building the QUBO for the UCP The strategy for defining the QUBO elements qij is to use quadratic terms that give higher objective values for expensive and forbidden states and lower objective values for cheaper and allowed states, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Forbidden and allowed in this context refers to the adherence to constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' As introduced in section 2, we want the following conditions and technical constraints to be fulfilled in the UCP: minimizing costs matching power demand obey minimum and maximum power generation constraints obey minup and mindown times We start the QUBO formulation of the UCP by introducing a resolution of the power supply in the form of a binary representation of numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Since we work in a discrete model, it is not possible for genk(t) to take continuous values between mingenk and maxgenk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' However, we can approximate the continuous case arbitrarily well by introducing a discretization of the interval [mingenk, maxgenk] and we write maxgenk − mingenk = B � b=1 pk,b with pk,b = 2b−1 · (maxgenk − mingenk) 2B − 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' (2) Here, the number of bits used for the discretization is B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Furthermore, we introduce the variables xp k,b(t) for b ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=', B} for each timestep t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' The variable xp k,b(t) describes how much the power supply of unit k contributes to the sum (2) depending on level b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' We also introduce the binary variables x1/0 k (t), which describe whether unit k is on or off in timestep t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Then we can write genk(t) = mingenk · x1/0 k (t) + B � b=1 pk,b · xp k,b(t) = mingenk · x1/0 k (t) + ⟨pk|xp k(t)⟩ (3) with pk = (pk,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' , pk,B) and xp k(t) = (xp k,1(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' , xp k,B(t)), which suits the QUBO formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Increasing the resolution B increases the number R(B) = 2B of possible power supply realizations and it improves the fineness F(B) = maxgenk − mingenk 2B − 1 of the power range as can be seen in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' The introduction of an additional variable xs k(t) per unit and timestep is required to describe the start of the power production of a unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' If unit k is off in timestep t − 1 and if it is turned on in timestep t, then we set the start-variable xs k(t) = 1 and xs k(t) = 0 else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' 1In the following, we write vectors (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' , xn)T as row vectors (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' , xn) to simplify notation and to avoid ambiguities with the number T of timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' 3 Table 1: Discretization of the interval [x, y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Number R(B) of power supply realizations and the relative fineness F(B)/(y − x) as functions of the resolution parameter B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' B R(B) F(B)/(y − x) 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='3333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' 3 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='1429.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' 4 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='0607.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' 6 64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='0159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' 10 1024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' 20 1048576 ≈ 1 · 10−6 For timestep t, we choose the corresponding part of the full solution vector to be arranged like x(t) = (xp(t), x1/0(t), xs(t)), where xp(t) = (xp 1(t), xp 2(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' , xp N(t)) encodes the power supply stages of all units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' The vectors x1/0(t) = (x1/0 1 (t), x1/0 2 (t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' , x1/0 N (t)) and xs(t) = (xs 1(t), xs 2(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' , xs N(t)) contain the on/off and start information, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Each element xp k(t) has the structure xp k(t) = (xp k,1(t), xp k,2(t), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' , xp k,B(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' The introduction of x1/0(t) enables us to implement the minimum power constraint, which comes at the cost of increasing the problem size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' The full solution vector for T timesteps is given by x = (x(1), x(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' , x(T )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' (4) The length of this vector is given by dim(x) = T N(B + 2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' it grows linearly with the number of timesteps T , units N and resolution B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' The next step is to build the QUBO matrix Q by translating the UCP into matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' We do this step by step and implement the individual problem specification via defining corresponding penalty terms, which are added together at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' The minimal costs for the optimal power unit allocation include the variable costs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' the running cost per unit time, as well as start costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' A suitable penalty term is given by P cost · T � t=1 N � k=1 varcostk · � mingenk · x1/0 k (t) + ⟨pk | xp k(t)⟩ � + startcostk · xs k(t), (5) where P cost is a penalty parameter that has to be adjusted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Note that this form is equivalent to equation (1), since for binary variables we can use the equality x = x2 to substitute linear terms with quadratic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' It is also possible to directly introduce a quadratic cost term by squaring (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' We obtain P cost · � T � t=1 N � k=1 varcostk · � mingenk · x1/0 k (t) + ⟨pk | xp k(t)⟩ � + startcostk · xs k(t) �2 and this enlarges the differences of the objective function for different states by increasing the range of the energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' We quantify this by defining κk(x(t)) = varcostk · � mingenk · x1/0 k (t) + ⟨pk | xp k(t)⟩ � + startcostk · xs k(t) and observe that κk(x(t)) ∈ [0, κmax k ] where κmax k = varcostk · maxgenk + startcostk is the upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' It is now obvious that P cost · T � t=1 N � k=1 κk(x(t)) < P cost � T � t=1 N � k=1 κk(x(t)) �2 if �T t=1 �N k=1 κk(x(t)) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Since the penalty term can only take non-negative values, we can rescale the costs to ensure that this is always true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Hence, the energy spectrum in the case of the quadratic term has a larger range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Also, if we additionally define κmax = max{κmax 1 , κmax 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' , κmax K }, the following inequalities are true: P cost · T � t=1 N � k=1 κk(x(t)) < P costT Nκmax P cost · � T � t=1 N � k=1 κk(x(t)) �2 < P costT 2N 2(κmax)2 4 This shows that the energy range is potentially larger in the case of a quadratic cost term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' From now on, we choose to continue with the quadratic cost term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' The next step is to match the power demand in every timestep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' We require the supply resulting from an optimal power unit schedule to match the forecasted energy amount for all timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' As described in section 2, we use an effective demand d(t) in every timestep and find that a suitable penalty term is given by P demand · T � t=1 � N � k=1 mingenk · x1/0 k (t) + ⟨pk | xp k(t)⟩ − d(t) �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' (7) Again, P demand is a penalty strength which has to be chosen properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Note that this gives multiple terms of the form d(t)2, which do not include x and therefore cannot take the form of equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' However, we can subtract these constant terms from (7) and this leads to a constant energy shift for all states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Consequently, minimizing this new function also minimizes (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' We now add a penalty to respect the technical restriction of minimum run times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' We have specified the minimum run time of unit k via minupk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' When unit k is turned on in timestep t, the earliest shutdown of this unit is possible after timestep t + minupk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' the unit has to run for minupk timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' With the penalty strength P minup, we use the penalty term P minup · T � t=1 N � k=1 xs k(t) � minupk · xs k(t) − t+minupk−1 � τ=t x1/0 k (τ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' If the minimum run time condition is fulfilled, the penalty vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Note that in the summation over τ we only consider elements up to at most τ = T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' The penalty terms for minimum down times work in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' When unit k is turned off in timestep t, the earliest possible start of this unit is after timestep t + mindownk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=', the unit has to be shut down for mindownk timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Introducing the penalty strength P mindown, we choose P mindown · T � t=1 N � k=1 � x1/0 k (t − 1) − x1/0 k (t) + xs k(t) � t+mindownk−1 � τ=t x1/0 k (τ) as the corresponding term in the QUBO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' As in the case of minimum run time, fulfilling the minimum down time condition gives a zero penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Note that we do not sum over elements for values of t − 1 and τ that are outside of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' , T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' The minimum and maximum power supply of each power generator are automatically met due to equation (3), if we can ensure that whenever at least one of the xp k,b(t) is non-zero, the variable x1/0 k (t) is also set to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' This requires a penalty term of the form P inter1 · T � t=1 N � k=1 B � b=1 xp k,b(t) � 1 − x1/0 k (t) � , where P inter1 denotes the penalty strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' This term interrelates the variables xp k,b(t) and x1/0 k (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Finally, setting the start variable is required to ensure that we correctly connect the start variable xs k(t) to the on/off-variable x1/0 k (t) of unit k such that we have xs k(t) = � 1 for x1/0 k (t − 1) = 0 and x1/0 k (t) = 1 0 else A suitable penalty term is P inter2 · T � t=1 N � k=1 � � x1/0 k (t + 1) − xs k(t + 1) �2 + x1/0 k (t) · � xs k(t + 1) − x1/0 k (t + 1) � � with the corresponding penalty strength P inter2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Here, we do not sum over elements for values of t + 1 and τ that are outside of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' , T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='We can now write the full QUBO as the sum of the penalty terms above: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='Q(x) =P cost ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='� T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='k=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='varcostk · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='mingenk · x1/0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='(t) + ⟨pk | xp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='k(t)⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='+ startcostk · xs ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='τ=t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='x1/0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='(τ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='+P mindown · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='T ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='τ=t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='x1/0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='(τ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='+P inter1 · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='t=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='N ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='k=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='B ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='b=1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='xp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='b(t) � 1 − x1/0 k (t) � +P inter2 · T � t=1 N � k=1 � � x1/0 k (t + 1) − xs k(t + 1) �2 + x1/0 k (t) · � xs k(t + 1) − x1/0 k (t + 1) � � (8) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content='3 Tuning penalty parameters In this section we derive relations between the different penalty strength parameters that appear in equation (8) to ensure that the model correctly prioritizes different problem constraints and relates good solutions of the UCP to low energy states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' The first requirement is that we want to avoid demand mismatches in favor of cost savings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Therefore, consider a solution x that matches the demand for all timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Further, a second solution is given which equals the first one except that in time t′ there is a supply deficit of ∆gen(t′) = gen(t′) − d(t′) < 0 which comes at the cost saving ∆κ(t′) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' Using equation (8) and the Kronecker delta notation δt,t′, the corresponding condition for the difference of these two solutions is P cost · � T � t=1 N � k=1 κk(x(t)) �2 < P cost · � T � t=1 � N � k=1 κk(x(t)) + ∆κ(t′)δt,t′ ��2 + P demand∆gen(t′)2 We now exploit the following inequality: With ǫ < 0 < |ǫ| < z it follows that (z + ǫ)2 < (z + ǫ)(z − ǫ) = z2 − ǫ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/aNAzT4oBgHgl3EQfK_vP/content/2301.01108v1.pdf'} +page_content=' We use this to estimate the right hand side and with a multiplication with N 2 and we obtain: P cost · � T � t=1 N � k=1 κk(x(t)) �2

> +f i r s t V i d e o P a r t . m4s +The cat command is used to read the data of the following +files and to write them in descending order to the defined +output file. In case the output file does not exist it will be +created. The number of segments that have to be concatenated + +depends on the individual segment length as well as the length +and design of the test sequences. In this study, test sequences +were set to 30 seconds and the segment length during encoding +was set to 5 seconds. Therefore, all test sequences that were +designed to display the stalling event, or switch, in the middle +of the test sequence required concatenating four segments. +The initialization segment is followed by three segments each +representing 5 seconds of content, totaling a length of 15 +seconds per concatenation. The test sequences for the multiple +quality switch, which represented a different temporal event +pattern, only required three segments per concatenated file. +Again the initialization segment is followed by two segments +resulting in an output representing 10 seconds of content. After +concatenating, the output file had to be transformed to .mp4 +file format with the following FFmpeg command: +$ ffmpeg − i +f i r s t V i d e o P a r t . m4s −c copy +f i r s t V i d e o P a r t . mp4 +First, the path to the input file is specified and then its data +is copied in the described output format, in this case, .mp4. +The -c copy option will prevent FFmpeg to re-encode the input +data [23]. +The process of concatenation and transformation to .mp4 had +to be repeated for every individual part that was necessary +to construct the test sequences. The high amplitude quality +switch as well as both stalling events were constructed from +two individual parts each representing 15 seconds of content. +The multiple quality switch, as it displays three different +quality levels, was constructed from three individual parts +each representing 10 seconds of content. In addition, audio +segments had to be concatenated and transformed in the same +manner. Once this process had been concluded, FFmpeg was +used to merge the individual video and audio parts: +$ ffmpeg −y − i +f i r s t V i d e o P a r t . mp4 − i +f i r s t A u d i o P a r t . mp4 −c : v copy −c : a copy +f i r s t P a r t . mp4 +Initially, the y-flag command is used to give global permission +to overwrite the output. Then two inputs are defined and +mapped, the first representing the video and the second the +audio source. As before the copy command is used to ensure +that inputs are not re-encoded. Lastly, the output filename and +format are defined [23]. This process also had to be repeated +until all individual audio and video parts, which were needed +for the designed test sequences, were combined. For the +imitation of a stalling event, an additional buffering sequence +was needed. To create a realistic scenario FFmpeg was used +to cut the last frame of the GOP previous to the stall, which +was then inserted as the background for the buffering wheel +video. This was achieved by the following two commands: +$ ffmpeg − s s e o f +−3 − i +f i r s t S t a l l P a r t . mp4 − +update 1 −q : v 1 lastFrame . png +$ ffmpeg −y − f r a m e r a t e +24 −loop 1 − i +lastFrame +. png − i +loadingWheel . mp4 − f i l t e r +c o m p l e x +’ [ 1 ] format =argb , +color channelm ixer =aa =0.5[ ol +] ; [ 0 ] [ ol ] o v erlay ’ − t +1 +f i n a l B u f f e r E f f e c t . +mp4 +The first command creates the screen capture of the last +frame. The sseof option tells FFmpeg to seek and output all +frames specified by the position, in this case, three seconds, +relative to the end of the input file. To ensure that only +the last frame is saved the update option is changed from +default 0 to 1. This setting will continuously overwrite the +data in the specified output file, ultimately leaving only the +last frame. The encoding quality is set to 1, which represents +the highest quality level. The captured frame is set to be in .png +file format [23]. The second command inserts the previously +captured frame into the buffering wheel video. The framerate +of the output is set to 24 frames per second and the image +file is specified to loop throughout the length of the output +file. Additionally, options to create a complex filtergraph are +defined. First, the color model of the second input referred to +as indexed 1, is set to Alpha Red Green Blue (ARGB). Then +its value for the alpha channel is set to 0.5, which makes the +input source semi-transparent [?]. Following, the first input, +marked as indexed 0, is set as the main input on which the +second input will be overlaid. Then output length is set to +one second and the output name and format are defined [23]. +Once the buffering sequence for all content types was created, +all previously prepared parts were concatenated with FFmpeg +to create the final test sequences. Therefore the following +command was used: +$ ffmpeg −y −f +concat − s a f e +0 − i +l i s t . t x t +−c +copy +f i n a l T e s t S e q u e n c e . mp4 +First, global overwrite permission was defined. The f-flag +option enforces the file format of the concatenated sequences +on the output. The safe option was set to 0 to disable safe +mode, which allows the input of all files. The input specifies +the path to a .txt file which contains the relative paths to the +required files for the test sequences in concatenation order. As +an example the structure of a .txt file used for the two-second +stall test sequence is given: +f i l e +’ s t a l l i n g +e v e n t / f i r s t P a r t . mp4 ’ +f i l e +’ s t a l l i n g +e v e n t / f i n a l B u f f e r E f f e c t . mp4 ’ +f i l e +’ s t a l l i n g +e v e n t / f i n a l B u f f e r E f f e c t . mp4 ’ +f i l e +’ s t a l l i n g +e v e n t / secondPart . mp4 ’ +Lastly, the input data is copied without further encoding to the +specified output .mp4 format. This concatenation process had +to be repeated for each of the defined test sequences of this +study [23]. +D. Effect Design +To answer the research question and to measure the different +effects of stalling events, multiple quality switches and high +amplitude quality switches on QoE the following effects +are proposed: 1-second stalling event Figure 2(a), 2-second +stalling event Figure 2(b), a high amplitude quality switch +from 1080p to 240p Figure 2(c), and a multiple quality switch +from 1080p to 240p to 720p Figure 2(d). The decision for +introducing the 1-second stall was based on a study by De +Pessemier et al., which quantified the effects of stalling events +[25]. Within this study, the median of a single stall event + +fluctuated around one second. The 2-second stall event was +chosen to analyze the effect difference between a 1-second +stall compared to a 2-second stall. The quality levels for the +high amplitude as well as the multiple quality switch were +based on a study that investigated, in particular, the impact of +different switching patterns on QoE [26]. The study indicated +that both events would resolve in worse QoE than other events. +To avoid additional influences of effects such as the primacy +or recency effect, which assume that the first or last seconds +of a video has a higher impact on the viewers’ QoE, the stall +event was placed in the middle of each test sequence [27]. +E. Graphical User Interface +In order to display the study to the participants, a website- +based Graphical User Interface (GUI) was developed. The +website was based on HTML, CSS, and JavaScript. In- +dexedDB, a low-level API that can pre-cache data on the +client-side, was used to prevent unwanted impairments [28]. +The caching process of the test sequences would immediately +commence when the participant opened the website and con- +tinued in the background without any disturbance. This was +implemented to reduce potential waiting times for the users +later in the study. In the backend, a node.js express server was +used to request the prepared test sequences from the hosting +server and to handle the connection to the database. The test +sequences were stored using Amazon S3 and, from there, +requested to the website. For deployment, Amazon’s EC2 +instances were chosen. In accordance with the best practice +framework, consideration was given to the size of the instance. +Ultimately, a small instance type was chosen, which was +expected to be sufficient for the extent of this study. Upon +opening the website, participants would see an introductory +text that explained the study procedure and an input field that +asked for an MTurk ID. After submission, an example video +sequence would be displayed to familiarize the participants +with the player design. The video player, implemented using +the HTML video tag, was minimized to the viewing field +and all additional customization options such as fullscreen, +start/stop, and audio options were deactivated. Furthermore, +the width of the video player was fixed to 1280px. This +measure was taken to ensure that every participant would +have the same experience from a technical perspective. A +customized overlay was cast on each test sequence to allow +participants to pause if necessary. Once the playback of a test +sequence was started, the video player would position itself +in the middle of the screen and the background would turn +grey as suggested in the ITU-T standard P.910 [29]. After the +test sequence finished, the background would turn white, and +only then would the questionnaire appear. This functionality +was implemented to ensure that participants could not simply +click through the study without watching the test sequences. +As suggested in [15] this measure would also ensure that the +proper completion of the study would require the same time +as a cheating attempt. Each questionnaire contained a content- +related question, which was treated as the gold standard data +and the 5-point quality scale scores. +As the main aim of this study was set on comparing +the resulting quality of the chosen effect types, the ACR +methodology was chosen. The test sequences were shown +in the following order: Big Buck Bunny, Elephants Dream, +The Swiss Account, and Tears of Steel (Figure 3). While the +content order remained the same between participants, the +effects were randomly chosen by an algorithm implemented +with JavaScript. The algorithm varied combinations of effect +and content types but ensured that each participant would +watch each effect once. Upon completion, the results would be +sent to the express server and stored in the MongoDB database. +The following data was stored: MTurk ID, timestamp of +submission, answers to the questionnaire, and an automatically +generated UUID for each user. The UUID would also be sent +back to the website and displayed to the participant. Mobile, +as well as tablet devices, were excluded from the study. If a +participant tried to access the website via these devices, an +error message would appear suggesting to switch to a desktop +or laptop device. Similarly, any browser other than Google +Chrome was excluded from the study as trials showed that +they did not support the manual bitrate switching within the +manipulated sequences. A similar error message would be +displayed in this case. +F. Amazon Mechanical Turk +Amazon Mechanical Turk (MTurk) is a crowdsourcing +marketplace which enables companies and individuals to out- +source tasks virtually [30]. The platform allows for unlimited +campaign creation and customized settings. Each campaign +must have a short description text, a set time frame, and al- +located compensation. Other elements that can be customized +are among other geographical locations and minimum user +rating. A trial campaign with 5 test subjects was created +to test functionality and user reliability. The timeframe was +set to 15 minutes and compensation to 0,50 $. To receive +compensation, users had to input their MTurk ID in the study’s +interface as well as paste the UUID which they received after +completion back on the MTurk platform. This measure was +implemented to ensure that users did complete the study. It +also allowed cross-checking between the database and the +MTurk platform to filter out double participation. The initial +test confirmed the functionality of the campaign. Despite the +given functionality of the interface and the MTurk setup, +none of the five datasets provided accurate answers to the +gold standard data. For further testing purposes, a second +campaign with 10 participants was started. In addition to the +prior settings, this campaign was only accessible to users with +an acceptance rating of 95% and higher and limited to 10 +minutes. Upon initial analysis, the second campaign appeared +successful as the gold standard data was answered correctly. +Therefore, it was assumed that participants in the second +campaign watched the content properly. Hence a third and +last campaign with 60 additional participants and the same +settings as prior was initiated. +During the in-depth analysis of the MTurk results, the fol- +lowing issues could be discovered: For one, all effects achieved + +(a) 1s stall event at 15s +(b) 2s stall event at 15s +(c) High amplitude quality switch +(d) Multiple quality switch +Fig. 2: Representation of the One and Two Second Stall Events and High Amplitude Quality Switch (1080p-240p) and Multiple +Quality switches (1080p-240p-480p) +Fig. 3: Graphical Representation of the ACR Stimulus Presentation in the Subjective Study - BBBr, Big Buck Bunny test +sequence at random effect r - EDr, Elephants Dream test sequence at random effect r - SWACCr, The Swiss Account test +sequence at random effect r - TOSr, Tears of Steel test sequence at random effect r +relatively high grading. Secondly, there were many instances in +which the participants rated all effect types equally, indicating +no difference between effect types. Based on these findings, +the following assumptions were made: a 5-point scale does +not give participants enough variety to judge the effects with a +meaningful difference. Some participants were only interested +in concluding the survey as fast as possible to receive the +incentive. Therefore, they did not take the time to properly +watch and evaluate the test sequences, so no differences in +effect types are noted. +G. Follow-Up Study +In order to respond to the stated observations, a follow- +up study was designed. Three major changes were made to +consider the previous findings and assumptions. First of all, +the 5-point rating scale was changed to the recommended 9- +point scale to give the participants a higher discriminative +rating power. Secondly, to respond to the equal ratings between +effect types, four additional test sequences were introduced. +Therefore, every participant would rate 8 sequences and ex- +perience each effect type twice. This change introduced an +additional reliability measure as participants who displayed +a significant variance within an effect type could be further +investigated. The additional sequences were taken from the +same four movies to prevent further potential content effects. +The last change concerned the participants. Participants in the +follow-up study received no compensation. Instead, they were +taken from a pool of volunteers to diminish the chances of +unreliable results due to an incentive. To replace MTurk’s time +frame setting two additional variables were introduced and +saved in the database: timestamp upon watching the first test +sequence and timestamp of submission. The introduction of +the rating scale on the introductory page and the scale beneath +each test sequence was adapted accordingly Figure 5. +IV. STATISTICAL ANALYSIS AND FINDINGS +The following section will summarize the statistical results +for both studies, once with participants which received an in- +centive and, secondly, with volunteers. For the statistical anal- +ysis, a One-Way ANOVA was conducted to validate significant +variances between the different effect groups. For the analysis +within effect types, a two-sample t-test was performed. All +statistical tests used an alpha level of 0,05. Two POST HOC +tests were used to validate the significant difference. For one, +the Bonferetti test with a corrected alpha value. Secondly, the +Tukey B test. In addition to significant variances between +effect types, the results were also checked for differences +between content types. For reasons of simplicity, figures and +tables in the following chapter will describe the effects with +the following abbreviations: 1-second stall as Os, 2-second +stall as Ts, multiple quality switch as Ms, and high amplitude +quality switch as Ha. +A. MTurk Evaluation +In total, 72 entries were submitted to the database. It can +be assumed that the additional two participants completed +the study but did not manage to input the UUID back on +MTurk. Therefore, they were not counted as participants by +MTurk but still submitted their results to the database. For this +reason, these two as well as 10 other datasets were excluded + +5000 +4000 +(kbps) +3000 +ls stall event +Bitrate ( +2000 +1000 +0 +0 +10 +20 +30 +Time (in seconds)5000 +4000 +Bitrate (kbps) +2s stall event +3000 +2000 +1000 +0 +0 +10 +20 +30 +Time (in seconds)5000 +4000 +Bitrate (kbps) +3000 +2000 +1000 +0 +0 +10 +20 +30 +Time (in seconds)5000 +4000 +Bitrate (kbps) +3000 +2000 +1000 +0 +0 +10 +20 +30 +Time (in seconds)Grey +Grey +Grey +Grey +Grey +Example +BBBr +EDr +SWACCr +TOSr +Introduction +30 seconds +30 seconds +voting +30 seconds +voting +30 seconds +voting +30 seconds +voting +Thank you +&UUIDFig. 4: Graphical Representation of the ACR Stimulus Presentation in the Follow-Up Study - BBB1r & BBB2r, First and +Second Big Buck Bunny test sequence at random effect r - ED1r & ED2r, First and Second Elephants Dream test sequence +at random effect r - SWACC1r & SWACC2r, First and Second The Swiss Account test sequence at random effect r - TOS1r +& TOS2r, First and Second Tears of Steel test sequence at random effect r +Fig. 5: GUI after Test Sequence Playback displaying Ques- +tionnaire +from analysis due to wrong gold standard data or double +participation. The ratings of the remaining 60 participants were +then checked for significance with a one-way ANOVA. The +evaluation, as seen in Table III showed a minor significance be- +tween the effect groups. Therefore two POST HOC tests were +conducted in which all effects were weighed against each other +(Table IV). Following the Bonferroni method, the corrected +alpha value was calculated, resulting in α = 0, 008333, which +indicates no significance between the effect groups. The same +results were achieved with the Tukey B POST HOC test with a +Q-value of Q = 3, 66 and a resulting T-value of T = 0, 41583. +B. Follow-up Study Evaluation +For the follow-up study, 43 data sets were collected. Accord- +ing to the ITU-T P.910 recommendation, sample sizes larger +than 40 merely result in minimal differences [29]. Therefore, +the 43 datasets were assumed sufficient for statistical purposes. +None of the participants in the follow-up study had any prior +experience with video or image ratings. The dataset analysis +showed that all participants answered the gold standard data +correctly. However, 6 participants were excluded as their study +completion time varied two standard deviations from the mean, +indicating that they were not paying close attention to the test. +A two-sample t-test was performed for each effect type to +check for significant differences between the first rating of an +effect contrary to the second rating. No significant difference +was found. According to the resulting MOS, effects ranked +in the following descending order: 1-second stall, multiple +quality switch, 2-second stall, and high amplitude quality +switch (Figure 1). The results of the 9-point scale were mapped +to a 5-point MOS based on the following formula, where NP +represents the rating value on the 9-point scale: +MOS = NP +2 ++ 0, 5 +Equally to the previous study, the variance between the +effect types was checked with a One-Way ANOVA, which +indicated a significant variance (Table III). The corrected alpha +remained the same for the Bonferroni POST HOC method +with α = 0, 008333. In the case of the 1-second stall and +the high amplitude quality switch, as well as the multiple +quality switch and the high amplitude quality switch, P < +corrected α = 0, 00833, indicating significance. The Tukey B +POST HOC test confirmed these results as T > Absolute Mean +Difference (Table IV). Therefore it can be assumed that there +was a significant difference between the high amplitude quality +switch and 1-second stall and the high amplitude quality +switch and the multiple quality switch events. In both cases, +the high amplitude quality switch performed significantly +worse. In addition, the results were checked for significant +variance between content types. The one-way ANOVA indi- +cated significance (Table III). The results of the Tukey B POST +HOC (T = 0, 764) confirmed a significant difference between +Big Buck Bunny and the other movie sequences. Throughout +all effect types, the Big Buck Bunny clips achieved the highest +MOS (Figure 1). +V. CONCLUSIONS AND FUTURE WORK +This paper investigated whether stalling events have the +highest impact on decreasing the viewer’s QoE compared +to multiple or high amplitude quality switches. The results +confirm that when compared to multiple and high amplitude +quality switches, stalling events do not have the worst impact + +Grey +Grey +Grey +Grey +Grey +Grey +Grey +Grey +Grey +Example +BBB1r +ED1r +SWACC1r +TOS1r +BBB2r +ED2r +SWACC2r +TOS2r +Introduction +30 s +30 s voting +30 s voting +30svoting +30 svoting +30 s voting +30 s voting +30s voting +30 s voting Thank you +&UUIDWhat could you see in the video? +O Forest/Jungle +O Airplane +O sking +O Bouldering +How would you describe your watching experience of this video on a scale from 1 to 9 (1 being "poor" and 9 being "excellent")? +0102 +03 +04 +0 5 +06 +07 +0809 +ixanTABLE III: Analysis of Variance (One-Way ANOVA, between +Groups) of the different Effect and Content Types of the +Subjective Results of MTurk and Laboratory Participants +Evaluation +df 1 +Mean Square +F 2 +P 3 +F 4 +Effects (MTurk) +3 +2.070 +2.673 +0.048 +2.642 +Contents (MTurk) +3 +1.348 +1.720 +0.163 +2.642 +Effects (Lab) +3 +4.721 +5.758 +0.00077 +2.635 +Contents (Lab) +3 +22.382 +6.900 +0.00016 +2.635 +1 Degree of freedom, number of total categories minus 1 +2 F-value, the significance of variance between two populations +3 P-value, significant if P < 0.05 (chosen α level) +4 Critical F-value, if F > Fcrit then significance +TABLE IV: Post hoc (Bonferroni, corrected α = 0, 0083) and +Tukey B (T = 0, 415) for MTurk and Laboratory Participants +between Effect Types +Environment +I 1 +J 1 +P 2 +Absolute Mean Difference 3 +MTurk +Ha +Ms +0.649 +0.083 +MTurk +Ha +Os +0.066 +0.3 +MTurk +Ha +Ts +0.018 +0.4 +MTurk +Ms +Os +0.161 +0.216 +MTurk +Ms +Ts +0.049 +0.316 +MTurk +Os +Ts +0.458 +0.1 +Laboratory +Ha +Ms +0.00099 +0.486 +Laboratory +Ha +Os +0.00012 +0.574 +Laboratory +Ha +Ts +0.0253 +0.358 +Laboratory +Ms +Os +0.526 +0.087 +Laboratory +Ms +Ts +0.399 +0.128 +Laboratory +Os +Ts +0.159 +0.216 +1 I & J, different effect types +2 Two-tail t-test +3 | I − J | +on QoE. 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Yang, “A method of qoe evaluation +for adaptive streaming based on bitrate distribution,” in 2014 IEEE +International Conference on Communications Workshops (ICC), 2014, +pp. 551–556. +[28] “Indexeddb,” +https://www.w3.org/TR/IndexedDB/, +last +accessed: +17.03.2022. +[29] “Recommendation p.910, subjective video quality assessment methods +for multimedia applications,” International Telecommunication Union, +Standard. +[30] “Amazon mechanical turk,” https://www.mturk.com/, last accessed: +17.03.2022. + diff --git a/atFRT4oBgHgl3EQfQze_/content/tmp_files/load_file.txt b/atFRT4oBgHgl3EQfQze_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f9e29569f2d70bd4c2bce38da13649d0a330cb48 --- /dev/null +++ b/atFRT4oBgHgl3EQfQze_/content/tmp_files/load_file.txt @@ -0,0 +1,672 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf,len=671 +page_content='Towards Better Quality of Experience in HTTP Adaptive Streaming Babak Taraghi Christian Doppler Laboratory ATHENA Institute of Information Technology (ITEC) Alpen-Adria-Universit¨at Klagenfurt Klagenfurt, Austria babak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='taraghi@aau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='at Selina Zo¨e Haack Christian Doppler Laboratory ATHENA Institute of Information Technology (ITEC) Alpen-Adria-Universit¨at Klagenfurt Klagenfurt, Austria zohaack@edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='aau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='at Christian Timmerer Christian Doppler Laboratory ATHENA Institute of Information Technology (ITEC) Alpen-Adria-Universit¨at Klagenfurt Klagenfurt, Austria christian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='timmerer@aau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='at Abstract—HTTP Adaptive Streaming (HAS) is nowadays a popular solution for multimedia delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The novelty of HAS lies in the possibility of continuously adapting the streaming session to current network conditions, facilitated by Adaptive Bitrate (ABR) algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Various popular streaming and Video on Demand services such as Netflix, Amazon Prime Video, and Twitch use this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Given this broad consumer base, ABR algorithms continuously improve to increase user satisfaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The insights for these improvements are, among others, gathered within the research area of Quality of Experience (QoE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Within this field, various researchers have dedicated their works to identifying potential impairments and testing their impact on viewers’ QoE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Two frequently discussed visual impairments influencing QoE are stalling events and quality switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' So far, it is commonly assumed that those stalling events have the worst impact on QoE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' This paper challenged this belief and reviewed this assumption by comparing stalling events with multiple quality and high amplitude quality switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Two subjective studies were con- ducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' During the first subjective study, participants received a monetary incentive, while the second subjective study was carried out with volunteers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The statistical analysis demonstrated that stalling events do not result in the worst degradation of QoE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' These findings suggest that a reevaluation of the effect of stalling events in QoE research is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Therefore, these findings may be used for further research and to improve current adaptation strategies in ABR algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Index Terms—HTTP Adaptive Streaming, Quality of Experi- ence, Subjective Evaluation, Crowdsourcing I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' INTRODUCTION HAS has been disrupting the industry since its first release by Move Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Various researchers are focusing on opti- mizing the adaptation logic that brought HAS its popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' A significant part of the development of HAS is credited to Quality of Experience (QoE) research, which aims to improve the consumers’ experience [1]–[5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' These studies aim to identify parameters that positively and negatively impact the users’ experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' A stalling event happens when the player buffer is empty and the player can not continue the playback, indicated often with a spinning wheel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The quality switch is the player function in which the media representation will be changed to continue the playback with lower or higher media quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Following the focus of this paper, three main findings should be pointed out: Stalling events are generally assumed to have the worst impact on QoE [6], [7] Multiple quality switches have a negative effect on QoE [1], [8] High amplitude quality switches decrease the QoE [1], [3] Nevertheless, no research has compared multiple quality switches, high amplitude quality switches, and stalling events, which have been proven to impact QoE negatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' This paper has conducted evaluations and analyses to provide further insights into these areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Following the introduction, the paper is divided into the following sections: Section II provides an overview of the background of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Section III focuses on the research methodology and study setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' And in Section IV we present the paper’s analysis of results and findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Finally, in Sec- tion V we conclude the paper with possible future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' BACKGROUND OVERVIEW A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' HTTP Adaptive Streaming HAS uses the HTTP protocol to deliver data via the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The usage of this protocol comes with the significant advantage of accessing the whole architecture built around it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' That includes proxies, which redirect the internet traffic, caches, which improve performance, and Content Delivery Networks (CDNs), which store content closer to the end-user [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The Moving Picture Expert Group developed Dynamic Adaptive Streaming over HTTP (MPEG-DASH) over two years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' In 2011, after submitting multiple drafts, it was eventu- ally published as an international standard (ISO/IEC 23009-1), and since its publication, it has been revised numerous times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Quality of Experience Before the term QoE was introduced, various kinds of research had been conducted under the roof of Quality of Service (QoS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' A good QoS score, however, does not nec- essarily translate to user satisfaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Therefore a shift in research from a quantifiable network to a subjective user- centered approach was necessary [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' European Network on arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='13523v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='MM] 31 Jan 2023 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='622 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='115 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='027 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='088 BBB ED SWACC TOS Mean Opinion Score (MOS) Conetnt Types 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='858 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='345 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='432 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='216 Ha Ms Os Ts Mean Opinion Score (MOS) Effect Types Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' 1: Subjective Study Results with Standard Error Bars between the Content Types (Left) and between the Effect Types (Right) Quality of Experience in Multimedia Systems and Services (Qualinet) defines QoE as follows: ”Quality of Experience (QoE) is the degree of delight or annoyance of the user of an application or service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' It results from the fulfillment of his or her expectations with respect to the utility and/or enjoyment of the application or service in the light of the user’s personality and current state.” [10] III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' METHODOLOGY AND STUDY SETUP A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Crowdsourced Approach The decision to conduct this study via crowdsourcing was mainly based on two reasons: it generally requires fewer resources and the ongoing COVID-19 pandemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Selecting, inviting, and training participants in a lab environment requires many resources such as time, lab space, and equipment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' In crowdsourcing, participants can be quickly recruited from a pool of workers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' In addition, they can conduct the study at their convenience with their equipment, which makes it less resource extensive for participants and researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Furthermore, at the time of conducting this research, the COVID-19 pandemic was still ongoing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Therefore, inviting people to a closed lab environment would have been socially irresponsible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' As the usage of crowdsourcing platforms in QoE studies is still limited, this approach will be explained shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Crowdsourcing is generally used to outsource tasks to a global pool of workers with different skills and backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' There are various advantages such as a larger diversity in test subjects, quick and efficient turnover, and, as mentioned earlier, less resource extensive than lab testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' That includes time, monetary and technological resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' There are differ- ent platform types as well as task types in crowdsourcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Independent of the platform type, crowdsourcing platforms mainly function as collectors and distributors of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Tasks such as the sequence rating used in this study are categorized as so-called microtasks that only take a couple of minutes to conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' To ensure that a suitable user base is attracted to the tasks, most platforms can further filter for different parameters such as system devices or specific skills.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' While crowdsourcing offers many advantages, it also introduces various challenges to the researcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' A common challenge is the task’s potential misinterpretation due to unclear instruction and the lack of a test moderator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Therefore, the evaluation instruction should be as clear and detailed as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Another issue that should be considered in crowdsourcing is the effect of the incentive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Some participants might cheat or not fulfill the task correctly to gain the incentive faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' To prevent and filter the submission of such results, Hossfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='proposes different techniques in their paper about crowdsourcing best practices [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' One is to implement ”gold standard data”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' This refers to introducing questions to which the answer is already known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' This method allows verifying whether the participants paid attention to the initial instructions and are answering the study questions accordingly and truthfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' There have been different works about the implementation of gold standard data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Some suggest instantly revealing the correctness of the data [12] while others use it in the post-evaluation procedure [13], [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' In addition, proper task design was also shown to impact cheat prevention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' A study revealed that participants were less likely to cheat if the task was designed so that cheating would require the same amount of time as conducting the study properly [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Best practices also suggest considering technical implementations, such as preventing unwanted impairments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' It is advised to implement a measure that could cache the se- quences before playback to ensure that participants with lower bandwidth accessibility would not experience any unplanned impairments [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' In addition, it is recommended to consider the selection of web server size as crowdsourcing platforms attract thousands, sometimes millions, of workers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' therefore, it could be possible that hundreds of people access the study simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Ultimately, regarding implementation, it is advised to use a standard scripting language such as JavaScript to ensure system and device compatibility as some browsers might not support uncommon ones [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' In this study, the platform Amazon Mechanical Turk (MTurk) was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The specific functionalities of MTurk are further explained in Section III-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Dataset Selection The sequences for the study were chosen from four different original sources (Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The first one is Big Buck Bunny, an animated movie from the Blender Foundation which stars animals as main characters and displays a moderate amount of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Furthermore, the short film includes large homoge- neous backgrounds and bright settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Secondly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Elephants TABLE I: Test Sequences Name Source Timestamp Order Big Buck Bunny Blender Foundation 01:55 - 02:25 1 Big Buck Bunny Blender Foundation 05:45 - 06:15 2 Elephants Dream Blender Foundation 04:00 - 04:30 3 The Swiss Account Louder Than 11 41:40 - 42:10 4 Tears of Steel Blender Foundation 07:25 - 07:55 5 Big Buck Bunny,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' retrieved from [16] Elephants Dream,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' retrieved from [17] The Swiss Account,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' retrieved from [18] Tears of Steel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' retrieved from [19] Dream,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' also part of the Blender portfolio,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' portrays a man and a woman walking through a strange world,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' potentially displaying the growing insanity of the world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The movie is slow-paced and takes mainly place in a dark setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The third test sequence was selected from The Swiss Account, a self-made documentary about bouldering that includes many close-ups and detailed shots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Last but not least, a sequence was taken from Tears of Steel, a high-motion action movie with various special effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The movie starred real actors alongside animated machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The variety in content was chosen to reduce the potential impact of content type, which can influence the ratings if a participant has a general like or dislike for one specific content type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Displaying different sequences also decreases the reduction in interest, resulting from watching the same content multiple times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Each of the prepared sequences was 30 seconds in length and therefore only an excerpt of the original content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The exact timestamps and the order in which the sequences were displayed during the subjective study are summarized in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' All of the selected videos are accessible under the Creative Commons license.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Sequence Preparation Following the best practice framework of Hossfeld et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='the sequences were prepared in advance [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' After selecting the original content introduced in the previous section, the source content was checked via ffprobe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Afterwards, they were encoded via FFmpeg with the following command: $ ffmpeg − i i n p u t f i l e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' mp4 \\ −map 0: v −map 0: v −map 0: v −map 0: a :0 \\ −c : a aac −c : v libx264 −x264opts ’ k e y i n t =24: min − k e y i n t =24: no− s ce n e c ut ’ \\ −dash segment type mp4 − s e g d u r a t i o n 5 − u s e t i m e l i n e 0 − use template 1 −keyint min 24 −g 24 \\ − init seg name \\$Bandwidth\\$ / i n i t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' m4s −media seg name \\$Bandwidth\\$ / seg −\\$Number%01d \\$ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' m4s \\ −b : v :0 4300k −s : v :0 1920 x1080 \\ −b : v :1 1050k −s : v :1 640 x480 \\ −b : v :2 235k −s : v :2 320 x240 \\ −b : a :3 128k − ar 48000 −ac 2 \\ − a d a p t a t i o n s e t s ” id =0 , streams =v id =1 , streams = a ” −f dash −r 24 ma n if e st .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' mpd Initially, the path to the original file is input, and four streams are mapped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The first three represent video streams to enable bitrate switching, and the last one represents the TABLE II: Bitrate Encoding Stream Bitrate Resolution Type Codec Length Stream0 4300kbps 1920x1080 video x264 5s Stream1 1050kbps 854x480 video x264 5s Stream2 235kbps 426x240 video x264 5s Stream3 128kbps 48000Hz audio aac 5s audio stream.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The audio stream is encoded with Advanced Audio Coding (AAC), which was introduced shortly in Sub- section ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' of the Terminology Section (Section ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Libx264, a platform-independent encoder for the H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='264/ MPEG-4 AVC video format, was used for the video streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' In addition, settings were customized to set the maximum size of a Group of Pictures (GOP) to 24 and to prevent scene cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The prevention of scene cuts ensures that the encoder does not use its automated I-frame detection algorithm to add additional I- frames to the encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' As I-frames are more expensive, it is recommended to limit the amount of inserted I-frames when encoding streams [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The ”dash-segment-type mp4” sets the file format to ISOBMFF, the ISO/IEC standard format for media segments [21], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' In addition, the segment duration is set to 5, which means the algorithm could decide to switch to a different quality every 5 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Furthermore, the segment template was enabled while the segment timeline was disabled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' These settings ensure that the previously set segment duration is treated as the actual duration time and not as the minimum duration time [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' In addition, two templated variables in dash are set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Line 6 specifies the naming convention for the initialization segment, whereas line 7 describes the naming convention for the following media segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Media segments are set to be named in descending order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The following four lines define the bitrate levels for the defined streams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Stream 0, 1, and 2 represent the video streams and are mapped with the corresponding bitrate and frame size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The videos were encoded at three different resolutions: 1080p, 720p and 240p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The bitrate ladder used in this study was based on a standard by Netflix [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Stream 3 corresponds to the audio and describes the bitrate and two additional parameters: the audio sample rate in hertz and the number of audio channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The exact specifications of each stream, such as bitrate, resolution, codec, and segment length, can be seen in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The last two lines focus on DASH-specific settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' First, the mapped streams are divided into two adaptation sets: audio and video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Following, the DASH muxer is called upon to create the individual segments and the manifest file needed for dash playback [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' After the encoding, FFmpeg was also used to manipulate the selected video and audio segments to create the test sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' As a first step the needed segments for each quality level of the individual test sequence had to be concatenated as follows: $ cat i n i t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' m4s seg −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' m4s seg −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' m4s >> f i r s t V i d e o P a r t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' m4s The cat command is used to read the data of the following files and to write them in descending order to the defined output file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' In case the output file does not exist it will be created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The number of segments that have to be concatenated depends on the individual segment length as well as the length and design of the test sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' In this study, test sequences were set to 30 seconds and the segment length during encoding was set to 5 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Therefore, all test sequences that were designed to display the stalling event, or switch, in the middle of the test sequence required concatenating four segments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The initialization segment is followed by three segments each representing 5 seconds of content, totaling a length of 15 seconds per concatenation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The test sequences for the multiple quality switch, which represented a different temporal event pattern, only required three segments per concatenated file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Again the initialization segment is followed by two segments resulting in an output representing 10 seconds of content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' After concatenating, the output file had to be transformed to .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='mp4 file format with the following FFmpeg command: $ ffmpeg − i f i r s t V i d e o P a r t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' m4s −c copy f i r s t V i d e o P a r t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' mp4 First, the path to the input file is specified and then its data is copied in the described output format, in this case, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='mp4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The -c copy option will prevent FFmpeg to re-encode the input data [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The process of concatenation and transformation to .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='mp4 had to be repeated for every individual part that was necessary to construct the test sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The high amplitude quality switch as well as both stalling events were constructed from two individual parts each representing 15 seconds of content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The multiple quality switch, as it displays three different quality levels, was constructed from three individual parts each representing 10 seconds of content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' In addition, audio segments had to be concatenated and transformed in the same manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Once this process had been concluded, FFmpeg was used to merge the individual video and audio parts: $ ffmpeg −y − i f i r s t V i d e o P a r t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' mp4 − i f i r s t A u d i o P a r t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' mp4 −c : v copy −c : a copy f i r s t P a r t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' mp4 Initially, the y-flag command is used to give global permission to overwrite the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Then two inputs are defined and mapped, the first representing the video and the second the audio source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' As before the copy command is used to ensure that inputs are not re-encoded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Lastly, the output filename and format are defined [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' This process also had to be repeated until all individual audio and video parts, which were needed for the designed test sequences, were combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' For the imitation of a stalling event, an additional buffering sequence was needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' To create a realistic scenario FFmpeg was used to cut the last frame of the GOP previous to the stall, which was then inserted as the background for the buffering wheel video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' This was achieved by the following two commands: $ ffmpeg − s s e o f −3 − i f i r s t S t a l l P a r t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' mp4 − update 1 −q : v 1 lastFrame .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' png $ ffmpeg −y − f r a m e r a t e 24 −loop 1 − i lastFrame .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' png − i loadingWheel .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' mp4 − f i l t e r c o m p l e x ’ [ 1 ] format =argb , color channelm ixer =aa =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='5[ ol ] ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' [ 0 ] [ ol ] o v erlay ’ − t 1 f i n a l B u f f e r E f f e c t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' mp4 The first command creates the screen capture of the last frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The sseof option tells FFmpeg to seek and output all frames specified by the position, in this case, three seconds, relative to the end of the input file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' To ensure that only the last frame is saved the update option is changed from default 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' This setting will continuously overwrite the data in the specified output file, ultimately leaving only the last frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The encoding quality is set to 1, which represents the highest quality level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The captured frame is set to be in .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='png file format [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The second command inserts the previously captured frame into the buffering wheel video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The framerate of the output is set to 24 frames per second and the image file is specified to loop throughout the length of the output file.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Additionally, options to create a complex filtergraph are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' First, the color model of the second input referred to as indexed 1, is set to Alpha Red Green Blue (ARGB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Then its value for the alpha channel is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='5, which makes the input source semi-transparent [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Following, the first input, marked as indexed 0, is set as the main input on which the second input will be overlaid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Then output length is set to one second and the output name and format are defined [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Once the buffering sequence for all content types was created, all previously prepared parts were concatenated with FFmpeg to create the final test sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Therefore the following command was used: $ ffmpeg −y −f concat − s a f e 0 − i l i s t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' t x t −c copy f i n a l T e s t S e q u e n c e .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' mp4 First, global overwrite permission was defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The f-flag option enforces the file format of the concatenated sequences on the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The safe option was set to 0 to disable safe mode, which allows the input of all files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The input specifies the path to a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='txt file which contains the relative paths to the required files for the test sequences in concatenation order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' As an example the structure of a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='txt file used for the two-second stall test sequence is given: f i l e ’ s t a l l i n g e v e n t / f i r s t P a r t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' mp4 ’ f i l e ’ s t a l l i n g e v e n t / f i n a l B u f f e r E f f e c t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' mp4 ’ f i l e ’ s t a l l i n g e v e n t / f i n a l B u f f e r E f f e c t .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' mp4 ’ f i l e ’ s t a l l i n g e v e n t / secondPart .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' mp4 ’ Lastly, the input data is copied without further encoding to the specified output .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='mp4 format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' This concatenation process had to be repeated for each of the defined test sequences of this study [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Effect Design To answer the research question and to measure the different effects of stalling events, multiple quality switches and high amplitude quality switches on QoE the following effects are proposed: 1-second stalling event Figure 2(a), 2-second stalling event Figure 2(b), a high amplitude quality switch from 1080p to 240p Figure 2(c), and a multiple quality switch from 1080p to 240p to 720p Figure 2(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The decision for introducing the 1-second stall was based on a study by De Pessemier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=', which quantified the effects of stalling events [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Within this study, the median of a single stall event fluctuated around one second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The 2-second stall event was chosen to analyze the effect difference between a 1-second stall compared to a 2-second stall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The quality levels for the high amplitude as well as the multiple quality switch were based on a study that investigated, in particular, the impact of different switching patterns on QoE [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The study indicated that both events would resolve in worse QoE than other events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' To avoid additional influences of effects such as the primacy or recency effect, which assume that the first or last seconds of a video has a higher impact on the viewers’ QoE, the stall event was placed in the middle of each test sequence [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Graphical User Interface In order to display the study to the participants, a website- based Graphical User Interface (GUI) was developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The website was based on HTML, CSS, and JavaScript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' In- dexedDB, a low-level API that can pre-cache data on the client-side, was used to prevent unwanted impairments [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The caching process of the test sequences would immediately commence when the participant opened the website and con- tinued in the background without any disturbance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' This was implemented to reduce potential waiting times for the users later in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' In the backend, a node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='js express server was used to request the prepared test sequences from the hosting server and to handle the connection to the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The test sequences were stored using Amazon S3 and, from there, requested to the website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' For deployment, Amazon’s EC2 instances were chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' In accordance with the best practice framework, consideration was given to the size of the instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Ultimately, a small instance type was chosen, which was expected to be sufficient for the extent of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Upon opening the website, participants would see an introductory text that explained the study procedure and an input field that asked for an MTurk ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' After submission, an example video sequence would be displayed to familiarize the participants with the player design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The video player, implemented using the HTML video tag, was minimized to the viewing field and all additional customization options such as fullscreen, start/stop, and audio options were deactivated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Furthermore, the width of the video player was fixed to 1280px.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' This measure was taken to ensure that every participant would have the same experience from a technical perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' A customized overlay was cast on each test sequence to allow participants to pause if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Once the playback of a test sequence was started, the video player would position itself in the middle of the screen and the background would turn grey as suggested in the ITU-T standard P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='910 [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' After the test sequence finished, the background would turn white, and only then would the questionnaire appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' This functionality was implemented to ensure that participants could not simply click through the study without watching the test sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' As suggested in [15] this measure would also ensure that the proper completion of the study would require the same time as a cheating attempt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Each questionnaire contained a content- related question, which was treated as the gold standard data and the 5-point quality scale scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' As the main aim of this study was set on comparing the resulting quality of the chosen effect types, the ACR methodology was chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The test sequences were shown in the following order: Big Buck Bunny, Elephants Dream, The Swiss Account, and Tears of Steel (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' While the content order remained the same between participants, the effects were randomly chosen by an algorithm implemented with JavaScript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The algorithm varied combinations of effect and content types but ensured that each participant would watch each effect once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Upon completion, the results would be sent to the express server and stored in the MongoDB database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The following data was stored: MTurk ID, timestamp of submission, answers to the questionnaire, and an automatically generated UUID for each user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The UUID would also be sent back to the website and displayed to the participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Mobile, as well as tablet devices, were excluded from the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' If a participant tried to access the website via these devices, an error message would appear suggesting to switch to a desktop or laptop device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Similarly, any browser other than Google Chrome was excluded from the study as trials showed that they did not support the manual bitrate switching within the manipulated sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' A similar error message would be displayed in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Amazon Mechanical Turk Amazon Mechanical Turk (MTurk) is a crowdsourcing marketplace which enables companies and individuals to out- source tasks virtually [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The platform allows for unlimited campaign creation and customized settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Each campaign must have a short description text, a set time frame, and al- located compensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Other elements that can be customized are among other geographical locations and minimum user rating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' A trial campaign with 5 test subjects was created to test functionality and user reliability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The timeframe was set to 15 minutes and compensation to 0,50 $.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' To receive compensation, users had to input their MTurk ID in the study’s interface as well as paste the UUID which they received after completion back on the MTurk platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' This measure was implemented to ensure that users did complete the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' It also allowed cross-checking between the database and the MTurk platform to filter out double participation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The initial test confirmed the functionality of the campaign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Despite the given functionality of the interface and the MTurk setup, none of the five datasets provided accurate answers to the gold standard data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' For further testing purposes, a second campaign with 10 participants was started.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' In addition to the prior settings, this campaign was only accessible to users with an acceptance rating of 95% and higher and limited to 10 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Upon initial analysis, the second campaign appeared successful as the gold standard data was answered correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Therefore, it was assumed that participants in the second campaign watched the content properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Hence a third and last campaign with 60 additional participants and the same settings as prior was initiated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' During the in-depth analysis of the MTurk results, the fol- lowing issues could be discovered: For one, all effects achieved (a) 1s stall event at 15s (b) 2s stall event at 15s (c) High amplitude quality switch (d) Multiple quality switch Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' 2: Representation of the One and Two Second Stall Events and High Amplitude Quality Switch (1080p-240p) and Multiple Quality switches (1080p-240p-480p) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' 3: Graphical Representation of the ACR Stimulus Presentation in the Subjective Study - BBBr, Big Buck Bunny test sequence at random effect r - EDr, Elephants Dream test sequence at random effect r - SWACCr, The Swiss Account test sequence at random effect r - TOSr, Tears of Steel test sequence at random effect r relatively high grading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Secondly, there were many instances in which the participants rated all effect types equally, indicating no difference between effect types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Based on these findings, the following assumptions were made: a 5-point scale does not give participants enough variety to judge the effects with a meaningful difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Some participants were only interested in concluding the survey as fast as possible to receive the incentive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Therefore, they did not take the time to properly watch and evaluate the test sequences, so no differences in effect types are noted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Follow-Up Study In order to respond to the stated observations, a follow- up study was designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Three major changes were made to consider the previous findings and assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' First of all, the 5-point rating scale was changed to the recommended 9- point scale to give the participants a higher discriminative rating power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Secondly, to respond to the equal ratings between effect types, four additional test sequences were introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Therefore, every participant would rate 8 sequences and ex- perience each effect type twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' This change introduced an additional reliability measure as participants who displayed a significant variance within an effect type could be further investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The additional sequences were taken from the same four movies to prevent further potential content effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The last change concerned the participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Participants in the follow-up study received no compensation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Instead, they were taken from a pool of volunteers to diminish the chances of unreliable results due to an incentive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' To replace MTurk’s time frame setting two additional variables were introduced and saved in the database: timestamp upon watching the first test sequence and timestamp of submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The introduction of the rating scale on the introductory page and the scale beneath each test sequence was adapted accordingly Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' STATISTICAL ANALYSIS AND FINDINGS The following section will summarize the statistical results for both studies, once with participants which received an in- centive and, secondly, with volunteers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' For the statistical anal- ysis, a One-Way ANOVA was conducted to validate significant variances between the different effect groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' For the analysis within effect types, a two-sample t-test was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' All statistical tests used an alpha level of 0,05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Two POST HOC tests were used to validate the significant difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' For one, the Bonferetti test with a corrected alpha value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Secondly, the Tukey B test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' In addition to significant variances between effect types, the results were also checked for differences between content types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' For reasons of simplicity, figures and tables in the following chapter will describe the effects with the following abbreviations: 1-second stall as Os, 2-second stall as Ts, multiple quality switch as Ms, and high amplitude quality switch as Ha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' MTurk Evaluation In total, 72 entries were submitted to the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' It can be assumed that the additional two participants completed the study but did not manage to input the UUID back on MTurk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Therefore, they were not counted as participants by MTurk but still submitted their results to the database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' For this reason,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' these two as well as 10 other datasets were excluded ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='(kbps) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='ls stall event ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='Bitrate ( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='Time (in seconds)5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='Bitrate (kbps) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='2s stall event ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='3000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='2000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='1000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='Time (in seconds)5000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='4000 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='Bitrate (kbps) ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='Grey ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='Grey ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='Grey ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='Example ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='BBBr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='EDr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='SWACCr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='TOSr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='Introduction ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='30 seconds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='30 seconds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='voting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='30 seconds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='voting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='30 seconds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='voting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='30 seconds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='voting ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='Thank you ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='&UUIDFig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' 4: Graphical Representation of the ACR Stimulus Presentation in the Follow-Up Study - BBB1r & BBB2r, First and Second Big Buck Bunny test sequence at random effect r - ED1r & ED2r, First and Second Elephants Dream test sequence at random effect r - SWACC1r & SWACC2r, First and Second The Swiss Account test sequence at random effect r - TOS1r & TOS2r, First and Second Tears of Steel test sequence at random effect r Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' 5: GUI after Test Sequence Playback displaying Ques- tionnaire from analysis due to wrong gold standard data or double participation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The ratings of the remaining 60 participants were then checked for significance with a one-way ANOVA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The evaluation, as seen in Table III showed a minor significance be- tween the effect groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Therefore two POST HOC tests were conducted in which all effects were weighed against each other (Table IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Following the Bonferroni method, the corrected alpha value was calculated, resulting in α = 0, 008333, which indicates no significance between the effect groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The same results were achieved with the Tukey B POST HOC test with a Q-value of Q = 3, 66 and a resulting T-value of T = 0, 41583.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Follow-up Study Evaluation For the follow-up study, 43 data sets were collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Accord- ing to the ITU-T P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='910 recommendation, sample sizes larger than 40 merely result in minimal differences [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Therefore, the 43 datasets were assumed sufficient for statistical purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' None of the participants in the follow-up study had any prior experience with video or image ratings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The dataset analysis showed that all participants answered the gold standard data correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' However, 6 participants were excluded as their study completion time varied two standard deviations from the mean, indicating that they were not paying close attention to the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' A two-sample t-test was performed for each effect type to check for significant differences between the first rating of an effect contrary to the second rating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' No significant difference was found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' According to the resulting MOS, effects ranked in the following descending order: 1-second stall, multiple quality switch, 2-second stall, and high amplitude quality switch (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The results of the 9-point scale were mapped to a 5-point MOS based on the following formula, where NP represents the rating value on the 9-point scale: MOS = NP 2 + 0, 5 Equally to the previous study, the variance between the effect types was checked with a One-Way ANOVA, which indicated a significant variance (Table III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The corrected alpha remained the same for the Bonferroni POST HOC method with α = 0, 008333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' In the case of the 1-second stall and the high amplitude quality switch, as well as the multiple quality switch and the high amplitude quality switch, P < corrected α = 0, 00833, indicating significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The Tukey B POST HOC test confirmed these results as T > Absolute Mean Difference (Table IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Therefore it can be assumed that there was a significant difference between the high amplitude quality switch and 1-second stall and the high amplitude quality switch and the multiple quality switch events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' In both cases, the high amplitude quality switch performed significantly worse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' In addition, the results were checked for significant variance between content types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The one-way ANOVA indi- cated significance (Table III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The results of the Tukey B POST HOC (T = 0, 764) confirmed a significant difference between Big Buck Bunny and the other movie sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' Throughout all effect types, the Big Buck Bunny clips achieved the highest MOS (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' CONCLUSIONS AND FUTURE WORK This paper investigated whether stalling events have the highest impact on decreasing the viewer’s QoE compared to multiple or high amplitude quality switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The results confirm that when compared to multiple and high amplitude quality switches, stalling events do not have the worst impact Grey Grey Grey Grey Grey Grey Grey Grey Grey Example BBB1r ED1r SWACC1r TOS1r BBB2r ED2r SWACC2r TOS2r Introduction 30 s 30 s voting 30 s voting 30svoting 30 svoting 30 s voting 30 s voting 30s voting 30 s voting Thank you &UUIDWhat could you see in the video?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' O Forest/Jungle O Airplane O sking O Bouldering How would you describe your watching experience of this video on a scale from 1 to 9 (1 being "poor" and 9 being "excellent")?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' 0102 03 04 0 5 06 07 0809 ixanTABLE III: Analysis of Variance (One-Way ANOVA, between Groups) of the different Effect and Content Types of the Subjective Results of MTurk and Laboratory Participants Evaluation df 1 Mean Square F 2 P 3 F 4 Effects (MTurk) 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='070 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='673 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='048 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='642 Contents (MTurk) 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='348 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='720 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='163 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='642 Effects (Lab) 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='721 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='758 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='00077 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='635 Contents (Lab) 3 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='382 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='00016 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='635 1 Degree of freedom, number of total categories minus 1 2 F-value, the significance of variance between two populations 3 P-value, significant if P < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='05 (chosen α level) 4 Critical F-value, if F > Fcrit then significance TABLE IV: Post hoc (Bonferroni, corrected α = 0, 0083) and Tukey B (T = 0, 415) for MTurk and Laboratory Participants between Effect Types Environment I 1 J 1 P 2 Absolute Mean Difference 3 MTurk Ha Ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='649 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='083 MTurk Ha Os 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='066 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='3 MTurk Ha Ts 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='4 MTurk Ms Os 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='161 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='216 MTurk Ms Ts 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='049 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='316 MTurk Os Ts 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='458 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='1 Laboratory Ha Ms 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='00099 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='486 Laboratory Ha Os 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='00012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='574 Laboratory Ha Ts 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='0253 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='358 Laboratory Ms Os 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='526 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='087 Laboratory Ms Ts 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='399 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='128 Laboratory Os Ts 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='159 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='216 1 I & J, different effect types 2 Two-tail t-test 3 | I − J | on QoE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' The results suggest ABR algorithms should consider introducing short stall events instead of high amplitude quality switches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' However, further studies with different stalling du- rations, switching patterns, and quality switching amplitudes are advised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' ACKNOWLEDGMENT The financial support of the Austrian Federal Ministry for Digital and Economic Affairs, the National Foundation for Research, Technology and Development, and the Christian Doppler Research Association is gratefully acknowledged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content=' https://athena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='itec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='aau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atFRT4oBgHgl3EQfQze_/content/2301.13523v1.pdf'} +page_content='at/.' 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Cities, University of Seoul, Seoul 02504, Korea +We study the commensuration torques and layer sliding energetics of alternating twist trilayer +graphene (t3G) and twisted bilayer graphene on hexagonal boron nitride (t2G/BN) that have two +superposed moire interfaces. Lattice relaxations for typical graphene twist angles of ∼ 1◦ in t3G +or t2G/BN are found to break the out-of-plane layer mirror symmetry, give rise to layer rotation +energy local minima dips of the order of ∼ 10−1 meV/atom at double moire alignment angles, and +have sliding energy landscape minima between top-bottom layers of comparable magnitude. Moire +superlubricity is restored for twist angles as small as ∼ 0.03◦ away from alignment resulting in +suppression of sliding energies by several orders of magnitude of typically ∼ 10−4 meV/atom, hence +indicating the precedence of rotation over sliding in the double moire commensuration process. +Introduction – Experimental realization of moire ma- +terials by layering two graphene-like 2D material systems +with different lattice constants and/or with a finite twist +angle [1, 2] have been furthered by combining two moire +patterns to form the so-called double or super moire +systems [3–11]. Crystals with long moire pattern peri- +ods allows to access certain physical observables at low +magnetic fields and gate carrier densities that would not +be accessible otherwise [12–14]. +This behavior can be +used for example in 30◦ twist-angle quasi-crystals [15– +19] to generate moire quasicrystals at experimentally ac- +cessible charge carrier densities using h-BN encapsulated +graphene [9, 20]. Most theories describing double moire +systems have so far used the simplest commensurate dou- +ble moire geometries to explain the observed physics. +For instance, in alternating twist trilayer graphene (t3G) +with twisted middle layer giving rise to two aligned moire +patterns, important in the context of flat band supercon- +ductivity [21–31], the most stable geometry corresponds +to superposed top and bottom layers. +Likewise com- +mensurate double moire geometries have been assumed +when studying twisted bilayer graphene on h-BN where a +spontaneous anomalous Hall effect was measured [32–37]. +However, the assumption that equal period and aligned +double moire pattern systems are energetically favored +has not been yet confirmed. +Here we analyze the atomic structure of t3G and +twisted bilayer graphene on hexagonal boron nitride +(t2G/BN) to show that double moire systems generate +torques that tend to lock the systems into commensurate +moire patterns and favor a specific sliding geometry. We +find that mirror symmetry breaking layer corrugations +are required for a correct total energy minimization, in- +cluding the AAA-stacked t3G systems where we use the +overline to indicate relative sliding geometries between +the layers regardless of their twist angles. We observe +that the specific sliding atomic structures between the +top and next nearest layer only matters when the moire +patterns are commensurate, since the energies are al- +most sliding-independent for incommensurate moire pat- +terns leading to superlubricity away from commensura- +tion, similar to the superlubricity behavior in single inter- +face twisted bilayer systems studied in the literature [38– +44]. +Systems and methods – We consider double moire +systems with two moire interfaces consisting of graphene +trilayers and t2G on hexagonal boron nitride. We illus- +trate schematically in Fig. 1 (a) all three different sys- +tems considered, namely t3G, t2G/BN type I and type +II depending on the twist angle sense of the bottom hBN +layer contacting graphene. The layer numerals 1, 2 and +3 correspond to bottom, middle and top layers respec- +tively. The middle layer 2 is taken as the reference frame +with zero twist angle and we use θ12 and θ32 labels to +represent the actual twist angles of the bottom and top +layers. The atomic structure relaxation is carried out us- +ing LAMMPS [45]. We use the REBO2 force-field [46] for +the intralayer interactions of graphene and EXTEP [47] +for those of hexagonal boron nitride, whose equilibrium +geometries are aG =2.4602˚A +and aBN =2.50576˚A +re- +spectively. The interlayer force fields are based on EXX- +RPA-informed [48] DRIP [49] parametrizations [50] and +we used both the fire and CG minimization scheme [51] +with a time step of 0.0001 ps for the former and 0.001 ps +for the latter and a stopping tolerance on the forces of +10−5 eV/˚A. In order to capture the tiny angle variations +with respect to the doubly commensurate angles we use +the approach outlined in Ref. [52] and Ref. [53] to find +the commensurate cell for a double moire system, see +Appendix A for details and summary of the integers rep- +resenting our commensurate cells. The incommensurate +moire systems are approximated by taking commensurate +simulation cells containing several repetitions the moire +unit cells or moirons. Due to the small angle differences +and the high sensitivity of the results on internal strains, +we are at times bound to choose large simulation sys- +tems containing millions of atoms. The lattice constant +variation tolerance is capped at 0.03% to minimize inter- +nal strains that lead to energy differences of the order of +0.005 meV/atom that are between one to two orders of +magnitude smaller than the energy differences of the or- +der 0.1 meV/atom required to resolve the local minima in +arXiv:2301.04105v1 [cond-mat.mes-hall] 10 Jan 2023 + +2 +FIG. 1. (color online) (a) Schematic figure illustrating the different t3G and t2G/BN systems considered here. θij = θi − θj +are the twist angles of the i-th layer with respect to the j-th layer where their particular choices can lead to commensurate +and incommensurate moire patterns. +(b) Corrugation effect that breaks the mirror-symmetry of AAA stacked t3G with +θ12 = θ32 = 1.53◦. (c) Interlayer distance differences at different local stacking positions and (c) bending corrugations in the +middle layer 2. The sketches at the bottom of the panels indicate the specific local stackings AAA, AAB and AAC for the +three-layer systems while the two-letter conventions refer to local sliding between two layers. Gray, green and orange colors +refer to the A and B sublattices of graphene. Blue and purple refer to the boron and nitrogen atoms in hBN in t2G/BN or the +A and B sublattices in a t3G system. +our energy curves. The stability of the atomic structure +relies on the total energy given as the sum +Etot = Eel + Epot +(1) +where we can distinguish the elastic energy Eel += +� +i Ei +el/2 that resists the deformation due to the strains, +and the potential energy Epot = � +i Ei +pot/2 that triggers +the formation of the moire pattern strains taken as sum +of contributions from each atomic site i and where the +division by 2 accounts for double counting. We can define +the local elastic Ei +el, potential Ei +pot and interface Ei +IFmn +energies as +Ei +el = +� +j∈layer i +φij +(2) +Ei +pot = +� +j /∈layer i +φij = +� +j∈ any layer +φij − Ei +el +(3) +Ei +IFmn = +� +j /∈ layer i +j ∈ layer n or m +φij +mn +(4) +where φij represents the pair-wise potentials between +atoms i and j. Our calculations show that the elastic +energy contributions in Eq. (2) are about one order mag- +nitude smaller than the potential and interface energies +in Eqs. (3-4), and therefore is only a small fraction of the +total energy in Eq. (1) that dictates the stability of our +systems. The interface energies are essentially the poten- +tial energies referred to a particular pair of layers. It will +be interesting to note how this interface energy changes +from one system to another by examining the interface +energy differences for two different relaxed atomic struc- +tures +∆EIFmn(r) = E3L +IFmn(r) − E2L +IFmn(r) +(5) +where E3L +IFmn at a given point is obtained relaxing simul- +taneously all three layers of t3G and then considering +the bilayer atomic positions for the considered mn inter- +face, while the E2L +IFmn interface energy is obtained using +the t2G relaxed atomic positions of the two mn layers +that form the interface. In Eq. (5) we have removed the +i-index dependence in Ei +IFmn in Eq. (4) using the posi- +tion vector r instead by interpolating the data from the +closest i-sublattice points. +Another quantity of interest is the torque constant +that we define as the derivative of the total energy as +a function of twist angle similar to the proposals in +Refs. [42, 44, 54] but focusing here on the rotation of +the top layer with respect to the middle layer +k± = dEtot +dθ32 +(6) +where its positive or negative values tend to either reduce +or increase the value of θ32 towards the commensurate +moire geometry. + +(a) +(b) +t3G +012 +(012 = 032 = 1.53) +Z3 1 +032 +1△C32 +Z2 1 +△C21 +Z1 1 +(c) △C32 = △C12 +(A) +t3G +0.10 f +A +△C32 = △C12 +0.05 +t2G/BN +(Type I) +0.00 +(A) +AADW1 AB DW3 BADW2AA +(d) Z2 +1.5 +Z2 +t2G/BN +Top +1.0 +(Type II) +Middle +0.5 +Bottom +0 +0.0 +AAA +AAB +AAC +AADW1 ABDW3 BADW2AA +L3 +AA +DW2 +BA +DW3 +AB +DW1 +AA +L2 +L3 +FIG. 2. (color online) (a) Energies of t3G double moire systems for three different values of θ12 where θ32 is varied from ∼ 0.1◦ +to ∼ 2.2◦. For t3G systems the commensurate moire patterns are obtained when θ12 = θ32 (indicated by vertical lines) and +leads to a local energy dip. The x-symbols represent the total energies when the top interface has AB-stacking. The dashed +lines are obtained by using polynomial interpolation away from the commensuration dips. On the right hand we show total +energies for different sliding of the top layer for select commensurate angles θ12 = θ32. (b) Similar plots for t2G/BN (Type I +and Type II) where the total energy dips are obtained for two different values of θ32, corresponding to angles where L12 = L32 +(solid vertical line) and L12 = L32/2 (dashed vertical line). (c) Intepolated interface energy differences ∆EIFmn(r) of Eq. (5) +for the 3 systems considered illustrating the local energy gain/penalty when a single moire comes into contact with a second +moire interface. The energy difference densities are plotted along a straight line that connects the opposite diagonal corners +of the moire cell through different local stacking positions where the most stable stacking arrangement is highlighted with a +thicker line. The lower the energy, the more stable is the double moire geometry locally. The negative energies correspond to a +stabilizing gain in energy while positive energies indicate a destabilizing energy penalty. The interface energy differences, and +therefore the potential energy differences, make up the dominant contributions of the total energy differences since the elastic +energy contributions are one order of magnitude smaller. + +(a) +t3G +012 = 1.08° +012 = 1.53° +12 = 2.0° +1.25 +AAA +AAC +(meV) +0.75 +AAA AAB +0.50 +12 = 1.08° +AAB +'tot +012 = 1.53° +X +E +0.25 +12 = 2.00° +X +0.00 +AAA +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +032 () +(meV/atom) 0.58 +0.88 0.88 +1.11 1.06 +1.22 +(b) +t2G/BN +Type I +Type II +(032 = 1.12) +(032 = 1.12) +0.8 +AAA +(meV /atom) +0.6 +AAC +0.4 +101 +AAB +Type I (012 = + 0.56) +E +0.2 +Type II (012 = - 0.58) +0.0 +AAA +0.0 +0.5 +1.0 +1.5 +032 () +(meV/atom) 0.64 +0.79 +0.61 +0.82 +(c) +(meV) +t3G +(meV) +t2G/BN Type I +(meV) +t2G/BN Type II +10 +2 +0 +0 +0 +-10 +-10 +-2 +AAA +-20 +-20 +AAA +AAA +AAB +AAB +AAB +-30 +AAC +-30 +AAC +2 +-1 +-2 +21 +10 +10 +0 +0: +0 +-10 +-10 +-1 +-2 +-20 +-20 +AAA +ACA +ABA +AAA +AAA +ACB +ABC +AAA +AAA +ABA +ACA +AAA +AAB +ACB +ABB +AAB +AAB +ACC +ABA +AAB +AAB +ABB +ACB +AAB +AAC +ACA +ABB +AAC +AAC +ABC +ACC +AAC4 +Results – We begin by testing the main assumption +of our study for t3G, namely that angles leading to com- +mensurate moire systems will be energetically more fa- +vorable than angle combinations leading to incommen- +surate moire systems. For this purpose we fix the angle +θ12 between the two bottom layers to three select values +of θ12 = 1.08◦, 1.54◦ and 2.0◦ to sample typical largest +magic angles values in tNG systems for N = 2, 3, ∞ [55] +while we vary θ32 between the two top layers from 0.1◦ to +2.3◦. Commensuration for alternating twist is naturally +achieved when θ12 = θ32 since the lattice constants of +all three layers are equal. The find numerically that the +relaxed atomic structures generally favor z-axis corruga- +tions, where Fig. 1(c) shows the typical local stacking- +dependent interlayer distance variation of the order of +∼ 0.1 ˚A while Fig. 1(d) illustrates for the middle layer +the z-axis corrugations of the order of ∼ 1.75 ˚A roughly +one half of the average interlayer distance. +The main results of this work are shown in Fig. 2(a) +where we plot the total energy per atom as a function of +θ32 after the atomic relaxation. For all three θ12 values +considered, we find total energy local minima at the com- +mensurate twist angles provided that we allow a bending +corrugation as illustrated in Fig. 1(b). In fact, the to- +tal energies reported in Table I indicate that the mirror- +symmetric t3G with a completely flat middle layer has a +higher total energy compared to the corrugated atomic +structure. There we list the integrated total, elastic, po- +tential, and interface energies taken as the sum over all +atomic sites i, with proper double counting correction, +of the energies given in Eqs. (1-4). In Fig. 2(a) we show +that the commensurate moire systems are most stable +in the AAA-stacking when there is no relative sliding +between the three layers. In fact, the energy difference +between the AAA at the local minima dip and AAB max- +ima represented is equal to 0.18 meV/atom near 1◦ and +drops to 0.06 meV/atom for an angle of 3.47◦ (not shown +here) and are comparable to the magnitude of the energy +dips due to the rotation. +Our sliding dependent total +energy plots indicate that there are barrier-free sliding +paths leading to the global minimum at AAA-stacking. +We then show a similar analysis for t2G/BN where we +fix the substrate angle between G and hBN at θ12 = 0.56◦ +for type I or −0.58◦ for type II and we allow θ32 between +both graphene layers to change up to a value of 1.5◦ to +achieve moire periods that satisfy pLM +tBG = q LM +tGBN [56] +where p, q are integers. For q = 1 we observe two dips +in the energy curve that correspond to θ32 = 0.56◦ for +p = 2 with a G/G moire pattern period twice as large as +the G/BN and θ32 = 1.12◦ for p = 1 with equal periods. +The details of the atomic relaxation giving rise to total +energy dips in Fig. 2(a) and (b) for t3G and t2G/BN can +be further understood through the line plots in Fig. 2(c) +calculated using Eq. (5) where we focus on the inter- +face energy difference between 3-layer and 2-layer relaxed +systems. We can thus quantify the energy gain/penalty +Etot +Eel +EIF12 +EIF23 +t3G (AAA) +−7.42537 −7.39458 −0.01542 −0.01541 +t3G (AAA, ms) +−7.42534 −7.39459 −0.01538 −0.01538 +t3G (AAB) +−7.42510 −7.39471 −0.01521 −0.01516 +t2G/BN-I (AAA) +−7.19472 −7.16454 −0.01458 −0.01559 +t2G/BN-I (AAB) +−7.19451 −7.16466 −0.01442 −0.01542 +t2G/BN-I (AAC) +−7.19467 −7.16455 −0.01456 −0.01556 +t2G/BN-II (AAA) −7.19561 −7.16554 −0.01452 −0.01556 +t2G/BN-II (AAB) −7.19562 −7.16553 −0.01453 −0.01556 +t2G/BN-II (AAC) −7.19581 −7.16556 −0.01463 −0.01563 +TABLE I. Sliding dependent total, elastic, and interface en- +ergies in for θ12 = θ32 = 1.5385◦ for t3G, θ12 = 0.56◦, θ32 = +1.12◦ for t2G/BN-I, and θ12 = −0.58◦, θ32 = 1.12◦ for +t2G/BN-II for systems containing 8322, 15490 and 15492 +atoms respectively. +These numbers are used to renormal- +ize and report the energies in eV/atom. We note that the +mirror-symmetric (ms) geometry has a higher total energy by +0.03 meV/atom. +(negative and positive values, respectively) one achieves +when putting two moire patterns in contact with each +other. For t3G the most stable 3-layer configuration is +the AAA-stacking that we achieve when the unstable +AA-stacking of each bilayer are stacked on top of each +other and the stable AB/BA stackings are simultane- +ously stacked on top of each other. The interface energy +plots in Fig. 2(c) illustrate the energy differences between +single and double moire pattern atomic structures. For +AAA-stacking the interface energy difference plot shows +that its energy gain is most pronounced at local stacking +regions in between the AAA and ACA/ABA regions. In +contrast the AAB-stacking has a generally unfavorable +interface energy for most local stackings. +The double moire energetics becomes more com- +plicated when we have an heterogeneous interface in +t2G/BN where the most stable (unstable) AB/BA (AA) +local stacking geometries in t2G combines with the AC +(AA/AB) stackings of GBN. For type I trilayers the ener- +getically stable geometry is AAA while for type II trilay- +ers it is AAC. We notice that the strongest local energy +penalties due to interfering moire patterns do not neces- +sarily happen at the high symmetry local stacking geome- +tries and the global energy minimization does not follow +simple rules of thumb consisting in combining together +the local stackings that are energetically most favorable +or unfavorable as in the t3G case. +The conclusions on stability energetics that we can +draw based on the energy density difference line cuts in +Fig. 2 are consistent with the interface and total ener- +gies that predict the AAA, AAA and AAC stacking as +the most stable geometries for t3G, t2G/BN type I and +t2G/BN type II, systems respectively. +To further ex- +plore the stabilization of the commensurate moire sys- +tems versus the incommensurate ones we show in Fig. 3 +the local stacking distribution maps for AAA and AAB + +5 +stackings of t3G following the conventions outlined in +Ref. [52] for each one of the interfaces. +Here we can +FIG. 3. +Illustration of the stacking redistribution in t3G +for two double moire configurations obtained using the con- +ventions outlined in Ref. [27] using the displacement vectors +between layers m and n. (a) Local stacking maps for AAA and +AAB stacking geometries for θ12 = θ32 = 1.53◦, and the ratios +of the local stacking areas of the left panels. The AAA stack- +ing shows a smaller AA local stacking area due to greater local +rotations that reduces this energetically unfavorable stacking +when compared with the AAB stacking going from 7.7% to +12.2% respectively. (b) Local stacking maps for θ12 = 1.53 +and θ32 = 1.1◦ for sliding geometries and corresponding local +stacking area ratios. Although the relaxation profiles are dif- +ferent, the ratios for local AA, AB and BA stackings remain +nearly the same for different sliding geometries. +compare how the local stacking distributions change for +the commensurate and incommensurate moire interfaces, +and how the favorable and unfavorable sliding geom- +etry influences the strain profiles. +For commensurate +moire patterns and stable AAA-stacking the energeti- +cally unfavorable AA local stacking that makes about +7.7% of the total area increases to 12.2% for the less sta- +ble AAB-stacking. As soon as we move away from the +same period commensurate moire geometries and have +different moire pattern periods, for instance the combi- +nation of 1.54◦ − 1.1◦ with 1/7 moire length ratios or +1.54◦ − 1.5◦ with 1/39 ratios that can be considered +approximations of the incommensurate moire patterns. +For these geometries the relative distribution of the AA, +AB or BA local stacking areas become practically insen- +sitive to the relative sliding of the top layer resulting +in almost the same local stacking area ratios for both +t3G (θ12 = θ32) +θ32 (◦) +1.08 +1.53 +2.00 +k−(θ32) +(meV/(atom·rad)) +−42.12 +−18.34 +−8.513 +k+(θ32) +(meV/(atom·rad)) +75.06 +53.04 +36.85 +Eb(θ32) +(meV/atom) +0.156322 0.135053 0.103796 +t2G/BN +Type I +(θ12 = +0.56◦) +Type II +(θ12 = −0.58◦) +θ32 (◦) +0.56 +1.12 +0.56 +1.12 +k−(θ32) +(meV/(atom·rad)) +−40.01 +−9.69 +−7.124 +−14.33 +k+(θ32) +(meV/(atom·rad)) +48.01 +35.52 +54.54 +40.46 +Eb(θ32) +(meV/atom) +0.024273 0.058352 0.039000 0.094661 +TABLE II. Torque constants k± = dEtot/dθ32 in units of +meV/(atom · rad) as defined in Eq. (6) evaluated to the left +(k−) and right (k+) of the respective local minima at the +commensurate angles θ32, and the binding energy Eb(θ32) es- +timated as the difference between the smoothly interpolated +polynomial curve and the respective minima in Fig. 2(a)-(b). +AAA and AAB stackings. We can thus conclude that +the sliding-dependent energy landscape of the outer lay- +ers will have non-negligible gradients only when we have +equal period commensurate moire patterns. Sliding en- +ergy landscapes of 10−1 meV/atom for commensurate +double moire systems where ∆θ = θ32 − θ12 = 0◦ quickly +drops to 10−4 meV/atom even for marginally small twist +angles away from commensuration of ∆θ ≃ 0.03◦ to down +to 10−5 meV/atom for ∆θ ≃ 0.5◦, indicating high moire +superlubricity away from exact commensuration. This is +consistent with the fact that both structures shown in +Fig. 3 for the 1.54◦ − 1.1◦ combination have nearly the +same local stacking area ratios, indicating in turn that +they will have weak interlayer sliding force gradients. +The torque constants obtained using Eq. 6 from the +total energy curves are summarized in Table II together +with the binding energies Eb defined as the energy gain +achieved due to commensuration that we define as the +difference between dashed interpolated line and the ac- +tual total energy at commensuration in Fig. 2(a) and (b). +The calculated torque magnitudes are generally larger +for smaller angles and the signs of k+ and k− torques +tend to bring the incommensurate moire systems back to +commensuration as a positive/negative torque value will +decrease/increase the θ32. +The analysis we have presented so far relied on free- +standing trilayer systems. In order to assess the impact a +substrate would have on our results, we have performed +the following checks. +For t3G, we have checked that +adding a rigid hBN substrate layer with a twist angle +of 3.41◦ dampens but does not completely remove the + +(a) +L12 (12 = 1.53°) +L23 (032 = 1.53°) +L12 +L23 +t3G +40 +33.3 +33.2 +33.3 +33.2 +(AAA) +30 +AAA +20 +7.7 +7.7 +10 +0 +AA +AB +BA AA +AB +BA +L12 (012 = 1.53°) +L23 (032 = 1.53°) +L12 +L23 +t3G +40 +36.3 +36.0 +(AAB) +30 +22.7 +22.9 +AAB +12.2 +13.4 +10 +0 +AA +AB +BA AA +AB +BA +(b) +L12 (012 = 1.53°) +L23 (032 = 1.1°) +L12 +L23 +t3G +40 +35.435.4 +31.2 +31.2 +(AAA +30 +AAA +10.7 +10 +6.5 +AA +AB +BA AA +AB +BA +L12 (012 = 1.53°) +L23 (032 = 1.1°) +L12 +L23 +t3G +40 +35.435.4 +31.231.2 +(AAB +30 +AAB +10.7 +10 +6.5 +0 +AA +AB +BA AA +AB +BA +AA +AB +BA +DW1 +DW2 +DW36 +bending corrugation observed for the mirror symmetry +broken commensurate case. Indeed, the maximum bend- +ing corrugation goes down for the suspended t3G system +from 1.70 ˚A as seen in Fig. 1(d) to 0.25, 0.35, and 0.45 ˚A +for L1, L2 and L3, respectively, when we add a rigid +substrate layer in contact with L1. If we don’t fix the +additional substrate layer, the bending corrugation actu- +ally increases by up to 3.79 ˚A, hence a realistic substrate +simulation, outside of the scope of this study, involving +many more layers [55, 57] would probably give a max- +imum bending corrugation somewhere in the middle of +those two values. For t2G on hBN, we have checked that +adding a fixed aligned hNB substrate below the existing +hBN layer does not modify the qualitative behaviors and +the system still shows local energy dips at the commen- +surate angles. We thus expect our conclusions based on +free-standing systems to hold under more realistic exper- +imental conditions when substrates are present. +Conclusions – We have shown the tendency of dou- +ble moire systems to spontaneously form commensurate +moire patterns with rational p/q moire length ratios and +align their angles, as illustrated in the alternating twist +trilayer graphene for p = q = 1 and twisted bilayer +graphene on hexagonal boron nitride for p = 1, q = 1, 2 +systems, and found that the effect quickly diminishes for +q > 1. The binding energy gained during the alignment +of the twist angle near θ32 ≃ 1◦, 1.5◦ and 2◦ is of the order +of ∼ 0.2, 0.17, 0.13 meV/atom respectively which is com- +parable in magnitude with the energy differences result- +ing from the relative sliding of the top and bottom layers +for commensurate moire geometries. These are several +orders of magnitude larger than the sliding-dependent +energy changes for incommensurate moire geometries in- +dicating the existence of superlubricity as soon as the +system is marginally twisted away from commensuration +already for angles as small as ∼ 0.03◦. +The rotation +torque constants presented have been evaluated for differ- +ent fixed bottom layers twist angles θ12. In experiments, +once θ12 is fixed we propose that targeting θ32 angles that +are equal or slightly larger than the value that yields com- +mensurate double moire patterns will more easily tend to +lock the system into moire commensuration, while target- +ing a smaller angle may result in the system rotating back +to the trivial zero-alignment commensurate phase upon +thermal annealing. The qualitative conclusions based on +the t3G and t2G/BN systems explored in this work are +expected to apply for a variety of other twisted layered +materials that will be investigated in future work. +Acknowledgments +– +This +work +was +supported +by +the +Korean +NRF +through +the +Grants +No. +2020R1A2C3009142 (N.L.), 2020R1A5A1016518 (J.A.), +2021R1A6A3A13045898 (Y.P.), and Samsung Science +and Technology Foundation Grant No. SSTF-BA1802- +06 (J.J.). We acknowledge computational support from +KISTI Grant No. +KSC-2022-CRE-0514 and by the +resources of Urban Big data and AI Institute (UBAI) +at UOS. J.J. also acknowledges support by the Korean +Ministry of Land, Infrastructure and Transport(MOLIT) +from the Innovative Talent Education Program for Smart +Cities. +∗ jeiljung@uos.ac.kr +[1] C. R. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, +Y. Gao, J. Katoch, M. Ishigami, P. Moon, M. Koshino, +T. Taniguchi, K. Watanabe, K. L. Shepard, J. Hone, and +P. 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Hod, ACS +Nano 13, 7603 (2019). +APPENDIX A +Commensurate moire superlattices can be built in gen- +eral based on 4 integer indices using the conventions in +Ref. [53], as exemplified in Ref. [52] for twisted bilayer +graphene systems. For double moire systems, the same +approach requires the definition of 6 integers, namely +(i, j, i′, j′, i′′, j′′). +These integers define the following +three transformation matrices where Mi with i = 1, 2, 3 + +8 +correspond to L1, L2 and L3 respectively: +M1 = +� +i +j +−j i + j +� +, +M2 = +� +i′ +j′ +−j′ i′ + j′ +� +, +M3 = +� +i′′ +j′′ +−j′′ i′′ + j′′ +� +. +(7) +These matrices relate the lattice vectors r1 and r2 to the +lattice vectors a1 and a2 of the respective layers through +� +r1 +r2 +� += M1 · +� +a1 +a2 +� += M2 · +� +a′ +1 +a′ +2 +� += M3 · +� +a′′ +1 +a′′ +2 +� +(8) +The lattice mismatch αmn and twist angle θmn between +the layers m and n can be related to these integers as +follows +α12 = |a1| +|a′ +1| = +� +i′2 + j′2 + i′j′ +i2 + j2 + ij , +α32 = |a′′ +1| +|a′ +1| = +� +i′2 + j′2 + i′j′ +i′′2 + j′′2 + i′′j′′ , +θ12 = θ1 − θ2 = cos−1 +�2ii′ + 2jj′ + ij′ + ji′ +2α12(i2 + j2 + ij) +� +, +θ32 = θ3 − θ2 = cos−1 +�2i′′i′ + 2j′′j′ + i′′j′ + j′′i′ +2α32(i′′2 + j′′2 + i′′j′′) +� +, +(9) +where we assume the middle L2 layer as the untwisted +reference layer. In Table. III, we summarize the six inte- +gers used to generate the systems represented in Fig. 2(a) +and (b) for the t3G and the two t2G/BN systems. +APPENDIX B +We illutrate here the difference between the local +torque maps for a commensurate and an incommensu- +rate moire system calculated through +τs = (rs − rcm) × Fs +(10) +where s is the sublattice index and cm refers to the center +of mass of the dimer formed by the neighboring A and +B sublattices, and where Fs is the interface component +of the force acting on an atom extracted at the end of +the LAMMPS minimization by subtracting the intralayer +forces. The left panels show the local torque maps for +commensurate moire pattern cases where the moire cell +has been repeated 15 times for a more direct compari- +son with the right hand panels that have the same size. +On the right panel, we illustrate the same torques for an +incommensurate moire configuration modeled through a +FIG. 4. +Local torque τs defined in Eq. 10 for each layer in +t3G at the site i ∈ sublattice s are illustrated when the two +moire interfaces are commensurate (θ12 = θ32 = 1.53◦) (left) +and incommensurate (θ12 = 1.53◦, θ32 = 1.64◦) (right). We +observe the presence of a super-moire pattern with a longer +period for the incommensurate moire case that gives rise to +the total energy differences with respect to the doubly com- +mensurate moire case. +multiple moirons system with longer super-moire period. +In the middle layer L2 we see that a large region of the +atoms feel the same torque as is seen for the commen- +surate phase, suggesting that within the region confined +by long period triangular patches we largely recover the +commensurate moire phase behavior seen for θ23 = 1.53◦. + +Commensurate +Incommensurate +(023 = 1.53°) +(023 = 1.64°) +(eV . A) +0.2054 +0.1026 +0 +0.1026 +-0.2054 +0.2054 +L +L +0.1026 +0 +-0.1026 +-0.2054 +0.2054 +0.1026 +0 +-0.1026 +-0.20549 +t3G +θ12 (◦) +θ32 (◦) +(i j i′ j′ i′′ j′′) +aL3(˚A) +# atoms +λ +1.084549 0.098591 +341 330 330 341 331 340 +2.460226 +2026246 11λ1 +0.299180 +899 870 870 899 878 891 +2.460278 14083050 29λ1 +0.500560 +403 390 390 403 396 397 +2.460300 +2829990 13λ1 +0.699713 +961 930 930 961 950 941 +2.460291 16092466 31λ1 +0.898632 1085 1050 1050 1085 1079 1056 2.460253 20513502 35λ1 +1.049565 +961 930 930 961 960 931 +2.460204 16092846 31λ1 +1.084549 +31 30 30 31 31 30 +2.460190 +16746 +λ1 +1.150275 +1023 990 990 1023 1025 988 +2.460162 18236534 33λ1 +1.200741 +868 840 840 868 871 837 +2.460138 13129050 28λ1 +1.491197 +248 240 240 248 251 237 +2.459963 +1071810 +8λ1 +1.951946 +155 150 150 155 159 146 +2.459556 +418722 +5λ1 +1.538500 0.099248 +682 651 651 682 653 680 +2.460244 +7997326 31λ2 +0.299135 +792 756 756 792 763 785 +2.460329 10784906 36λ2 +0.498959 +814 777 777 814 789 802 +2.460384 11392218 37λ2 +0.699314 +242 231 231 242 236 237 +2.460410 +1006902 11λ2 +0.901887 +638 609 609 638 626 621 +2.460405 +6998394 29λ2 +1.098945 +154 147 147 154 152 149 +2.460371 +407758 +7λ2 +1.201971 +704 672 672 704 697 679 +2.460342 +8521378 32λ2 +1.301824 +286 273 273 286 284 275 +2.460305 +1406374 13λ2 +1.398649 +242 231 231 242 241 232 +2.460263 +1006942 11λ2 +1.499055 +858 819 819 858 857 820 +2.460212 12657686 39λ2 +1.538500 +22 21 21 22 22 21 +2.460190 +8322 +λ2 +1.577943 +858 819 819 858 859 818 +2.460167 12657842 39λ2 +1.600032 +550 525 525 550 551 524 +2.460153 +5201302 25λ2 +1.700421 +418 399 399 418 420 397 +2.460087 +3004326 19λ2 +1.794869 +132 126 126 132 133 125 +2.460018 +299606 +6λ2 +1.900421 +374 357 357 374 378 353 +2.459932 +2405226 17λ2 +1.999934 +220 210 210 220 223 207 +2.459844 +832278 +10λ2 +2.004628 0.200433 +170 160 160 170 161 169 +2.460326 +490182 +10λ3 +0.801831 +255 240 240 255 246 249 +2.460551 +1102842 15λ3 +1.202796 +255 240 240 255 249 246 +2.460551 +1102842 15λ3 +1.397199 +561 528 528 561 551 538 +2.460508 +5337818 33λ3 +1.603742 +85 80 80 85 84 81 +2.460431 +122542 +5λ3 +1.700933 +561 528 528 561 556 533 +2.460384 +5337998 33λ3 +1.799055 +663 624 624 663 659 628 +2.460329 +7455662 39λ3 +1.899139 +323 304 304 323 322 305 +2.460265 +1769586 19λ3 +1.950459 +629 592 592 629 628 593 +2.460230 +6710766 37λ3 +2.004628 +17 16 16 17 17 16 +2.460190 +4902 +λ3 +2.056017 +663 624 624 663 664 623 +2.460150 +7456022 39λ3 +2.100064 +357 336 336 357 358 335 +2.460115 +2161826 21λ3 +2.198572 +527 496 496 527 530 493 +2.460030 +4711026 31λ3 +2.301520 +459 432 432 459 463 428 +2.459934 +3573806 27λ3 +t2G/BN Type I +θ12 (◦) +θ32 (◦) +(i j i′ j′ i′′ j′′) +aL3(˚A) +# atoms +λ +0.560656 0.200227 +812 812 812 840 817 835 +2.460259 12143930 28λ4 +0.400464 +406 406 406 420 411 415 +2.460298 +3035950 14λ4 +0.501637 1102 1102 1102 1140 1119 1123 2.460306 22366846 38λ4 +0.546279 1131 1131 1131 1170 1150 1151 2.460308 23559530 39λ4 +0.560656 +58 58 58 60 59 59 +2.460308 +61958 +2λ4 +0.583082 +725 725 725 750 738 737 +2.460308 +9680938 25λ4 +0.600704 +812 812 812 840 827 825 +2.460307 12143770 28λ4 +0.700824 +232 232 232 240 237 235 +2.460300 +991330 +8λ4 +0.800943 +203 203 203 210 208 205 +2.460286 +758990 +7λ4 +0.897056 +145 145 145 150 149 146 +2.460265 +387242 +5λ4 +1.000094 1073 1073 1073 1110 1106 1077 2.460235 21205546 37λ4 +1.079783 +783 783 783 810 809 784 +2.460207 11292158 27λ4 +1.121311 +29 29 29 30 30 29 +2.460190 +15490 +λ4 +1.159974 +841 841 841 870 871 840 +2.460173 13027150 29λ4 +1.229816 +899 899 899 930 933 896 +2.460140 14886094 31λ4 +1.300703 +725 725 725 750 754 721 +2.460103 +9681482 25λ4 +1.495028 +261 261 261 270 273 258 +2.459981 +1254762 +9λ4 +1.761923 +203 203 203 210 214 199 +2.459767 +759098 +7λ4 +2.018092 +145 145 145 150 154 141 +2.459512 +387322 +5λ4 +t2G/BN Type II +θ12 (◦) +θ32 (◦) +(i j i′ j′ i′′ j′′) +aL3(˚A) +# atoms +λ +−0.579874 0.200227 +840 784 840 812 835 817 +2.460259 12145498 28λ5 +0.537294 1440 1344 1440 1392 1417 1415 2.460308 35692418 48λ5 +0.560656 +60 56 60 58 59 59 +2.460308 +61966 +2λ5 +0.584017 1440 1344 1440 1392 1415 1417 2.460308 35692418 48λ5 +0.600704 +840 784 840 812 825 827 +2.460307 12145338 28λ5 +0.700824 +240 224 240 232 235 237 +2.460300 +991458 +8λ5 +0.800943 +210 196 210 203 205 208 +2.460286 +759088 +7λ5 +0.897056 +150 140 150 145 146 149 +2.460265 +387292 +5λ5 +1.000094 1110 1036 1110 1073 1077 1106 2.460235 21208284 37λ5 +1.079783 +810 756 810 783 784 809 +2.460207 11293616 27λ5 +1.121311 +30 28 30 29 29 30 +2.460190 +15492 +λ5 +1.159974 +870 812 870 841 840 871 +2.460173 13028832 29λ5 +1.229816 +930 868 930 899 896 933 +2.460140 14888016 31λ5 +1.300703 +750 700 750 725 721 754 +2.460103 +9682732 25λ5 +1.495028 +270 252 270 261 258 273 +2.459981 +1254924 +9λ5 +1.761923 +210 196 210 203 199 214 +2.459767 +759196 +7λ5 +2.018092 +150 140 150 145 141 154 +2.459512 +387372 +5λ5 +2.099900 1530 1428 1530 1479 1434 1574 2.460238 40294166 51λ5 +2.299955 1470 1372 1470 1421 1369 1521 2.460025 37197972 49λ5 +TABLE III. Details about the commensurate cells that are used for our simulations on the t3G (left) and t2G/BN (right) +systems where the first column contains θ12, the second column summarizes the θ32 for each of the corresponding θ12 values, +the third column contains the six integers as defined in Ref. [52, 53] where the first two integers control the lattice vectors of +the first layer, the next two integers define the lattice of the second layer and the final two integers orient the lattice vectors of +the top layer following Eq. (8). The fourth column contains the slightly strained lattice constant aL3 for the third layer which +is different from the unstrained lattice constants of 2.4602 ˚A for L1 and L2. The fifth column contains the number of atoms +and the final column represents the super-moire length or commensuration cell length λ as a multiple of the commensurate cell +moire length λi when θ12 = θ32, where λ1 = 129.97 ˚A, λ2 = 91.62 ˚A and λ3 = 70.32 ˚A for θ12 = 1.0845◦, 1.5385◦ and 2.0046◦ +respectively for t3G and λ4 = λ5 = 125.71 ˚A for t2G/BN Type I and t2G/BN Type II. We highlight the commensurate angle +configurations from the main text in bold while the other entries are commensurate approximations of the incommensurate +angle combinations. + diff --git a/dNE2T4oBgHgl3EQfwwjn/content/tmp_files/load_file.txt b/dNE2T4oBgHgl3EQfwwjn/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fd27576b9d91d89b5379fef99a13031122e0207e --- /dev/null +++ b/dNE2T4oBgHgl3EQfwwjn/content/tmp_files/load_file.txt @@ -0,0 +1,1194 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf,len=1193 +page_content='Commensuration torques and lubricity in double moire systems Nicolas Leconte,1 Youngju Park,1 Jiaqi An,1 and Jeil Jung1, 2, ∗ 1Department of Physics, University of Seoul, Seoul 02504, Korea 2Department of Smart Cities, University of Seoul, Seoul 02504, Korea We study the commensuration torques and layer sliding energetics of alternating twist trilayer graphene (t3G) and twisted bilayer graphene on hexagonal boron nitride (t2G/BN) that have two superposed moire interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Lattice relaxations for typical graphene twist angles of ∼ 1◦ in t3G or t2G/BN are found to break the out-of-plane layer mirror symmetry, give rise to layer rotation energy local minima dips of the order of ∼ 10−1 meV/atom at double moire alignment angles, and have sliding energy landscape minima between top-bottom layers of comparable magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Moire superlubricity is restored for twist angles as small as ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='03◦ away from alignment resulting in suppression of sliding energies by several orders of magnitude of typically ∼ 10−4 meV/atom, hence indicating the precedence of rotation over sliding in the double moire commensuration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Introduction – Experimental realization of moire ma- terials by layering two graphene-like 2D material systems with different lattice constants and/or with a finite twist angle [1, 2] have been furthered by combining two moire patterns to form the so-called double or super moire systems [3–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Crystals with long moire pattern peri- ods allows to access certain physical observables at low magnetic fields and gate carrier densities that would not be accessible otherwise [12–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' This behavior can be used for example in 30◦ twist-angle quasi-crystals [15– 19] to generate moire quasicrystals at experimentally ac- cessible charge carrier densities using h-BN encapsulated graphene [9, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Most theories describing double moire systems have so far used the simplest commensurate dou- ble moire geometries to explain the observed physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' For instance, in alternating twist trilayer graphene (t3G) with twisted middle layer giving rise to two aligned moire patterns, important in the context of flat band supercon- ductivity [21–31], the most stable geometry corresponds to superposed top and bottom layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Likewise com- mensurate double moire geometries have been assumed when studying twisted bilayer graphene on h-BN where a spontaneous anomalous Hall effect was measured [32–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' However, the assumption that equal period and aligned double moire pattern systems are energetically favored has not been yet confirmed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Here we analyze the atomic structure of t3G and twisted bilayer graphene on hexagonal boron nitride (t2G/BN) to show that double moire systems generate torques that tend to lock the systems into commensurate moire patterns and favor a specific sliding geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' We find that mirror symmetry breaking layer corrugations are required for a correct total energy minimization, in- cluding the AAA-stacked t3G systems where we use the overline to indicate relative sliding geometries between the layers regardless of their twist angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' We observe that the specific sliding atomic structures between the top and next nearest layer only matters when the moire patterns are commensurate, since the energies are al- most sliding-independent for incommensurate moire pat- terns leading to superlubricity away from commensura- tion, similar to the superlubricity behavior in single inter- face twisted bilayer systems studied in the literature [38– 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Systems and methods – We consider double moire systems with two moire interfaces consisting of graphene trilayers and t2G on hexagonal boron nitride.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' We illus- trate schematically in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' 1 (a) all three different sys- tems considered, namely t3G, t2G/BN type I and type II depending on the twist angle sense of the bottom hBN layer contacting graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' The layer numerals 1, 2 and 3 correspond to bottom, middle and top layers respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' The middle layer 2 is taken as the reference frame with zero twist angle and we use θ12 and θ32 labels to represent the actual twist angles of the bottom and top layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' The atomic structure relaxation is carried out us- ing LAMMPS [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' We use the REBO2 force-field [46] for the intralayer interactions of graphene and EXTEP [47] for those of hexagonal boron nitride, whose equilibrium geometries are aG =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='4602˚A and aBN =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='50576˚A re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' The interlayer force fields are based on EXX- RPA-informed [48] DRIP [49] parametrizations [50] and we used both the fire and CG minimization scheme [51] with a time step of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='0001 ps for the former and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='001 ps for the latter and a stopping tolerance on the forces of 10−5 eV/˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' In order to capture the tiny angle variations with respect to the doubly commensurate angles we use the approach outlined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' [52] and Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' [53] to find the commensurate cell for a double moire system, see Appendix A for details and summary of the integers rep- resenting our commensurate cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' The incommensurate moire systems are approximated by taking commensurate simulation cells containing several repetitions the moire unit cells or moirons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Due to the small angle differences and the high sensitivity of the results on internal strains, we are at times bound to choose large simulation sys- tems containing millions of atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' The lattice constant variation tolerance is capped at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='03% to minimize inter- nal strains that lead to energy differences of the order of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='005 meV/atom that are between one to two orders of magnitude smaller than the energy differences of the or- der 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='1 meV/atom required to resolve the local minima in arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='04105v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='mes-hall] 10 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' (color online) (a) Schematic figure illustrating the different t3G and t2G/BN systems considered here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' θij = θi − θj are the twist angles of the i-th layer with respect to the j-th layer where their particular choices can lead to commensurate and incommensurate moire patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' (b) Corrugation effect that breaks the mirror-symmetry of AAA stacked t3G with θ12 = θ32 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='53◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' (c) Interlayer distance differences at different local stacking positions and (c) bending corrugations in the middle layer 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' The sketches at the bottom of the panels indicate the specific local stackings AAA, AAB and AAC for the three-layer systems while the two-letter conventions refer to local sliding between two layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Gray, green and orange colors refer to the A and B sublattices of graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Blue and purple refer to the boron and nitrogen atoms in hBN in t2G/BN or the A and B sublattices in a t3G system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' our energy curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' The stability of the atomic structure relies on the total energy given as the sum Etot = Eel + Epot (1) where we can distinguish the elastic energy Eel = � i Ei el/2 that resists the deformation due to the strains, and the potential energy Epot = � i Ei pot/2 that triggers the formation of the moire pattern strains taken as sum of contributions from each atomic site i and where the division by 2 accounts for double counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' We can define the local elastic Ei el, potential Ei pot and interface Ei IFmn energies as Ei el = � j∈layer i φij (2) Ei pot = � j /∈layer i φij = � j∈ any layer φij − Ei el (3) Ei IFmn = � j /∈ layer i j ∈ layer n or m φij mn (4) where φij represents the pair-wise potentials between atoms i and j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Our calculations show that the elastic energy contributions in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' (2) are about one order mag- nitude smaller than the potential and interface energies in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' (3-4), and therefore is only a small fraction of the total energy in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' (1) that dictates the stability of our systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' The interface energies are essentially the poten- tial energies referred to a particular pair of layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' It will be interesting to note how this interface energy changes from one system to another by examining the interface energy differences for two different relaxed atomic struc- tures ∆EIFmn(r) = E3L IFmn(r) − E2L IFmn(r) (5) where E3L IFmn at a given point is obtained relaxing simul- taneously all three layers of t3G and then considering the bilayer atomic positions for the considered mn inter- face,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' while the E2L IFmn interface energy is obtained using the t2G relaxed atomic positions of the two mn layers that form the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' (5) we have removed the i-index dependence in Ei IFmn in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' (4) using the posi- tion vector r instead by interpolating the data from the closest i-sublattice points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Another quantity of interest is the torque constant that we define as the derivative of the total energy as a function of twist angle similar to the proposals in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' [42, 44, 54] but focusing here on the rotation of the top layer with respect to the middle layer k± = dEtot dθ32 (6) where its positive or negative values tend to either reduce or increase the value of θ32 towards the commensurate moire geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' (a) (b) t3G 012 (012 = 032 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='53) Z3 1 032 1△C32 Z2 1 △C21 Z1 1 (c) △C32 = △C12 (A) t3G 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='10 f A △C32 = △C12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='05 t2G/BN (Type I) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='00 (A) AADW1 AB DW3 BADW2AA (d) Z2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='5 Z2 t2G/BN Top 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='0 (Type II) Middle 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='5 Bottom 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='0 AAA AAB AAC AADW1 ABDW3 BADW2AA L3 AA DW2 BA DW3 AB DW1 AA L2 L3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' (color online) (a) Energies of t3G double moire systems for three different values of θ12 where θ32 is varied from ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='1◦ to ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='2◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' For t3G systems the commensurate moire patterns are obtained when θ12 = θ32 (indicated by vertical lines) and leads to a local energy dip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' The x-symbols represent the total energies when the top interface has AB-stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' The dashed lines are obtained by using polynomial interpolation away from the commensuration dips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' On the right hand we show total energies for different sliding of the top layer for select commensurate angles θ12 = θ32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' (b) Similar plots for t2G/BN (Type I and Type II) where the total energy dips are obtained for two different values of θ32, corresponding to angles where L12 = L32 (solid vertical line) and L12 = L32/2 (dashed vertical line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' (c) Intepolated interface energy differences ∆EIFmn(r) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' (5) for the 3 systems considered illustrating the local energy gain/penalty when a single moire comes into contact with a second moire interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' The energy difference densities are plotted along a straight line that connects the opposite diagonal corners of the moire cell through different local stacking positions where the most stable stacking arrangement is highlighted with a thicker line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' The lower the energy, the more stable is the double moire geometry locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' The negative energies correspond to a stabilizing gain in energy while positive energies indicate a destabilizing energy penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' The interface energy differences, and therefore the potential energy differences, make up the dominant contributions of the total energy differences since the elastic energy contributions are one order of magnitude smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' (a) t3G 012 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='08° 012 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='53° 12 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='0° 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='25 AAA AAC (meV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='75 AAA AAB 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='50 12 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content="08° AAB 'tot 012 = 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='53° X E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='25 12 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='00° X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='00 AAA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='5 032 () (meV/atom) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='88 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='22 (b) t2G/BN Type I Type II (032 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='12) (032 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='12) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='8 AAA (meV /atom) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='6 AAC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='4 101 AAB Type I (012 = + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='56) E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='2 Type II (012 = - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='58) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='0 AAA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='5 032 () (meV/atom) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='64 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='82 (c) (meV) t3G (meV) t2G/BN Type I (meV) t2G/BN Type II 10 2 0 0 0 10 10 2 AAA 20 20 AAA AAA AAB AAB AAB 30 AAC 30 AAC 2 1 2 21 10 10 0 0: 0 10 10 1 2 20 20 AAA ACA ABA AAA AAA ACB ABC AAA AAA ABA ACA AAA AAB ACB ABB AAB AAB ACC ABA AAB AAB ABB ACB AAB AAC ACA ABB AAC AAC ABC ACC AAC4 Results – We begin by testing the main assumption of our study for t3G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' namely that angles leading to com- mensurate moire systems will be energetically more fa- vorable than angle combinations leading to incommen- surate moire systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' For this purpose we fix the angle θ12 between the two bottom layers to three select values of θ12 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='08◦, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='54◦ and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='0◦ to sample typical largest magic angles values in tNG systems for N = 2, 3, ∞ [55] while we vary θ32 between the two top layers from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='1◦ to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='3◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Commensuration for alternating twist is naturally achieved when θ12 = θ32 since the lattice constants of all three layers are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' The find numerically that the relaxed atomic structures generally favor z-axis corruga- tions, where Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' 1(c) shows the typical local stacking- dependent interlayer distance variation of the order of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='1 ˚A while Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' 1(d) illustrates for the middle layer the z-axis corrugations of the order of ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='75 ˚A roughly one half of the average interlayer distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' The main results of this work are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' 2(a) where we plot the total energy per atom as a function of θ32 after the atomic relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' For all three θ12 values considered, we find total energy local minima at the com- mensurate twist angles provided that we allow a bending corrugation as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' In fact, the to- tal energies reported in Table I indicate that the mirror- symmetric t3G with a completely flat middle layer has a higher total energy compared to the corrugated atomic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' There we list the integrated total, elastic, po- tential, and interface energies taken as the sum over all atomic sites i, with proper double counting correction, of the energies given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' (1-4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' 2(a) we show that the commensurate moire systems are most stable in the AAA-stacking when there is no relative sliding between the three layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' In fact, the energy difference between the AAA at the local minima dip and AAB max- ima represented is equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='18 meV/atom near 1◦ and drops to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='06 meV/atom for an angle of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='47◦ (not shown here) and are comparable to the magnitude of the energy dips due to the rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Our sliding dependent total energy plots indicate that there are barrier-free sliding paths leading to the global minimum at AAA-stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' We then show a similar analysis for t2G/BN where we fix the substrate angle between G and hBN at θ12 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='56◦ for type I or −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='58◦ for type II and we allow θ32 between both graphene layers to change up to a value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='5◦ to achieve moire periods that satisfy pLM tBG = q LM tGBN [56] where p, q are integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' For q = 1 we observe two dips in the energy curve that correspond to θ32 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='56◦ for p = 2 with a G/G moire pattern period twice as large as the G/BN and θ32 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='12◦ for p = 1 with equal periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' The details of the atomic relaxation giving rise to total energy dips in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' 2(a) and (b) for t3G and t2G/BN can be further understood through the line plots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' 2(c) calculated using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' (5) where we focus on the inter- face energy difference between 3-layer and 2-layer relaxed systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' We can thus quantify the energy gain/penalty Etot Eel EIF12 EIF23 t3G (AAA) −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='42537 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='39458 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='01542 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='01541 t3G (AAA, ms) −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='42534 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='39459 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='01538 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='01538 t3G (AAB) −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='42510 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='39471 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='01521 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='01516 t2G/BN-I (AAA) −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='19472 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='16454 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='01458 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='01559 t2G/BN-I (AAB) −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='19451 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='16466 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='01442 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='01542 t2G/BN-I (AAC) −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='19467 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='16455 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='01456 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='01556 t2G/BN-II (AAA) −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='19561 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='16554 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='01452 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='01556 t2G/BN-II (AAB) −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='19562 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='16553 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='01453 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='01556 t2G/BN-II (AAC) −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='19581 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='16556 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='01463 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='01563 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Sliding dependent total, elastic, and interface en- ergies in for θ12 = θ32 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='5385◦ for t3G, θ12 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='56◦, θ32 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='12◦ for t2G/BN-I, and θ12 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='58◦, θ32 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='12◦ for t2G/BN-II for systems containing 8322, 15490 and 15492 atoms respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' These numbers are used to renormal- ize and report the energies in eV/atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' We note that the mirror-symmetric (ms) geometry has a higher total energy by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='03 meV/atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' (negative and positive values, respectively) one achieves when putting two moire patterns in contact with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' For t3G the most stable 3-layer configuration is the AAA-stacking that we achieve when the unstable AA-stacking of each bilayer are stacked on top of each other and the stable AB/BA stackings are simultane- ously stacked on top of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' The interface energy plots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' 2(c) illustrate the energy differences between single and double moire pattern atomic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' For AAA-stacking the interface energy difference plot shows that its energy gain is most pronounced at local stacking regions in between the AAA and ACA/ABA regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' In contrast the AAB-stacking has a generally unfavorable interface energy for most local stackings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' The double moire energetics becomes more com- plicated when we have an heterogeneous interface in t2G/BN where the most stable (unstable) AB/BA (AA) local stacking geometries in t2G combines with the AC (AA/AB) stackings of GBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' For type I trilayers the ener- getically stable geometry is AAA while for type II trilay- ers it is AAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' We notice that the strongest local energy penalties due to interfering moire patterns do not neces- sarily happen at the high symmetry local stacking geome- tries and the global energy minimization does not follow simple rules of thumb consisting in combining together the local stackings that are energetically most favorable or unfavorable as in the t3G case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' The conclusions on stability energetics that we can draw based on the energy density difference line cuts in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' 2 are consistent with the interface and total ener- gies that predict the AAA, AAA and AAC stacking as the most stable geometries for t3G, t2G/BN type I and t2G/BN type II, systems respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' To further ex- plore the stabilization of the commensurate moire sys- tems versus the incommensurate ones we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' 3 the local stacking distribution maps for AAA and AAB 5 stackings of t3G following the conventions outlined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' [52] for each one of the interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Here we can FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Illustration of the stacking redistribution in t3G for two double moire configurations obtained using the con- ventions outlined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' [27] using the displacement vectors between layers m and n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' (a) Local stacking maps for AAA and AAB stacking geometries for θ12 = θ32 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='53◦, and the ratios of the local stacking areas of the left panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' The AAA stack- ing shows a smaller AA local stacking area due to greater local rotations that reduces this energetically unfavorable stacking when compared with the AAB stacking going from 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='7% to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='2% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' (b) Local stacking maps for θ12 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='53 and θ32 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='1◦ for sliding geometries and corresponding local stacking area ratios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Although the relaxation profiles are dif- ferent, the ratios for local AA, AB and BA stackings remain nearly the same for different sliding geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' compare how the local stacking distributions change for the commensurate and incommensurate moire interfaces, and how the favorable and unfavorable sliding geom- etry influences the strain profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' For commensurate moire patterns and stable AAA-stacking the energeti- cally unfavorable AA local stacking that makes about 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='7% of the total area increases to 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='2% for the less sta- ble AAB-stacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' As soon as we move away from the same period commensurate moire geometries and have different moire pattern periods, for instance the combi- nation of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='54◦ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='1◦ with 1/7 moire length ratios or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='54◦ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='5◦ with 1/39 ratios that can be considered approximations of the incommensurate moire patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' For these geometries the relative distribution of the AA, AB or BA local stacking areas become practically insen- sitive to the relative sliding of the top layer resulting in almost the same local stacking area ratios for both t3G (θ12 = θ32) θ32 (◦) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='53 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='00 k−(θ32) (meV/(atom·rad)) −42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='12 −18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='34 −8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='513 k+(θ32) (meV/(atom·rad)) 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='06 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='04 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='85 Eb(θ32) (meV/atom) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='156322 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='135053 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='103796 t2G/BN Type I (θ12 = +0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='56◦) Type II (θ12 = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='58◦) θ32 (◦) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='56 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='56 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='12 k−(θ32) (meV/(atom·rad)) −40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='01 −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='69 −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='124 −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='33 k+(θ32) (meV/(atom·rad)) 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='01 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='52 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='54 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='46 Eb(θ32) (meV/atom) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='024273 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='058352 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='039000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='094661 TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Torque constants k± = dEtot/dθ32 in units of meV/(atom · rad) as defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' (6) evaluated to the left (k−) and right (k+) of the respective local minima at the commensurate angles θ32, and the binding energy Eb(θ32) es- timated as the difference between the smoothly interpolated polynomial curve and the respective minima in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' 2(a)-(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' AAA and AAB stackings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' We can thus conclude that the sliding-dependent energy landscape of the outer lay- ers will have non-negligible gradients only when we have equal period commensurate moire patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Sliding en- ergy landscapes of 10−1 meV/atom for commensurate double moire systems where ∆θ = θ32 − θ12 = 0◦ quickly drops to 10−4 meV/atom even for marginally small twist angles away from commensuration of ∆θ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='03◦ to down to 10−5 meV/atom for ∆θ ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='5◦, indicating high moire superlubricity away from exact commensuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' This is consistent with the fact that both structures shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' 3 for the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='54◦ − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='1◦ combination have nearly the same local stacking area ratios, indicating in turn that they will have weak interlayer sliding force gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' The torque constants obtained using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' 6 from the total energy curves are summarized in Table II together with the binding energies Eb defined as the energy gain achieved due to commensuration that we define as the difference between dashed interpolated line and the ac- tual total energy at commensuration in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' 2(a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' The calculated torque magnitudes are generally larger for smaller angles and the signs of k+ and k− torques tend to bring the incommensurate moire systems back to commensuration as a positive/negative torque value will decrease/increase the θ32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' The analysis we have presented so far relied on free- standing trilayer systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' In order to assess the impact a substrate would have on our results, we have performed the following checks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' For t3G, we have checked that adding a rigid hBN substrate layer with a twist angle of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='41◦ dampens but does not completely remove the (a) L12 (12 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='53°) L23 (032 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='53°) L12 L23 t3G 40 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='3 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='2 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='3 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='2 (AAA) 30 AAA 20 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='7 10 0 AA AB BA AA AB BA L12 (012 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='53°) L23 (032 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='53°) L12 L23 t3G 40 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='3 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='0 (AAB) 30 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='7 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='9 AAB 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='2 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='4 10 0 AA AB BA AA AB BA (b) L12 (012 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='53°) L23 (032 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='1°) L12 L23 t3G 40 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='435.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='4 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='2 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='2 (AAA 30 AAA 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='7 10 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='5 AA AB BA AA AB BA L12 (012 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='53°) L23 (032 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='1°) L12 L23 t3G 40 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='435.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='4 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='2 (AAB 30 AAB 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='7 10 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='5 0 AA AB BA AA AB BA AA AB BA DW1 DW2 DW36 bending corrugation observed for the mirror symmetry broken commensurate case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Indeed, the maximum bend- ing corrugation goes down for the suspended t3G system from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='70 ˚A as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' 1(d) to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='35, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='45 ˚A for L1, L2 and L3, respectively, when we add a rigid substrate layer in contact with L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' If we don’t fix the additional substrate layer, the bending corrugation actu- ally increases by up to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='79 ˚A, hence a realistic substrate simulation, outside of the scope of this study, involving many more layers [55, 57] would probably give a max- imum bending corrugation somewhere in the middle of those two values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' For t2G on hBN, we have checked that adding a fixed aligned hNB substrate below the existing hBN layer does not modify the qualitative behaviors and the system still shows local energy dips at the commen- surate angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' We thus expect our conclusions based on free-standing systems to hold under more realistic exper- imental conditions when substrates are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Conclusions – We have shown the tendency of dou- ble moire systems to spontaneously form commensurate moire patterns with rational p/q moire length ratios and align their angles, as illustrated in the alternating twist trilayer graphene for p = q = 1 and twisted bilayer graphene on hexagonal boron nitride for p = 1, q = 1, 2 systems, and found that the effect quickly diminishes for q > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' The binding energy gained during the alignment of the twist angle near θ32 ≃ 1◦, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='5◦ and 2◦ is of the order of ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='17, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='13 meV/atom respectively which is com- parable in magnitude with the energy differences result- ing from the relative sliding of the top and bottom layers for commensurate moire geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' These are several orders of magnitude larger than the sliding-dependent energy changes for incommensurate moire geometries in- dicating the existence of superlubricity as soon as the system is marginally twisted away from commensuration already for angles as small as ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='03◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' The rotation torque constants presented have been evaluated for differ- ent fixed bottom layers twist angles θ12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' In experiments, once θ12 is fixed we propose that targeting θ32 angles that are equal or slightly larger than the value that yields com- mensurate double moire patterns will more easily tend to lock the system into moire commensuration, while target- ing a smaller angle may result in the system rotating back to the trivial zero-alignment commensurate phase upon thermal annealing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' The qualitative conclusions based on the t3G and t2G/BN systems explored in this work are expected to apply for a variety of other twisted layered materials that will be investigated in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Acknowledgments – This work was supported by the Korean NRF through the Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' 2020R1A2C3009142 (N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' ), 2020R1A5A1016518 (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' ), 2021R1A6A3A13045898 (Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' ), and Samsung Science and Technology Foundation Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' SSTF-BA1802- 06 (J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' We acknowledge computational support from KISTI Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' KSC-2022-CRE-0514 and by the resources of Urban Big data and AI Institute (UBAI) at UOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' also acknowledges support by the Korean Ministry of Land, Infrastructure and Transport(MOLIT) from the Innovative Talent Education Program for Smart Cities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' ∗ jeiljung@uos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='kr [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Dean, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Wang, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Maher, C.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' B 41, 11837 (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' [55] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Leconte, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Park, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' An, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Samudrala, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Jung, 2D Materials 9, 044002 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' [56] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Shin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Park, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Chittari, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Sun, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Jung, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' B 103, 075423 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' [57] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Mandelli, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Ouyang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Urbakh, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Hod, ACS Nano 13, 7603 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' APPENDIX A Commensurate moire superlattices can be built in gen- eral based on 4 integer indices using the conventions in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' [53], as exemplified in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' [52] for twisted bilayer graphene systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' For double moire systems, the same approach requires the definition of 6 integers, namely (i, j, i′, j′, i′′, j′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' These integers define the following three transformation matrices where Mi with i = 1, 2, 3 8 correspond to L1, L2 and L3 respectively: M1 = � i j −j i + j � , M2 = � i′ j′ −j′ i′ + j′ � , M3 = � i′′ j′′ −j′′ i′′ + j′′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' (7) These matrices relate the lattice vectors r1 and r2 to the lattice vectors a1 and a2 of the respective layers through � r1 r2 � = M1 · � a1 a2 � = M2 · � a′ 1 a′ 2 � = M3 · � a′′ 1 a′′ 2 � (8) The lattice mismatch αmn and twist angle θmn between the layers m and n can be related to these integers as follows α12 = |a1| |a′ 1| = � i′2 + j′2 + i′j′ i2 + j2 + ij ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' α32 = |a′′ 1| |a′ 1| = � i′2 + j′2 + i′j′ i′′2 + j′′2 + i′′j′′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' θ12 = θ1 − θ2 = cos−1 �2ii′ + 2jj′ + ij′ + ji′ 2α12(i2 + j2 + ij) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' θ32 = θ3 − θ2 = cos−1 �2i′′i′ + 2j′′j′ + i′′j′ + j′′i′ 2α32(i′′2 + j′′2 + i′′j′′) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' (9) where we assume the middle L2 layer as the untwisted reference layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' In Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' III, we summarize the six inte- gers used to generate the systems represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' 2(a) and (b) for the t3G and the two t2G/BN systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' APPENDIX B We illutrate here the difference between the local torque maps for a commensurate and an incommensu- rate moire system calculated through τs = (rs − rcm) × Fs (10) where s is the sublattice index and cm refers to the center of mass of the dimer formed by the neighboring A and B sublattices, and where Fs is the interface component of the force acting on an atom extracted at the end of the LAMMPS minimization by subtracting the intralayer forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' The left panels show the local torque maps for commensurate moire pattern cases where the moire cell has been repeated 15 times for a more direct compari- son with the right hand panels that have the same size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' On the right panel, we illustrate the same torques for an incommensurate moire configuration modeled through a FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Local torque τs defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' 10 for each layer in t3G at the site i ∈ sublattice s are illustrated when the two moire interfaces are commensurate (θ12 = θ32 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='53◦) (left) and incommensurate (θ12 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='53◦, θ32 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='64◦) (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' We observe the presence of a super-moire pattern with a longer period for the incommensurate moire case that gives rise to the total energy differences with respect to the doubly com- mensurate moire case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' multiple moirons system with longer super-moire period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' In the middle layer L2 we see that a large region of the atoms feel the same torque as is seen for the commen- surate phase, suggesting that within the region confined by long period triangular patches we largely recover the commensurate moire phase behavior seen for θ23 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='53◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Commensurate Incommensurate (023 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='53°) (023 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='64°) (eV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' A) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='2054 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='1026 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='1026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='2054 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='2054 L L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='1026 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='1026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='2054 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='2054 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='1026 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='099900 1530 1428 1530 1479 1434 1574 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='460238 40294166 51λ5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='299955 1470 1372 1470 1421 1369 1521 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='460025 37197972 49λ5 TABLE III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' Details about the commensurate cells that are used for our simulations on the t3G (left) and t2G/BN (right) systems where the first column contains θ12, the second column summarizes the θ32 for each of the corresponding θ12 values, the third column contains the six integers as defined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' [52, 53] where the first two integers control the lattice vectors of the first layer, the next two integers define the lattice of the second layer and the final two integers orient the lattice vectors of the top layer following Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' The fourth column contains the slightly strained lattice constant aL3 for the third layer which is different from the unstrained lattice constants of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='4602 ˚A for L1 and L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' The fifth column contains the number of atoms and the final column represents the super-moire length or commensuration cell length λ as a multiple of the commensurate cell moire length λi when θ12 = θ32, where λ1 = 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='97 ˚A, λ2 = 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='62 ˚A and λ3 = 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='32 ˚A for θ12 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='0845◦, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='5385◦ and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='0046◦ respectively for t3G and λ4 = λ5 = 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content='71 ˚A for t2G/BN Type I and t2G/BN Type II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} +page_content=' We highlight the commensurate angle configurations from the main text in bold while the other entries are commensurate approximations of the incommensurate angle combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNE2T4oBgHgl3EQfwwjn/content/2301.04105v1.pdf'} diff --git a/dNFIT4oBgHgl3EQfnyuw/content/tmp_files/2301.11316v1.pdf.txt b/dNFIT4oBgHgl3EQfnyuw/content/tmp_files/2301.11316v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..19950c78cf2cb7e268438b078f129ea54634cbf4 --- /dev/null +++ b/dNFIT4oBgHgl3EQfnyuw/content/tmp_files/2301.11316v1.pdf.txt @@ -0,0 +1,1869 @@ +Open Problems in Applied Deep Learning +Maziar Raissi +Department of Applied Mathematics, University of Colorado Boulder, +Boulder, Colorado, 80309, USA +Abstract +This work formulates the machine learning mechanism as a bi-level opti- +mization problem. The inner level optimization loop entails minimizing a +properly chosen loss function evaluated on the training data. This is nothing +but the well-studied training process in pursuit of optimal model parame- +ters. The outer level optimization loop is less well-studied and involves max- +imizing a properly chosen performance metric evaluated on the validation +data. This is what we call the “iteration process”, pursuing optimal model +hyper-parameters. Among many other degrees of freedom, this process en- +tails model engineering (e.g., neural network architecture design) and man- +agement, experiment tracking, dataset versioning and augmentation. The +iteration process could be automated via Automatic Machine Learning (Au- +toML) or left to the intuitions of machine learning students, engineers, and +researchers. Regardless of the route we take, there is a need to reduce the +computational cost of the iteration step and as a direct consequence reduce +the carbon footprint of developing artificial intelligence algorithms. Despite +the clean and unified mathematical formulation of the iteration step as a +bi-level optimization problem, its solutions are case specific and complex. +This work will consider such cases while increasing the level of complexity +from supervised learning to semi-supervised, self-supervised, unsupervised, +few-shot, federated, reinforcement, and physics-informed learning. As a con- +sequence of this exercise, this proposal surfaces a plethora of open problems +in the field, many of which can be addressed in parallel. +Keywords: +supervised learning, semi-supervised learning, self-supervised +learning, unsupervised learning, few-shot learning, federated learning, +reinforcement learning, and physics-informed learning +Preprint submitted to Journal Name +January 27, 2023 +arXiv:2301.11316v1 [cs.LG] 26 Jan 2023 + +1. Introduction +The general mechanism for building machine learning solutions is illus- +trated in Fig. 1 and outlined in the following. 1) Everything starts with data +as the “source code” for machine learning. 2) We would then write a model +to fit the data. 3) The model is then trained to maximize the likelihood +of the training data or minimize a distance/divergence between the training +data distribution and model predictions in the case of generative adversarial +networks. The training process typically entails multiple steps of stochastic +gradient descents (e.g., Adam optimizer [1]). 4) The model is then evaluated +using a properly chosen performance metric (e.g., accuracy, mean average +precision, inception score, etc.) on the validation data. 5) Next is the iter- +ation step where the aforementioned steps 1-4 need to be repeated tens of +thousands of times to find the most performant solution. This step entails +model engineering (e.g., neural network architecture design) and manage- +ment, experiment tracking, dataset versioning/augmentation, in addition to +seemingly “minor” details such as choosing learning rates and learning rate +schedules, batch sizes, weight decay and batch/weight/layer/group/spectral +normalizations, just to name a few. +The iteration step is a crucial piece +of the machine learning pipeline and is usually the most time and resource +consuming step while often being overlooked. This is what makes machine +learning difficult. 6) Before putting the model into production, we test it +one last time on some test data. 7) The final stage is serving the model +in production to millions of customers/users while constantly monitoring its +performance and re-training it if needed. +Step 5 (i.e., the iteration step) is the topic of this work. The jury is still +out whether the iteration step should be automated (i.e., AutoML) or left +to the intuitions of machine learning students, engineers, and researchers. +Regardless of the route we take, there is a need to reduce the computational +cost of the iteration step and as a direct consequence reduce the carbon foot- +print of developing artificial intelligence algorithms [2]. The computational +bottleneck of the iteration step is the training of each model to convergence, +only to measure its performance on the validation data which is then used +as the feedback signal to guide the iteration process. +2 + +Figure 1: The general mechanism for building machine learning solutions. +2. Background +In mathematical notations, we are trying to solve bi-level optimization +problems [3, 4] of the following form; +min +α +Mval(w∗(α), α) +s.t. +w∗(α) = arg min +w Ltrain(w, α). +(1) +Here, α denotes the “hyper-parameters” of the model in an abstract sense, en- +capsulating concepts as generic as learning rate (schedule), depth and width +of neural networks, discrete choices between CNNs, RNNs, or Transformers +[5] and their variants [6], presence or absence of different types of normal- +ization layers [7], to pad or not to pad, convolutional kernel sizes, to use +dropout or not, etc. Moreover, Mval denotes a performance metric such as +accuracy, mean average precision [8], Fr´echet Inception Distance [9], etc. The +loss function is denoted by Ltrain while the model parameters are represented +by w. Whereas Eq. 1 explains the iteration problem (see Fig. 1) in a clean +and unified mathematical formulation, the solution to this problem is com- +plex and case specific. +3 + +3. Training +1. Data +2. Model +5. Iteration +6. Testing +7. Production +4. ValidationFor some machine learning problems, the evaluation metric Mval is a dis- +crete and non-differentiable function of α (e.g., accuracy [10]). For some other +problems, it isn’t well-defined (e.g., domain adaptation [11], semi-supervised +[12], self-supervised and unsupervised learning [13, 14] as well as generative +models [15]) or is even non-existent (e.g., physics-informed deep learning +[16–19]). The hyper-parameters α could be continuous (learning rate, mo- +mentum, weight-decay, etc.) or discrete variables (e.g., the choice between a +regular convolution or a depth-wise-separable one [20, 21], number of layer- +s/channels, to use batch-norm [22] or not, etc.). Sometimes α is a function +of the current training epoch (e.g., learning rate schedule, progressive grow- +ing of GANs [23], etc.). More often than not, the loss function is a weighted +combination of multiple loss functions (e.g., multi-task learning [24], physics- +informed machine learning [25], etc.); Those weights could be part of α and +require special treatment to balance the trade-off between focusing on one +objective function versus another. The loss function Ltrain could itself be dis- +crete and non-differentiable (e.g., cost/reward in reinforcement learning [26]). +Therefore, the solutions to the problem specified in Eq. 1 and illustrated in +Fig. 1 are not as clean as its formulation and end up being case specific. +The first solution that comes to mind is to rely on the intuition of students, +researchers, and engineers obtained through tens of thousands of collective +trial and errors and knowledge sharing over the Internet (e.g., stackover- +flow.com), blog posts, and social media platforms. We can also build tools to +facilitate such experimentations. A good example is “Weights & Biases”, an +MLOps platform to help researchers build better models faster with experi- +ment tracking, dataset versioning, and model management. MLOps is an ac- +tive area of research both in academia and industry. In the cases where Mval +is a well-defined function, let it be discrete or non-differentiable, one could +use grid search (albeit for low dimensional α), random search [27], Bayesian +optimization [28], reinforcement learning [29–31] or evolutionary algorithms +[32, 33] to find α∗, even if α is discrete. The shortcoming of such approaches +for solving Eq. 1 is their extensive computational cost (e.g., 32,400-43,200 +GPU-Hours, the equivalent of using 450 GPUs for 3-4 days [34, 35]). Such +approaches have therefore a high carbon footprint [2]. +The bottleneck is +solving the inner optimization problem in Eq. 1 (i.e., the training loop) to +completion, only to measure the model’s performance on the validation data +which is then used as the feedback signal to guide the iteration process (i.e., +the outer optimization problem in Eq. 1). One solution to this problem +is to trade off computation for more memory consumption using parameter +4 + +sharing ideas [34]. This could result in up to 1000x faster solutions to Eq. 1 +(e.g., 16 GPU-Hours) in some cases. The idea is to warm-start the training +process using some shared parameters cached in memory, rather than start- +ing from random parameters (i.e., cold-starting). This could speed up the +convergence of the inner optimization problem in Eq. 1. Alternatively, one +could approximate the validation function Mval with a differentiable func- +tion (e.g., the loss function on the validation data Lval [35]). Doing so will +enable us to use plain-vanilla stochastic gradient descent algorithms (e.g., +the Adam optimizer [1]) to optimize over both the hyperparameters α and +parameters w; One could take a couple of gradient descent steps for w and +one step for α in an iterative fashion. +There is no need to solve the in- +ner optimization problem to completion. This will also result in significant +speed-up by trading computation for more memory consumption. As a com- +promise between purely automated solutions to Eq. 1 (i.e., AutoML) and the +expert-in-the-loop solutions (i.e., MLOps), we could also use a combination +of the two [36, 37]. Moreover, as an additional layer of complication, the per- +formance metric Mval(w∗(α), α) could pursue multiple competing objectives +such as minimizing the error rates (i.e., maximizing accuracy) and reducing +the computational and memory costs of the model, perhaps to make them +suitable for smaller devices such as mobile phones, tablets, or IoT (Internet +of Things) devices [38, 39]. Here, balancing the trade-off between optimizing +one objective versus the other is an open problem. This will introduce ad- +ditional hyper-parameters (lets call them “hyper-hyper-parameters”) for the +outer optimization problem in Eq. 1. Last but not least, data augmentation +policies [27, 31] could also be part of the search space (i.e., α). +This should give us enough background on the current state-of-the-art in +AutoML and help us outline the open problems that need to be addressed +collectively by the machine learning community. +3. Open Problems +As mentioned above, despite the clean and unified mathematical formu- +lation of the iteration step as a bi-level optimization problem (see Eq. 1 and +Fig. 1), its solutions are case specific and complex. In the following, we will +consider such cases while increasing the level of complexity from supervised +learning to semi-supervised, self-supervised, unsupervised, few-shot, feder- +ated, reinforcement, and physics-informed learning. As a consequence of this +5 + +exercise, this work exposes a multitude of open challenges in the field, many +of which can be addressed in parallel. +3.1. Supervised Learning +Most of the progress made over the last few years, ever-since the advent +of AutoML, falls under the umbrella of supervised learning, in particular (im- +age) classification [29–39]. These works use accuracy as their performance +metric or a differentiable approximation of it (e.g., the negative of the loss +function or the log-likelihood). As a low hanging fruit, we could investigate +if such techniques would still work in the face of other performance metrics, +such as precision, recall, F1 score, calibration [40], etc., for balanced and +unbalanced datasets. Finding differentiable approximations of such metrics +is particlarly interesting because we can employ similar techniques to the +ones used in [35] (i.e., using plain-vanilla stochastic gradient descent to opti- +mize over both hyper-parameters and parameters of our models). Such tech- +niques are interesting not only because of their computational efficiency but +also because they require very little “hyper-hyper-parameter” tuning. How- +ever, how to make such methods memory efficient is still an open question. +Moreover, methods such as Bayesian optimization, reinforcement learning, +evolutionary algorithms, and even parameter-sharing for solving Eq. 1 intro- +duce additional hyper-parameters (i.e., “hyper-hyper-parameter”); In simple +terms, we would like to avoid doing AutoAutoML, AutoAutoAutoML, etc., +and fragmenting our datasets beyond training, validation, and testing. We +could also study the effect of such approximations to different performance +metrics and provide theoretical upper-bounds on the loss of performance as +a result of such approximations. +Large Networks: Recently, we are witnessing a trend in computer vision +trying to replace convolutions neural networks with transformers [6, 41, 42] +or even multi-layer perceptrons [43], inspired by their success in language +[44–46]. The question is if the currently available techniques (see the back- +ground section) for solving Eq. 1 could generalize to such architectures and +improve their performance. It is worth noting that the techniques outlined in +the background section are primarily designed for convolutions neural net- +works. +Furthermore, when it comes to data-augmentation strategies, the +current techniques leverage only a single image [31], the question is if novel +data-augmentation strategies such as mix-up [47] and cut-mix [48] (leverag- +ing pairs of images) can be discovered as part of solving Eq. 1. Answering +6 + +these two questions would entail rethinking the design space (i.e., the space +in which α is assumed to live). Another interesting and fundamental question +is investigating the possibility of automating the discovery of learning rate +schedules such as the cosine learning rate schedule [49]. Here, the learning +rate is a function of the current epoch rather than being a constant. This +will significantly increase the complexity of the iteration problem (see Eq. 1). +Methods similar to the ones presented in [35] seem to have a good chance at +solving this problem because they don’t rely on solving the inner optimiza- +tion problem (i.e., training) in Eq. 1 to completion. This will allow us to +modify the learning rate in tandem with the training process per each epoch. +Small Networks: There are times when not only we are looking for the +most performant model but also we want the model to be as memory and +compute efficient as possible. This is an important stepping stone towards +democratizing artificial intelligence in anticipation of the future of Internet +of Things where a lot of our devices (e.g., cellphones, cars, security cameras, +refrigerators, air conditioners, etc.) will be intelligent. Such devices usually +have smaller compute capabilities and memory capacity than our comput- +ers in data-centers or on the cloud. To make them intelligent we need to +take their constraints into consideration. Mathematically speaking, Mval is +a weighted combination of at least two objectives; One is the performance +metric, while the other is about making the model more nimble and could +take different forms such at FLOPs, MAC (memory access cost), number +of parameters, latency and memory consumption of the target devices. This +last item necessitates a hardware-in-the-loop approach. The weights given to +each objective function are what we called “hyper-hyper-parameters” earlier +in this document. It is still an open question how to set such weights in or- +der to balance the tradeoff between optimizing one objective versus another. +We are therefore dealing with a multi-objective bi-level optimization prob- +lem. Here, ideas such as the ones proposed in [50] for multi-task learning +using uncertainty to weigh different objectives could be extended to solve +our multi-objective bi-level optimization problem. A similar multi-objective +optimization problem arises in physics-informed deep learning [17–19] where +we need to balance the trade-off between fitting the data and respecting the +law of physics modeled using ordinary and partial differential equations. Al- +ternatively, we could investigate the possibility of automating the process of +making pre-trained models smaller. This is an after-the-fact approach where +we would like to automate the discovery of methods such as knowledge distil- +7 + +lation [51], model pruning and compression [52–54], etc. This approach will +avoid the aforementioned multi-objective bi-level optimization problem and +will instead break the problem into two or more stages; In the first stage we +will be looking for the most performant model, regardless of its cost, while +in the second stage we will look for the best model compression strategies. +This will lead to a multi-stage (versus multi-objective) bi-level optimization +problem. +Robustness: It is a well-known fact that deep neural networks are vul- +nerable to adversarial and backdoor attacks [55–59]. We could be investi- +gating how much of this vulnerability can be attributed to the iteration step +(see Fig. 1 and Eq. 1) and whether automating the process can alleviate or +aggravate the problem. +Explainable AI: In the past few years, the machine learning community +has made a lot of progress in the emerging field of explainable and trustwor- +thy AI (see e.g., [40, 60–72] and the references therein) to the extend that +deep neural networks are no longer considered black boxes but rather gray +ones. +However, there has been very little effort in the literature (if any) +to explain the choices made as part of the iteration process (see Fig. 1). +The question is what features of the data, or rather the meta-data, explain +the choices we make for learning rate schedules, architecture designs, data- +augmentation, etc. What is the effect of noise in the data on such choices? +How important are the size and intrinsic dimensionality of the data? This a +place where the underlying structure of the data (i.e., the lower dimensional +manifold on which the data lives) could help us shed some light on these +fundamental questions. If successful, efforts in this direction could lead to a +new field, namely “Explainable AutoML”. +Transfer Learning: Moreover, in a parallel thrust, we could be studying +the impact of the iteration process (see Fig. 1) on the generalizability and +transferability of the learned features to downstream tasks. The field of deep +learning is largely driven by the ideas of Transfer learning to the extent that +we rarely train our models from scratch. Along the same lines, an impor- +tant question worthy of systematic studies is transferability (or lack thereof) +of data augmentation policies (e.g., AutoAugment and RandAugment) from +one dataset (e.g., ImageNet) to another (e.g., Pascal VOC). +8 + +Semantic Segmentation: When it comes to the task of semantic seg- +mentation, we typically start with neural networks pre-trained on a related +classification task and do transfer learning [73–81]. This is mainly because we +usually have access to smaller datasets for this task as labeling every single +pixel in an image is more cumbersome than labeling an entire image with a +single label. Referring back to Fig. 1, there is very little work on an iteration +phase dedicated to the semantic segmentation task. Here, the performance +metrics (e.g., pixel accuracy, mean accuracy, mean IU, frequency weighted +IU, etc.) are more complex than the accuracy metric often used for the clas- +sification task. Finding differentiable approximations of such metrics so that +we can employ similar techniques to the ones used in [35] is particularly in- +teresting. Moreover, when it comes to the task of semantic segmentation, we +need to not only capture the global information in an image (i.e., resolving +the “what” of the image) but also the local information (i.e., resolving the +“where” of the objects in the image). It would be interesting to investigate +how the iteration stage (see Fig. 1) would have an impact on the “what” +and “where” components of the semantic segmentation task. In particular, +would the iteration phase leverage tools such as atrous convolutions, short- +cut connections, conditional random fields, multi-scale aggregation, deep su- +pervision, deconvolutions, upsampling, attention mechanisms, etc.? If so, to +what extent? +Super-Resolution, Denoising, and Colorization: When it comes +to creative tasks such as super-resolution, denoising, colorization, and style +transfer where the output of a neural network is an image, it is very hard to +judge the quality of the generated images in a quantitative fashion. Avail- +able perormance metrics such as Peak Signal-to-Noise (PSNR), Structural +Similarity (SSIM), and Feature Similarity (FSIM) fall short of doing justice +to the task. +It is therefore very hard to measure progress in these fields +[82–94] and more importantly guide the iteration phase (see Fig. +1). +In +this regard, deep features extracted from deep neural networks (e.g., VGG) +trained on the ImageNet classification task show unreasonable empirical ef- +fectiveness as perceptual metrics [93]. However, there aren’t many works (if +any) that guide the iteration phase using such deep features (e.g., Learned +Perceptual Image Patch Similarity (LPIPS) metric) in an automated fash- +ion. In a related note, the tasks of super-resolution, denoising, colorization, +and style transfer usually entail balancing the trade-off between multiple loss +functions (e.g., reconstruction L1/L2 loss versus perceptual loss [83]). It is +9 + +still an open problem how to strike such a balance in the absence of univer- +sally accepted performance metrics. Here, ideas such as the ones proposed +in [50] for multi-task learning using uncertainty to weigh different objectives +could be extended to solve this problem. Alternatively, we could investigate +the possibility of guiding the iteration process (see Fig. 1) using human-in- +the-loop Reinforcement Learning algorithms where the reward signal comes +from the judgment of human beings; A human can easily take a look at an +image and associate a quality score to it, perhaps from 1-10. This is feasible +because with Reinforcement Learning we don’t need to differentiate through +the reward signals or the thought process of the human evaluator. Human- +in-the-loop techniques are gaining traction these days (see e.g., [95]) because +writing well-defined reward functions is very challenging if not impossible for +many real-word applications of Reinforcement Learning beyond games and +simulated environments. +Pose Estimation: A keyboard and a mouse are not the only means of +interacting with a computer; A key topic in the field of human-computer +interaction in particular and the meta verse in general is human pose estima- +tion [96–100]. Two evaluation metrics that could guide the iteration phase +(see Fig. 1) are Percentage of Correct Parts (PCP) and Percent of Detected +Joints (PDJ). PCP measures detection rate of limbs, where a limb is consid- +ered detected if the distance between the two predicted joint locations and +the true limb joint locations is at most half of the limb length. As for PDJ, a +joint is considered detected if the distance between the predicted vector and +the true joint is within a certain fraction of the torso (e.g., left shoulder and +right hip) diameter. A closely related metric is Percentage of Correct Key- +points (PCK) which measures the percentage of detections that fall within +a normalized distance of the ground truth. It would be interesting to inves- +tigate how the iteration stage (see Fig. 1) would leverage these metrics or +their differentiable approximations to come up with novel architectures (e.g., +stacked hourglass blocks, cascaded pyramid layers, part affinity fields, etc.) +in an automated fashion. This is not a well-studied topic as of this writing. +Optical Flow and Depth Estimation: The world around us is 3D +and evolving in time. On the one hand, depth estimation [101–105] allows +us to add a third dimension to our 2D images and has applications for self- +driving cars and robots. On the other hand, optical flows [106–108] enable +us to capture the evolution in time and the relationship between consecu- +10 + +tive frames in a video. In addition to applications for self-driving vehicles +and robotics, optical flows can be used as additional features for the task +of action recognition in videos. For optical flows, End Point Error (EPE) +is typically used as the performance metric guiding the iteration phase (see +Fig. 1). It is the Euclidean distance between the predicted flow vector and +the ground truth, averaged over all pixels. Here, the training data is usually +simulated because it is very hard to measure optical flows in the physical +world [106]. Inevitably, we have to rely on domain adaptation techniques +to close the reality gap (i.e., the gap between the real and simulated data +distributions) to the extent possible. It is therefore an intriguing research +question to study the effect of domain adaptation techniques on the iteration +phase (see Fig. 1) and vice versa. The task of depth estimation also suf- +fers from lack of enough labeled training data. Fortunately, there are ways +to leverage the underlying physics of the problem to perform unsupervised +monocular depth estimation with e.g., left-right consistency [103]. Here, the +training loss would involve a weighted combination of multiple individual +loss functions [103, 104]. Properly setting those weights in an automated +fashion is an open question. Here, we could utilize performance metrics such +as absolute relative distance, squared relative distance, or root mean square +error (RMSE) to guide the iteration stage (see Fig. 1). +Object Detection: Generally speaking, there are two major types of +object detectors: multi-stage (typically two) and one-stage detectors. With +multi-stage detectors [8, 109–117], the primary objective is to find the most +performant model (measured using mean average precision) while efficiency +(measured using frames per second) is a secondary objective. With single- +stage detectors [118–131], the primary objective is to find the most agile +model (measured using frames per second) while performance (measured us- +ing mean average precision) is a secondary objective. Referring back to equa- +tion 1, Mval is pursuing multiple (i.e., at least two) competing objectives for +the task of object detection. Balancing the trade-off between optimizing one +objective versus the other is an open problem. It is worth mentioning that +this problem is about what we called “hyper-hyper-parameters” in the itera- +tion phase and that we would like to avoid fragmenting our datasets beyond +training, validation and testing. As for hyper-parameters α, the design space +(the space in which α lives) is a much more complex one for object detection +compared to classification tasks. The input data could be in the form of im- +ages, image patches, image pyramids, etc. The backbone could be in the form +11 + +of VGG, ResNet, ResNeXt, Darknet, Hourglass Network, Transformers, etc. +The neck of the architecture could be in the form of FPN (Feature Pyramid +Network), PANet, Bi-FPN (Bi-directional FPN), etc. The head of the object +detection system could be a dense predictor (RPN, YOLO, SSD, RetinaNet, +FCOS) or a sparse one (Faster R-CNN, R-FCN). As for data augmentation +we could use CutMix, MixUp, Mosaic, Bluring, etc. For the loss functions, +we could use L1, L2, Smooth L1, or CIoU for the regression component of +the total loss function and MSE, binary or multi-class cross-entropy loss for +the classification portion of the total loss. Therefore, exploring the design +space in a systematic and automatic fashion as part of the iteration phase +(see Fig. 1) is a challenging task. Last but not least, all metrics are wrong, +some are useful. This is specially true when it comes to the object detection +task. There are heated debates in the literature about the appropriateness +of mean average precision (mAP) or its COCO style variants as valid perfor- +mance metrics for the object detection task (see e.g., [124]). +Face Recognition and Detection: Face detection is a special case of +the object detection and key point (pose) estimation topics that we covered +earlier in this document. Futhermore, face recognition (verification and iden- +tification) can be viewed as a close-set or an open-set problem, depending +on the type of available data. Close-set face recognition is nothing but a +classification task that we covered earlier in this document. Open-set face +recognition on the other hand is about metric learning, i.e., learning features +that are capable of pulling similar images together while pushing dissimilar +images apart. The literature on open-set face recognition spends a lot of time +designing new loss functions such as the triplet loss [132, 133], the center loss +[134], angular softmax loss [135], additive angular margin loss [136], etc. It +is therefore a natural question to ask if it is possible to automate the search +for appropriate loss functions as part of the iteration phase (see Fig. 1). To +guide the iteration process we could leverage the ROC (Receiver Operator +Characteristic) curves relating the true positive rate to the false positive rate. +Video & 3D data: As mentioned earlier in this document, the world +around us is evolving in time and is 3D. When it comes to videos, we could +think of at least two important applications, namely action recognition and +object tracking. Action recognition [137–146] is a classification task albeit +on a sequence of image frames in a video as the input data. Here the design +space (the space in which α lives, referring to equation 1) is more complex +12 + +compared to the design space for images. It would therefore be interesting to +see if ideas such as early/late/slow fusion, multi-streaming, using optical flows +as additional input features, 3D convolutions, trajectory pooling, or slow-fast +networks would survive the iteration phase (see Fig. 1). Furthermore, when +it comes to object tracking [147, 148], there is very little work on automating +the iteration phase (see Fig. 1) and studying its impact on the resulting +algorithms. Here, we could use evaluation metrics such as the center location +error and the bounding box overlap ratio to guide the iteration phase. In +a parallel thrust, performing object recognition, detection and segmentation +on 3D point cloud data is more challenging than doing so on images [149– +153]. These types of data (e.g., LIDAR data) appear naturally in self-driving +vehicles and robotics applications. +However, there is very little work on +automating the iteration phase (see Fig. 1) for performing object recognition, +detection and segmentation on 3D point cloud data. +3.2. Beyond Supervised Learning +It is now time to increase the level of complexity and move beyond super- +vised learning towards semi-supervised, self-supervised, unsupervised, few- +shot, federated, reinforcement, and physics-informed learning. We will be +approaching these topics from the perspective of the iteration phase (see +Fig. 1). +Natural Language Processing: For applications such as word vector +representations [154–158], text classification and sequence tagging [159–172], +translation [173–186] and language modeling [187–210], unlabeled text data is +available en masse thanks to the Internet – for example, the Common Crawl +project produces about 20TB of text data extracted from web pages each +month. This makes pre-training large language models on such data partic- +ularly attractive. Such models can then be fine-tuned on downstream tasks +(BERT family of models) or used in a few-shot setting (GPT type of models). +Here, the fundamental question is what performance metric(s) should we use +to guide the iteration phase (see Fig. 1) when it comes to training large lan- +guage models on unlabeled data. One idea is to use perplexity as a measure +of the goodness of language models. However, we don’t typically train large +language models for the sake of modeling the language but rather to use them +in some downstream applications such as text classification (e.g., sentiment +analysis), sequence tagging (e.g., named entity recognition), machine trans- +lation, program synthesis, question answering, summarization, semantic tex- +13 + +tual similarity, language comprehension, conversational response generation, +etc. For instance, it is a well-known observation that the common practice of +extracting sentence embeddings from the BERT language model, by average +pooling the last layer output vectors or using the output of the first token +(i.e., the [CLS] token), yields rather bad sentence embeddings [204], often +worse than averaging the GloVe vectors. This is why researchers came up +with the ideas of Sentence-BERT and Siamese BERT-Networks. Another +observation is that BERT in its original form cannot perform translation. +This is why researchers introduced BART, GPT, T5, etc. However, each one +of these contributions focus on a hand-full of downstream tasks to evaluate +the performance of their language modeling capabilities. Some focus on the +GLUE benchmark [211] of a suit of downstream tasks, some focus on the +BLUE score [212] for translation, etc. Perhaps a better strategy to guide +the iteration process (see Fig. 1) is to approach language modeling from a +multi-task learning perspective where Mval in equation 1 is a weighted com- +bination of the performance metrics for a multitude of downstream tasks. +Here, an open question is how to properly weigh one objective function ver- +sus the others. Here, ideas such as the ones proposed in [50] for multi-task +learning using uncertainty to weigh different objectives could be extended +to solve our multi-objective bi-level optimization problem. In addition, we +could include extra objectives in Mval to penalize the computational com- +plexity and memory consumption of the resulting language models. +The +idea is to come up with the smallest language model that is good at solving +a multitude of downstream tasks in an automated fashion. Of particular in- +terest are techniques similar to the ones used in [35] (i.e., using plain-vanilla +stochastic gradient descent to optimize over both hyper-parameters and pa- +rameters of our models). Last but not least, many of the ideas in natural +language processing can be extend to graphs (e.g., social networks) [213–222]. +Multimodal Learning: With multimodal learning [223–234], we are +taking baby steps towards human level artificial intelligence (i.e., artificial +general intelligence). For instance, if we look at an intelligent robot and say +“pick that up and put it on the table” while pointing at a box sitting on +the ground, to be able to execute the command correctly, the robot should +not only process speech and language but also should be able to use its vi- +sion system to understand what we mean when we say “that”. Moreover, a +common criticism to large language models is that even if they manage to +generate seemingly cohesive text, they have very little idea about what they +14 + +are actually talking about; For example, a language model trained only on +textual data has never seen images of airplanes, cars and ships, it has only +read about them on the internet. The field of multimodal learning is therefore +attracting the attention of a lot of great researchers both in academia and +the industry. Two important applications are translating images (or videos) +to text (e.g., image and video captioning) and vice versa (e.g., text to image +synthesis). Another equally important application is visual question answer- +ing. Moreover, training large language models both on images and textual +data is also showing some great promise. We will be approaching these top- +ics from the perspective of the iteration phase (see Fig. 1). When it comes +to translating images to text, we could use the BLUE score to guide the +iteration phase. However, there are heated debates in the literature if the +BLUE score is the best performance metric both for image captioning and +translation. Coming up with better metrics is an open problem. Moreover, +there is very little work (if any) on automating the iteration phase of trans- +lating images to texts. For the text to image synthesis type of tasks, like +any other creative task (e.g., super-resolution, denoising, colorization, style +transfer and generative adversarial networks), it is very hard to judge the +quality of the generated images in a quantitative fashion. Here, we could +use metrics such the Inception Score, the Fr´echet Inception Distance or the +Learned Perceptual Image Patch Similarity (LPIPS) metric to guide the it- +eration phase. As of this writing, there aren’t many works (if any) that have +done so before. Furthermore, when it comes to training large language mod- +els on unlabeled textual data as well as images, the fundamental question +is what performance metric(s) should we use to guide the iteration phase +(see Fig. 1). In particular, the question is how existing performance metrics +such as perplexity can be generalized to handle both text and image type +of data. Alternatively, similar to language models trained only on text, we +could define a set of downstream tasks involving both text and images as +benchmarks to guide the iteration phase. The idea is then to approach mul- +timodal modeling from a multi-task learning perspective. +Generative Networks: Generative models are either trained to max- +imize the likelihood (or rather its lower bound) of the training data (e.g., +Variational Auto-Encoders) or minimize a distance/divergence between the +training data distribution and model predictions (e.g., unconditional and con- +ditional Generative Adversarial Networks). A central question here is how +to measure the quality of the generated data in a quantitative fashion. For +15 + +images, we could use the Inception score or the Fr´echet Inception Distance +to guide the iteration process, despite all their imperfections. However, it +is not clear what performance metrics we should use for other types of data +such as text, speech, graphs (e.g., social networks), etc. Furthermore, it is +a well-known observation that training generative adversarial networks is an +unstable process and most of the contributions in this field are made towards +stabilizing this process by using different loss functions (e.g., Feature Match- +ing, Least Squares GANs, Wasserstein GANs, Hinge Loss, etc.), normaliza- +tion and regularization schemes (e.g., Gradient Penalty, Spectral Normal- +ization, Orthogonal Regularization, Adaptive Instance Normalization, Path +Length Regularization, etc.), architectures (e.g., DCGANs, Self-Attention +GANs, etc.) and training schedules (Progressive Growing, two time-scale +update rule, historical averaging of parameters, etc.). It would be interesting +to study if such techniques would survive an automated iteration process (see +Fig. 1). Of particular interest is the progressive growing idea because the +neural network architecture itself is a function of the current training epoch. +This will significantly increase the complexity of the iteration problem (see +Eq. 1). Methods similar to the ones presented in [35] seem to have a good +chance at solving this problem because they don’t rely on solving the inner +optimization problem (i.e., training) in Eq. 1 to completion. This will al- +low us to modify the architecture in tandem with the training process per +each epoch. Last but not least, when it comes to conditional GANs (e.g., +image-to-image translation), we typically try to minimize a total loss func- +tion being a weighted combination of multiple individual loss functions. It is +still an open problem how to come up with those weights in the absence of +validation data and appropriate performance metrics. Perhaps reformulating +the problems as a multi-task learning problem and using ideas similar to the +ones proposed in [50], that leverage uncertainty to weigh different objectives, +could help us address this problem. +Domain Adaptation: The concept of domain adaptation [11, 235–238] +is related to scenarios where we have a lot of labeled data (e.g., simulated +data) from a source domain and zero (or very few) labeled data (e.g., real +data) from a target domain. Such scenarios happen frequently in many engi- +neering fields (sometimes called multi-fidelity modeling [239–241] in fluid and +solid mechanics) including but not limited to self-driving cars and robotics. +Domain adaptation can help us close the so called reality gap between the +simulated and real data distributions. We are going to approach domain +16 + +adaptation from the perspective of the iteration phase (see Fig. 1). With +domain adaptation we would like to minimize the risk of making errors on the +target data, not necessarily the source data. Now, the question is how can we +measure the performance of our models on the target data in the absence of +any target labels (e.g., unsupervised domain adaptation) or in the presence +of very few of them (e.g., weakly-supervised domain adaptation). One idea is +to use unsupervised hyper-parameter selection techniques [11]. We can split +the source domain labeled data into a training set Strain and a validation set +Sval. Similarly, we can split the target domain unlabeled data into a training +set Ttrain and a validation set Tval. We can then use Strain and Ttrain to learn +a model using domain adaptation techniques. The trained model can now +be used to generated labels for the unlabeled data Ttrain. We then remove +the labels from Strain. We will then train a reverse model using Ttrain as the +source domain and Strain as the target domain. The reverse model can now +be evaluated on the validation set Sval. We could use this reserve validation +risk as a proxy for the true validation risk to guide the iteration process. +Few-shot Learning: When it comes to using a machine learning model +to serve millions of users perhaps over the internet, not only we need to take +care of the distributional shift between the training and test data (i.e., do- +main adaptation) but also we need to be able to handle new use cases and +more importantly new labels associated to such use cases. As an example, in +the context of a recommendation system [242–251], we can think of new items +(e.g., movies, products, etc.) to be recommended to new/existing users. This +is related to the topic of few-shot learning [252–254] where our models need +to be able to handle new labels given very few observations per each label. +For classification tasks, we can use N-way K-shot classification accuracy to +guide the iteration phase (see Fig. 1). It would therefore be interesting to +study the effect of an automated iteration stage on the resulting algorithms. +More importantly, it is still unclear what performance metrics we should use +for other applications of few-shot learning beyond classification such as se- +mantic segmentation, object detection, pose estimation, depth estimation, +etc. +Federated Learning: In addition to the publicly available data on the +internet, there is a wealth of data sitting on our privately-held devices (e.g., +cellphones, tablets, laptops, etc.). Our devices are also getting more power- +ful in their compute and data collection capabilities (e.g., multiple camera +17 + +lenses on the back of our cellphones). Federated Learning tries to leverage +such privately held data to train machine learning models while preserving +the privacy of the data. The idea is to bring the models to the data (rather +than bringing the data to the models on the cloud) and use the heteroge- +neous compute capabilities of user devices to train our models. What is being +communicated to the cloud is the parameters of our models or rather their +gradients. The field is still in its infancy and there are many open technical +and non-technical challenges (e.g., communication efficiency, the non-i.i.d na- +ture of the data, data pre-processing, training self-supervised models, privacy +preserving, being robust to backdoor attacks, being able to train models on +smaller devices, etc.) to be addressed before we can fully realize the potential +of federated learning. In this work, we are approaching federated learning +from the iteration perspective (see Fig. 1). Given the distributed nature of +the data over millions of user devices, the question is how can we evaluate +the performance of our models. One idea is to have the users of our models +give star ratings (perhaps out of five) to our models. We can then aggregate +these stars as a feedback signal to guide the iteration phase. Given that such +a performance metric is discrete and non-differentiable we can use methods +based on Reinforcement Learning to perform hyper-parameter selection. +Semi-Supervised & Self-Supervised Learning: Let us now move to- +wards the cases where we have access to a lot of unlabeled data and very few +labeled data, if any. We faced a similar situation in natural language pro- +cessing. However, with language we are working with discrete tokens which +makes it easier to perform self-supervision by defining next token predic- +tion (e.g., GPT style models) or masked token prediction tasks (e.g., BERT +style models). Discrete tokens make it possible to use the softmax function +as the last layer of a neural network (e.g., a Transformer architecture) and +turn the self-supervision problem into a classification one. However, image +and speech type of data are continuous signals. Normal, Laplace, or even +mixture of Gaussians for modeling the distribution of continuous random +variables are not as flexible as the softmax function is in modeling the dis- +tribution of discrete random variables. It is therefore required to rethink +the semi-supervised and self-supervised learning paradigms when it comes +to continuous signals (e.g., images and speech). In fact, this field has been +growing at an exponential rate over the past two or three years. One com- +mon theme emerging in the literature is to take a single image and augment +it into two different views. +These two views should then give consistent +18 + +representations once processed by the same neural network (or two similar +ones). Here, a central challenge that needs to be overcome is avoiding the +trivial solutions, either implicitly or explicitly. A network being supervised +by itself or by another similar network is prone to converging to a trivial +solution (e.g., a constant function ignoring its inputs altogether). We face +a similar problem with physics-informed neural networks (e.g., any constant +function is a solution to the Navier-Stokes equations) [17–19]. We are going +to approach this problem from the perspective of the iteration stage (see +Fig. 1). We need to either explicitly include a term in our training objective +function that encourages non-trivial solutions, or design our search space in +such a way that it includes mechanisms that have shown empirical success +in avoiding trivial solutions such as stop-gradients, predictor heads, model +averaging, contrastive losses, etc. Here, balancing the trade-off between the +consistency loss and avoiding the trivial solution is very delicate. Fortunately, +with semi-supervised learning [12, 41, 255–258] we have some labeled data +and performance metrics that we can leverage to guide the iteration phase. +However, with self-supervised learning [13, 14, 259–271], neither such labeled +data exists nor are there any universally accepted performance metrics. One +idea is to define a set of downstream tasks (e.g., object recognition, detec- +tion, and segmentation for images) to judge the transferablility of the learned +features. Here, how much importance we should give to each downstream +task is an open problem. +Speech: Similar to the language modeling paradigm for text, there is +an emerging trend over the past few years to model speech [272–276]. The +idea is that before we (as human beings) learn to read and write, we learn +to listen and speak. +This self-supervised learning paradigm for speech is +sometimes also called learning by listening. Speech being a continuous signal +inherits many of the challenges that we went over in the previous paragraph +on self-supervised learning for images. In addition, as of this writing, there +are only two well-defined downstream tasks, namely translating speech to +text [277–285] and vice versa [286–289], to guide the iteration stage (see +Fig. 1) using their respective performance metrics (i.e., label error rate for +speech recognition and subjective 5-scale mean opinion score in naturalness +for speech synthesis). The performance metric for speech synthesis, however, +requires human in the loop evaluators and leaves Reinforcement Learning +or evolutionary algorithms as the only options to guide the iteration phase. +Fortunately, there are some researchers in both academia and industry who +19 + +are trying to come up with more downstream tasks to judge the quality of +speech models. Even in the presence of such downstream tasks, we will need +to solve multi-objective bi-level optimisation problems of a form that gener- +alizes the one given in equation 1. This is still an open problem. +Reinforcement Learning: If we take a closer at the literature, many +of the success stories of Reinforcement Learning are for Games (e.g., Atari, +Chess, Shogi, Go, and StarCraft II) or in simulated environments (e.g., Ope- +nAI Gym, MuJoCo, etc.). For such cases, we have well-defined reward func- +tions and are able to interact with the environment as many times as we +like to collect enough experiences (i.e., data). This is not the case in the +real world due the well-known physical constraints. What makes Reinforce- +ment Learning difficult is 1) the need to collect plenty of experiences (i.e., +inefficient use of data), 2) lack of well-defined rewards signals to not only +guide the training process but also the iteration process (see Fig. 1) and 3) +the sheer number hyper-parameters (i.e., degrees of freedom). We will there- +fore approach Reinforcement Learning from the perspective of the iteration +process (see Fig. 1). A central question here whose answer can address (at +least partially) all three of the aforementioned challenges is how to properly +balance the trade-off between exploration and exploitation in the absence +of well-denied reward functions. Here, we will investigate the possibility of +using human-in-the-loop reward signals [95] (see also ChatGPT); A human +can easily take a look at the performance of a robot in the real world and give +it feedback. It is worth noting that this is different from imitation learning. +To deal with the data-inefficiency issue, we can reformulate Reinforcement +Learning as a multi-task learning problem; One task is to make the human +evaluator happy and the other is to encourage exploration. Balancing the +weights given to each objective is an open problem and has a direct impact +on becoming more data-efficient. +Physics-Informed Learning: So far, we have been working on the +brain of our artificial intelligent agents. If we take this brain, mount it on +a robot (e.g., a drone) and ask it to operate in the real physical world (e.g., +in a fluid), it will most definitely fail. This is because it has never learned +to respect the laws of physics (e.g., conservation of mass, momentum and +energy, gravity, etc.). If anything, it has learned to find loopholes of the +simulated environment and bypass such laws (e.g., go faster than the speed +of light), simply because it is trained only to maximize a reward signal or fit +20 + +the corresponding data. This motivated research in Physics-Informed Neural +Networks (PINNs) [17]. The field has been growing at an exponential rate +ever since its advent in 2019. PINNs can be used to solve a wide range of +problems involving (partial) differential equations, namely forward, inverse, +model discovery, surrogate modeling and uncertainty quantification. How- +ever, PINNs have an Achilles heel. Namely, how to balance the trade-off +between fitting the data and respecting the laws of physics in the absence of +validation data? We are therefore dealing with a multi-objective optimiza- +tion problem. Here, ideas such as the ones proposed in [50] for multi-task +learning using uncertainty to weigh different objectives could be extended +to solve our multi-objective optimization problem. Here, another challenge +that we need to overcome is avoiding trivial solutions (e.g., any constant +function is a solution to the Navier Stokes equations). A feasible strategy is +to explicitly include a term in our training objective function that encourages +non-trivial solutions. +4. Concluding Remarks +Artificial intelligence (AI) evangelizes the idea of automation. On the sur- +face, AI algorithms take the data, develop their own understanding of it, and +generate valuable insights and predictions – all without human intervention. +In truth, AI involves an enormous amount of repetitive manual operations, +all hidden behind the scenes. This is what we call the “iteration process”. +Among many other degrees of freedom, this process entails model engineer- +ing (e.g., neural network architecture design) and management, experiment +tracking, dataset versioning and augmentation. The iteration process is typ- +ically carried out by data engineers, data scientists, machine learning engi- +neers, and other highly-trained (and highly-paid) specialists. However, at +least part of their work can be streamlined by AutoML. In recent years, Au- +toML has demonstrated some promise in solving simple supervised learning +problems, in particular (image) classification. However, this does not mean +that AutoML will be successful in the face of more complex problems beyond +(image) classification. It remains to be seen and tested in practice. +21 + +References +[1] D. P. Kingma and J. Ba, “Adam: A method for stochastic optimiza- +tion,” arXiv preprint arXiv:1412.6980, 2014. +[2] E. Strubell, A. Ganesh, and A. 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Bae, “Hifi-gan: Generative adversarial net- +works for efficient and high fidelity speech synthesis,” Advances in Neu- +ral Information Processing Systems, vol. 33, pp. 17 022–17 033, 2020. +52 + diff --git a/dNFIT4oBgHgl3EQfnyuw/content/tmp_files/load_file.txt b/dNFIT4oBgHgl3EQfnyuw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..874b6524102ec78706fd4ff759d25ad1140286dd --- /dev/null +++ b/dNFIT4oBgHgl3EQfnyuw/content/tmp_files/load_file.txt @@ -0,0 +1,2928 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf,len=2927 +page_content='Open Problems in Applied Deep Learning Maziar Raissi Department of Applied Mathematics, University of Colorado Boulder, Boulder, Colorado, 80309, USA Abstract This work formulates the machine learning mechanism as a bi-level opti- mization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' The inner level optimization loop entails minimizing a properly chosen loss function evaluated on the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' This is nothing but the well-studied training process in pursuit of optimal model parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' The outer level optimization loop is less well-studied and involves max- imizing a properly chosen performance metric evaluated on the validation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' This is what we call the “iteration process”, pursuing optimal model hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Among many other degrees of freedom, this process en- tails model engineering (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', neural network architecture design) and man- agement, experiment tracking, dataset versioning and augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' The iteration process could be automated via Automatic Machine Learning (Au- toML) or left to the intuitions of machine learning students, engineers, and researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Regardless of the route we take, there is a need to reduce the computational cost of the iteration step and as a direct consequence reduce the carbon footprint of developing artificial intelligence algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Despite the clean and unified mathematical formulation of the iteration step as a bi-level optimization problem, its solutions are case specific and complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' This work will consider such cases while increasing the level of complexity from supervised learning to semi-supervised, self-supervised, unsupervised, few-shot, federated, reinforcement, and physics-informed learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' As a con- sequence of this exercise, this proposal surfaces a plethora of open problems in the field, many of which can be addressed in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Keywords: supervised learning, semi-supervised learning, self-supervised learning, unsupervised learning, few-shot learning, federated learning, reinforcement learning, and physics-informed learning Preprint submitted to Journal Name January 27, 2023 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='11316v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='LG] 26 Jan 2023 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Introduction The general mechanism for building machine learning solutions is illus- trated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1 and outlined in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1) Everything starts with data as the “source code” for machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 2) We would then write a model to fit the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 3) The model is then trained to maximize the likelihood of the training data or minimize a distance/divergence between the training data distribution and model predictions in the case of generative adversarial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' The training process typically entails multiple steps of stochastic gradient descents (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', Adam optimizer [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 4) The model is then evaluated using a properly chosen performance metric (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', accuracy, mean average precision, inception score, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=') on the validation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 5) Next is the iter- ation step where the aforementioned steps 1-4 need to be repeated tens of thousands of times to find the most performant solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' This step entails model engineering (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', neural network architecture design) and manage- ment, experiment tracking, dataset versioning/augmentation, in addition to seemingly “minor” details such as choosing learning rates and learning rate schedules, batch sizes, weight decay and batch/weight/layer/group/spectral normalizations, just to name a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' The iteration step is a crucial piece of the machine learning pipeline and is usually the most time and resource consuming step while often being overlooked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' This is what makes machine learning difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 6) Before putting the model into production, we test it one last time on some test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 7) The final stage is serving the model in production to millions of customers/users while constantly monitoring its performance and re-training it if needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Step 5 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', the iteration step) is the topic of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' The jury is still out whether the iteration step should be automated (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', AutoML) or left to the intuitions of machine learning students, engineers, and researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Regardless of the route we take, there is a need to reduce the computational cost of the iteration step and as a direct consequence reduce the carbon foot- print of developing artificial intelligence algorithms [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' The computational bottleneck of the iteration step is the training of each model to convergence, only to measure its performance on the validation data which is then used as the feedback signal to guide the iteration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 2 Figure 1: The general mechanism for building machine learning solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Background In mathematical notations, we are trying to solve bi-level optimization problems [3, 4] of the following form;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' min α Mval(w∗(α), α) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' w∗(α) = arg min w Ltrain(w, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' (1) Here, α denotes the “hyper-parameters” of the model in an abstract sense, en- capsulating concepts as generic as learning rate (schedule), depth and width of neural networks, discrete choices between CNNs, RNNs, or Transformers [5] and their variants [6], presence or absence of different types of normal- ization layers [7], to pad or not to pad, convolutional kernel sizes, to use dropout or not, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Moreover, Mval denotes a performance metric such as accuracy, mean average precision [8], Fr´echet Inception Distance [9], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' The loss function is denoted by Ltrain while the model parameters are represented by w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Whereas Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1 explains the iteration problem (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1) in a clean and unified mathematical formulation, the solution to this problem is com- plex and case specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Training 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Data 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Model 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Iteration 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Testing 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Production 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' ValidationFor some machine learning problems, the evaluation metric Mval is a dis- crete and non-differentiable function of α (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', accuracy [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' For some other problems, it isn’t well-defined (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', domain adaptation [11], semi-supervised [12], self-supervised and unsupervised learning [13, 14] as well as generative models [15]) or is even non-existent (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', physics-informed deep learning [16–19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' The hyper-parameters α could be continuous (learning rate, mo- mentum, weight-decay, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=') or discrete variables (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', the choice between a regular convolution or a depth-wise-separable one [20, 21], number of layer- s/channels, to use batch-norm [22] or not, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Sometimes α is a function of the current training epoch (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', learning rate schedule, progressive grow- ing of GANs [23], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' More often than not, the loss function is a weighted combination of multiple loss functions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', multi-task learning [24], physics- informed machine learning [25], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Those weights could be part of α and require special treatment to balance the trade-off between focusing on one objective function versus another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' The loss function Ltrain could itself be dis- crete and non-differentiable (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', cost/reward in reinforcement learning [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Therefore, the solutions to the problem specified in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1 and illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1 are not as clean as its formulation and end up being case specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' The first solution that comes to mind is to rely on the intuition of students, researchers, and engineers obtained through tens of thousands of collective trial and errors and knowledge sharing over the Internet (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', stackover- flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='com), blog posts, and social media platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' We can also build tools to facilitate such experimentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' A good example is “Weights & Biases”, an MLOps platform to help researchers build better models faster with experi- ment tracking, dataset versioning, and model management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' MLOps is an ac- tive area of research both in academia and industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' In the cases where Mval is a well-defined function, let it be discrete or non-differentiable, one could use grid search (albeit for low dimensional α), random search [27], Bayesian optimization [28], reinforcement learning [29–31] or evolutionary algorithms [32, 33] to find α∗, even if α is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' The shortcoming of such approaches for solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1 is their extensive computational cost (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', 32,400-43,200 GPU-Hours, the equivalent of using 450 GPUs for 3-4 days [34, 35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Such approaches have therefore a high carbon footprint [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' The bottleneck is solving the inner optimization problem in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', the training loop) to completion, only to measure the model’s performance on the validation data which is then used as the feedback signal to guide the iteration process (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', the outer optimization problem in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' One solution to this problem is to trade off computation for more memory consumption using parameter 4 sharing ideas [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' This could result in up to 1000x faster solutions to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', 16 GPU-Hours) in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' The idea is to warm-start the training process using some shared parameters cached in memory, rather than start- ing from random parameters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', cold-starting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' This could speed up the convergence of the inner optimization problem in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Alternatively, one could approximate the validation function Mval with a differentiable func- tion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', the loss function on the validation data Lval [35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Doing so will enable us to use plain-vanilla stochastic gradient descent algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', the Adam optimizer [1]) to optimize over both the hyperparameters α and parameters w;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' One could take a couple of gradient descent steps for w and one step for α in an iterative fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' There is no need to solve the in- ner optimization problem to completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' This will also result in significant speed-up by trading computation for more memory consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' As a com- promise between purely automated solutions to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', AutoML) and the expert-in-the-loop solutions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', MLOps), we could also use a combination of the two [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Moreover, as an additional layer of complication, the per- formance metric Mval(w∗(α), α) could pursue multiple competing objectives such as minimizing the error rates (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', maximizing accuracy) and reducing the computational and memory costs of the model, perhaps to make them suitable for smaller devices such as mobile phones, tablets, or IoT (Internet of Things) devices [38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Here, balancing the trade-off between optimizing one objective versus the other is an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' This will introduce ad- ditional hyper-parameters (lets call them “hyper-hyper-parameters”) for the outer optimization problem in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Last but not least, data augmentation policies [27, 31] could also be part of the search space (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' This should give us enough background on the current state-of-the-art in AutoML and help us outline the open problems that need to be addressed collectively by the machine learning community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Open Problems As mentioned above, despite the clean and unified mathematical formu- lation of the iteration step as a bi-level optimization problem (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1), its solutions are case specific and complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' In the following, we will consider such cases while increasing the level of complexity from supervised learning to semi-supervised, self-supervised, unsupervised, few-shot, feder- ated, reinforcement, and physics-informed learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' As a consequence of this 5 exercise, this work exposes a multitude of open challenges in the field, many of which can be addressed in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Supervised Learning Most of the progress made over the last few years, ever-since the advent of AutoML, falls under the umbrella of supervised learning, in particular (im- age) classification [29–39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' These works use accuracy as their performance metric or a differentiable approximation of it (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', the negative of the loss function or the log-likelihood).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' As a low hanging fruit, we could investigate if such techniques would still work in the face of other performance metrics, such as precision, recall, F1 score, calibration [40], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', for balanced and unbalanced datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Finding differentiable approximations of such metrics is particlarly interesting because we can employ similar techniques to the ones used in [35] (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', using plain-vanilla stochastic gradient descent to opti- mize over both hyper-parameters and parameters of our models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Such tech- niques are interesting not only because of their computational efficiency but also because they require very little “hyper-hyper-parameter” tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' How- ever, how to make such methods memory efficient is still an open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Moreover, methods such as Bayesian optimization, reinforcement learning, evolutionary algorithms, and even parameter-sharing for solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1 intro- duce additional hyper-parameters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', “hyper-hyper-parameter”);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' In simple terms, we would like to avoid doing AutoAutoML, AutoAutoAutoML, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', and fragmenting our datasets beyond training, validation, and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' We could also study the effect of such approximations to different performance metrics and provide theoretical upper-bounds on the loss of performance as a result of such approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Large Networks: Recently, we are witnessing a trend in computer vision trying to replace convolutions neural networks with transformers [6, 41, 42] or even multi-layer perceptrons [43], inspired by their success in language [44–46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' The question is if the currently available techniques (see the back- ground section) for solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1 could generalize to such architectures and improve their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' It is worth noting that the techniques outlined in the background section are primarily designed for convolutions neural net- works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Furthermore, when it comes to data-augmentation strategies, the current techniques leverage only a single image [31], the question is if novel data-augmentation strategies such as mix-up [47] and cut-mix [48] (leverag- ing pairs of images) can be discovered as part of solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Answering 6 these two questions would entail rethinking the design space (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', the space in which α is assumed to live).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Another interesting and fundamental question is investigating the possibility of automating the discovery of learning rate schedules such as the cosine learning rate schedule [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Here, the learning rate is a function of the current epoch rather than being a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' This will significantly increase the complexity of the iteration problem (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Methods similar to the ones presented in [35] seem to have a good chance at solving this problem because they don’t rely on solving the inner optimiza- tion problem (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', training) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1 to completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' This will allow us to modify the learning rate in tandem with the training process per each epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Small Networks: There are times when not only we are looking for the most performant model but also we want the model to be as memory and compute efficient as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' This is an important stepping stone towards democratizing artificial intelligence in anticipation of the future of Internet of Things where a lot of our devices (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', cellphones, cars, security cameras, refrigerators, air conditioners, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=') will be intelligent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Such devices usually have smaller compute capabilities and memory capacity than our comput- ers in data-centers or on the cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' To make them intelligent we need to take their constraints into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Mathematically speaking, Mval is a weighted combination of at least two objectives;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' One is the performance metric, while the other is about making the model more nimble and could take different forms such at FLOPs, MAC (memory access cost), number of parameters, latency and memory consumption of the target devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' This last item necessitates a hardware-in-the-loop approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' The weights given to each objective function are what we called “hyper-hyper-parameters” earlier in this document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' It is still an open question how to set such weights in or- der to balance the tradeoff between optimizing one objective versus another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' We are therefore dealing with a multi-objective bi-level optimization prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Here, ideas such as the ones proposed in [50] for multi-task learning using uncertainty to weigh different objectives could be extended to solve our multi-objective bi-level optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' A similar multi-objective optimization problem arises in physics-informed deep learning [17–19] where we need to balance the trade-off between fitting the data and respecting the law of physics modeled using ordinary and partial differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Al- ternatively, we could investigate the possibility of automating the process of making pre-trained models smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' This is an after-the-fact approach where we would like to automate the discovery of methods such as knowledge distil- 7 lation [51], model pruning and compression [52–54], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' This approach will avoid the aforementioned multi-objective bi-level optimization problem and will instead break the problem into two or more stages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' In the first stage we will be looking for the most performant model, regardless of its cost, while in the second stage we will look for the best model compression strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' This will lead to a multi-stage (versus multi-objective) bi-level optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Robustness: It is a well-known fact that deep neural networks are vul- nerable to adversarial and backdoor attacks [55–59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' We could be investi- gating how much of this vulnerability can be attributed to the iteration step (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1) and whether automating the process can alleviate or aggravate the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Explainable AI: In the past few years, the machine learning community has made a lot of progress in the emerging field of explainable and trustwor- thy AI (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', [40, 60–72] and the references therein) to the extend that deep neural networks are no longer considered black boxes but rather gray ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' However, there has been very little effort in the literature (if any) to explain the choices made as part of the iteration process (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' The question is what features of the data, or rather the meta-data, explain the choices we make for learning rate schedules, architecture designs, data- augmentation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' What is the effect of noise in the data on such choices?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' How important are the size and intrinsic dimensionality of the data?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' This a place where the underlying structure of the data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', the lower dimensional manifold on which the data lives) could help us shed some light on these fundamental questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' If successful, efforts in this direction could lead to a new field, namely “Explainable AutoML”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Transfer Learning: Moreover, in a parallel thrust, we could be studying the impact of the iteration process (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1) on the generalizability and transferability of the learned features to downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' The field of deep learning is largely driven by the ideas of Transfer learning to the extent that we rarely train our models from scratch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Along the same lines, an impor- tant question worthy of systematic studies is transferability (or lack thereof) of data augmentation policies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', AutoAugment and RandAugment) from one dataset (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', ImageNet) to another (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', Pascal VOC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 8 Semantic Segmentation: When it comes to the task of semantic seg- mentation, we typically start with neural networks pre-trained on a related classification task and do transfer learning [73–81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' This is mainly because we usually have access to smaller datasets for this task as labeling every single pixel in an image is more cumbersome than labeling an entire image with a single label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Referring back to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1, there is very little work on an iteration phase dedicated to the semantic segmentation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Here, the performance metrics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', pixel accuracy, mean accuracy, mean IU, frequency weighted IU, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=') are more complex than the accuracy metric often used for the clas- sification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Finding differentiable approximations of such metrics so that we can employ similar techniques to the ones used in [35] is particularly in- teresting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Moreover, when it comes to the task of semantic segmentation, we need to not only capture the global information in an image (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', resolving the “what” of the image) but also the local information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', resolving the “where” of the objects in the image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' It would be interesting to investigate how the iteration stage (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1) would have an impact on the “what” and “where” components of the semantic segmentation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' In particular, would the iteration phase leverage tools such as atrous convolutions, short- cut connections, conditional random fields, multi-scale aggregation, deep su- pervision, deconvolutions, upsampling, attention mechanisms, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' If so, to what extent?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Super-Resolution, Denoising, and Colorization: When it comes to creative tasks such as super-resolution, denoising, colorization, and style transfer where the output of a neural network is an image, it is very hard to judge the quality of the generated images in a quantitative fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Avail- able perormance metrics such as Peak Signal-to-Noise (PSNR), Structural Similarity (SSIM), and Feature Similarity (FSIM) fall short of doing justice to the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' It is therefore very hard to measure progress in these fields [82–94] and more importantly guide the iteration phase (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' In this regard, deep features extracted from deep neural networks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', VGG) trained on the ImageNet classification task show unreasonable empirical ef- fectiveness as perceptual metrics [93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' However, there aren’t many works (if any) that guide the iteration phase using such deep features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', Learned Perceptual Image Patch Similarity (LPIPS) metric) in an automated fash- ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' In a related note, the tasks of super-resolution, denoising, colorization, and style transfer usually entail balancing the trade-off between multiple loss functions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', reconstruction L1/L2 loss versus perceptual loss [83]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' It is 9 still an open problem how to strike such a balance in the absence of univer- sally accepted performance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Here, ideas such as the ones proposed in [50] for multi-task learning using uncertainty to weigh different objectives could be extended to solve this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Alternatively, we could investigate the possibility of guiding the iteration process (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1) using human-in- the-loop Reinforcement Learning algorithms where the reward signal comes from the judgment of human beings;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' A human can easily take a look at an image and associate a quality score to it, perhaps from 1-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' This is feasible because with Reinforcement Learning we don’t need to differentiate through the reward signals or the thought process of the human evaluator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Human- in-the-loop techniques are gaining traction these days (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', [95]) because writing well-defined reward functions is very challenging if not impossible for many real-word applications of Reinforcement Learning beyond games and simulated environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Pose Estimation: A keyboard and a mouse are not the only means of interacting with a computer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' A key topic in the field of human-computer interaction in particular and the meta verse in general is human pose estima- tion [96–100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Two evaluation metrics that could guide the iteration phase (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1) are Percentage of Correct Parts (PCP) and Percent of Detected Joints (PDJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' PCP measures detection rate of limbs, where a limb is consid- ered detected if the distance between the two predicted joint locations and the true limb joint locations is at most half of the limb length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' As for PDJ, a joint is considered detected if the distance between the predicted vector and the true joint is within a certain fraction of the torso (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', left shoulder and right hip) diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' A closely related metric is Percentage of Correct Key- points (PCK) which measures the percentage of detections that fall within a normalized distance of the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' It would be interesting to inves- tigate how the iteration stage (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1) would leverage these metrics or their differentiable approximations to come up with novel architectures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', stacked hourglass blocks, cascaded pyramid layers, part affinity fields, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=') in an automated fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' This is not a well-studied topic as of this writing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Optical Flow and Depth Estimation: The world around us is 3D and evolving in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' On the one hand, depth estimation [101–105] allows us to add a third dimension to our 2D images and has applications for self- driving cars and robots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' On the other hand, optical flows [106–108] enable us to capture the evolution in time and the relationship between consecu- 10 tive frames in a video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' In addition to applications for self-driving vehicles and robotics, optical flows can be used as additional features for the task of action recognition in videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' For optical flows, End Point Error (EPE) is typically used as the performance metric guiding the iteration phase (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' It is the Euclidean distance between the predicted flow vector and the ground truth, averaged over all pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Here, the training data is usually simulated because it is very hard to measure optical flows in the physical world [106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Inevitably, we have to rely on domain adaptation techniques to close the reality gap (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', the gap between the real and simulated data distributions) to the extent possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' It is therefore an intriguing research question to study the effect of domain adaptation techniques on the iteration phase (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1) and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' The task of depth estimation also suf- fers from lack of enough labeled training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Fortunately, there are ways to leverage the underlying physics of the problem to perform unsupervised monocular depth estimation with e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', left-right consistency [103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Here, the training loss would involve a weighted combination of multiple individual loss functions [103, 104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Properly setting those weights in an automated fashion is an open question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Here, we could utilize performance metrics such as absolute relative distance, squared relative distance, or root mean square error (RMSE) to guide the iteration stage (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Object Detection: Generally speaking, there are two major types of object detectors: multi-stage (typically two) and one-stage detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' With multi-stage detectors [8, 109–117], the primary objective is to find the most performant model (measured using mean average precision) while efficiency (measured using frames per second) is a secondary objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' With single- stage detectors [118–131], the primary objective is to find the most agile model (measured using frames per second) while performance (measured us- ing mean average precision) is a secondary objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Referring back to equa- tion 1, Mval is pursuing multiple (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', at least two) competing objectives for the task of object detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Balancing the trade-off between optimizing one objective versus the other is an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' It is worth mentioning that this problem is about what we called “hyper-hyper-parameters” in the itera- tion phase and that we would like to avoid fragmenting our datasets beyond training, validation and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' As for hyper-parameters α, the design space (the space in which α lives) is a much more complex one for object detection compared to classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' The input data could be in the form of im- ages, image patches, image pyramids, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' The backbone could be in the form 11 of VGG, ResNet, ResNeXt, Darknet, Hourglass Network, Transformers, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' The neck of the architecture could be in the form of FPN (Feature Pyramid Network), PANet, Bi-FPN (Bi-directional FPN), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' The head of the object detection system could be a dense predictor (RPN, YOLO, SSD, RetinaNet, FCOS) or a sparse one (Faster R-CNN, R-FCN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' As for data augmentation we could use CutMix, MixUp, Mosaic, Bluring, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' For the loss functions, we could use L1, L2, Smooth L1, or CIoU for the regression component of the total loss function and MSE, binary or multi-class cross-entropy loss for the classification portion of the total loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Therefore, exploring the design space in a systematic and automatic fashion as part of the iteration phase (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1) is a challenging task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Last but not least, all metrics are wrong, some are useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' This is specially true when it comes to the object detection task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' There are heated debates in the literature about the appropriateness of mean average precision (mAP) or its COCO style variants as valid perfor- mance metrics for the object detection task (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', [124]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Face Recognition and Detection: Face detection is a special case of the object detection and key point (pose) estimation topics that we covered earlier in this document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Futhermore, face recognition (verification and iden- tification) can be viewed as a close-set or an open-set problem, depending on the type of available data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Close-set face recognition is nothing but a classification task that we covered earlier in this document.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Open-set face recognition on the other hand is about metric learning, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', learning features that are capable of pulling similar images together while pushing dissimilar images apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' The literature on open-set face recognition spends a lot of time designing new loss functions such as the triplet loss [132, 133], the center loss [134], angular softmax loss [135], additive angular margin loss [136], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' It is therefore a natural question to ask if it is possible to automate the search for appropriate loss functions as part of the iteration phase (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' To guide the iteration process we could leverage the ROC (Receiver Operator Characteristic) curves relating the true positive rate to the false positive rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Video & 3D data: As mentioned earlier in this document, the world around us is evolving in time and is 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' When it comes to videos, we could think of at least two important applications, namely action recognition and object tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Action recognition [137–146] is a classification task albeit on a sequence of image frames in a video as the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Here the design space (the space in which α lives, referring to equation 1) is more complex 12 compared to the design space for images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' It would therefore be interesting to see if ideas such as early/late/slow fusion, multi-streaming, using optical flows as additional input features, 3D convolutions, trajectory pooling, or slow-fast networks would survive the iteration phase (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Furthermore, when it comes to object tracking [147, 148], there is very little work on automating the iteration phase (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1) and studying its impact on the resulting algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Here, we could use evaluation metrics such as the center location error and the bounding box overlap ratio to guide the iteration phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' In a parallel thrust, performing object recognition, detection and segmentation on 3D point cloud data is more challenging than doing so on images [149– 153].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' These types of data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', LIDAR data) appear naturally in self-driving vehicles and robotics applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' However, there is very little work on automating the iteration phase (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1) for performing object recognition, detection and segmentation on 3D point cloud data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Beyond Supervised Learning It is now time to increase the level of complexity and move beyond super- vised learning towards semi-supervised, self-supervised, unsupervised, few- shot, federated, reinforcement, and physics-informed learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' We will be approaching these topics from the perspective of the iteration phase (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Natural Language Processing: For applications such as word vector representations [154–158], text classification and sequence tagging [159–172], translation [173–186] and language modeling [187–210], unlabeled text data is available en masse thanks to the Internet – for example, the Common Crawl project produces about 20TB of text data extracted from web pages each month.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' This makes pre-training large language models on such data partic- ularly attractive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Such models can then be fine-tuned on downstream tasks (BERT family of models) or used in a few-shot setting (GPT type of models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Here, the fundamental question is what performance metric(s) should we use to guide the iteration phase (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1) when it comes to training large lan- guage models on unlabeled data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' One idea is to use perplexity as a measure of the goodness of language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' However, we don’t typically train large language models for the sake of modeling the language but rather to use them in some downstream applications such as text classification (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', sentiment analysis), sequence tagging (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', named entity recognition), machine trans- lation, program synthesis, question answering, summarization, semantic tex- 13 tual similarity, language comprehension, conversational response generation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' For instance, it is a well-known observation that the common practice of extracting sentence embeddings from the BERT language model, by average pooling the last layer output vectors or using the output of the first token (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', the [CLS] token), yields rather bad sentence embeddings [204], often worse than averaging the GloVe vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' This is why researchers came up with the ideas of Sentence-BERT and Siamese BERT-Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Another observation is that BERT in its original form cannot perform translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' This is why researchers introduced BART, GPT, T5, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' However, each one of these contributions focus on a hand-full of downstream tasks to evaluate the performance of their language modeling capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Some focus on the GLUE benchmark [211] of a suit of downstream tasks, some focus on the BLUE score [212] for translation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Perhaps a better strategy to guide the iteration process (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1) is to approach language modeling from a multi-task learning perspective where Mval in equation 1 is a weighted com- bination of the performance metrics for a multitude of downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Here, an open question is how to properly weigh one objective function ver- sus the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Here, ideas such as the ones proposed in [50] for multi-task learning using uncertainty to weigh different objectives could be extended to solve our multi-objective bi-level optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' In addition, we could include extra objectives in Mval to penalize the computational com- plexity and memory consumption of the resulting language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' The idea is to come up with the smallest language model that is good at solving a multitude of downstream tasks in an automated fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Of particular in- terest are techniques similar to the ones used in [35] (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', using plain-vanilla stochastic gradient descent to optimize over both hyper-parameters and pa- rameters of our models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Last but not least, many of the ideas in natural language processing can be extend to graphs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', social networks) [213–222].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Multimodal Learning: With multimodal learning [223–234], we are taking baby steps towards human level artificial intelligence (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', artificial general intelligence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' For instance, if we look at an intelligent robot and say “pick that up and put it on the table” while pointing at a box sitting on the ground, to be able to execute the command correctly, the robot should not only process speech and language but also should be able to use its vi- sion system to understand what we mean when we say “that”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Moreover, a common criticism to large language models is that even if they manage to generate seemingly cohesive text, they have very little idea about what they 14 are actually talking about;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' For example, a language model trained only on textual data has never seen images of airplanes, cars and ships, it has only read about them on the internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' The field of multimodal learning is therefore attracting the attention of a lot of great researchers both in academia and the industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Two important applications are translating images (or videos) to text (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', image and video captioning) and vice versa (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', text to image synthesis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Another equally important application is visual question answer- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Moreover, training large language models both on images and textual data is also showing some great promise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' We will be approaching these top- ics from the perspective of the iteration phase (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' When it comes to translating images to text, we could use the BLUE score to guide the iteration phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' However, there are heated debates in the literature if the BLUE score is the best performance metric both for image captioning and translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Coming up with better metrics is an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Moreover, there is very little work (if any) on automating the iteration phase of trans- lating images to texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' For the text to image synthesis type of tasks, like any other creative task (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', super-resolution, denoising, colorization, style transfer and generative adversarial networks), it is very hard to judge the quality of the generated images in a quantitative fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Here, we could use metrics such the Inception Score, the Fr´echet Inception Distance or the Learned Perceptual Image Patch Similarity (LPIPS) metric to guide the it- eration phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' As of this writing, there aren’t many works (if any) that have done so before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Furthermore, when it comes to training large language mod- els on unlabeled textual data as well as images, the fundamental question is what performance metric(s) should we use to guide the iteration phase (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' In particular, the question is how existing performance metrics such as perplexity can be generalized to handle both text and image type of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Alternatively, similar to language models trained only on text, we could define a set of downstream tasks involving both text and images as benchmarks to guide the iteration phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' The idea is then to approach mul- timodal modeling from a multi-task learning perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Generative Networks: Generative models are either trained to max- imize the likelihood (or rather its lower bound) of the training data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', Variational Auto-Encoders) or minimize a distance/divergence between the training data distribution and model predictions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', unconditional and con- ditional Generative Adversarial Networks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' A central question here is how to measure the quality of the generated data in a quantitative fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' For 15 images, we could use the Inception score or the Fr´echet Inception Distance to guide the iteration process, despite all their imperfections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' However, it is not clear what performance metrics we should use for other types of data such as text, speech, graphs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', social networks), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Furthermore, it is a well-known observation that training generative adversarial networks is an unstable process and most of the contributions in this field are made towards stabilizing this process by using different loss functions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', Feature Match- ing, Least Squares GANs, Wasserstein GANs, Hinge Loss, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' ), normaliza- tion and regularization schemes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', Gradient Penalty, Spectral Normal- ization, Orthogonal Regularization, Adaptive Instance Normalization, Path Length Regularization, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' ), architectures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', DCGANs, Self-Attention GANs, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=') and training schedules (Progressive Growing, two time-scale update rule, historical averaging of parameters, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' It would be interesting to study if such techniques would survive an automated iteration process (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Of particular interest is the progressive growing idea because the neural network architecture itself is a function of the current training epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' This will significantly increase the complexity of the iteration problem (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Methods similar to the ones presented in [35] seem to have a good chance at solving this problem because they don’t rely on solving the inner optimization problem (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', training) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1 to completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' This will al- low us to modify the architecture in tandem with the training process per each epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Last but not least, when it comes to conditional GANs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', image-to-image translation), we typically try to minimize a total loss func- tion being a weighted combination of multiple individual loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' It is still an open problem how to come up with those weights in the absence of validation data and appropriate performance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Perhaps reformulating the problems as a multi-task learning problem and using ideas similar to the ones proposed in [50], that leverage uncertainty to weigh different objectives, could help us address this problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Domain Adaptation: The concept of domain adaptation [11, 235–238] is related to scenarios where we have a lot of labeled data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', simulated data) from a source domain and zero (or very few) labeled data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', real data) from a target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Such scenarios happen frequently in many engi- neering fields (sometimes called multi-fidelity modeling [239–241] in fluid and solid mechanics) including but not limited to self-driving cars and robotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Domain adaptation can help us close the so called reality gap between the simulated and real data distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' We are going to approach domain 16 adaptation from the perspective of the iteration phase (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' With domain adaptation we would like to minimize the risk of making errors on the target data, not necessarily the source data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Now, the question is how can we measure the performance of our models on the target data in the absence of any target labels (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', unsupervised domain adaptation) or in the presence of very few of them (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', weakly-supervised domain adaptation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' One idea is to use unsupervised hyper-parameter selection techniques [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' We can split the source domain labeled data into a training set Strain and a validation set Sval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Similarly, we can split the target domain unlabeled data into a training set Ttrain and a validation set Tval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' We can then use Strain and Ttrain to learn a model using domain adaptation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' The trained model can now be used to generated labels for the unlabeled data Ttrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' We then remove the labels from Strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' We will then train a reverse model using Ttrain as the source domain and Strain as the target domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' The reverse model can now be evaluated on the validation set Sval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' We could use this reserve validation risk as a proxy for the true validation risk to guide the iteration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Few-shot Learning: When it comes to using a machine learning model to serve millions of users perhaps over the internet, not only we need to take care of the distributional shift between the training and test data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', do- main adaptation) but also we need to be able to handle new use cases and more importantly new labels associated to such use cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' As an example, in the context of a recommendation system [242–251], we can think of new items (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', movies, products, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=') to be recommended to new/existing users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' This is related to the topic of few-shot learning [252–254] where our models need to be able to handle new labels given very few observations per each label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' For classification tasks, we can use N-way K-shot classification accuracy to guide the iteration phase (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' It would therefore be interesting to study the effect of an automated iteration stage on the resulting algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' More importantly, it is still unclear what performance metrics we should use for other applications of few-shot learning beyond classification such as se- mantic segmentation, object detection, pose estimation, depth estimation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Federated Learning: In addition to the publicly available data on the internet, there is a wealth of data sitting on our privately-held devices (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', cellphones, tablets, laptops, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Our devices are also getting more power- ful in their compute and data collection capabilities (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', multiple camera 17 lenses on the back of our cellphones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Federated Learning tries to leverage such privately held data to train machine learning models while preserving the privacy of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' The idea is to bring the models to the data (rather than bringing the data to the models on the cloud) and use the heteroge- neous compute capabilities of user devices to train our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' What is being communicated to the cloud is the parameters of our models or rather their gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' The field is still in its infancy and there are many open technical and non-technical challenges (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', communication efficiency, the non-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='d na- ture of the data, data pre-processing, training self-supervised models, privacy preserving, being robust to backdoor attacks, being able to train models on smaller devices, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=') to be addressed before we can fully realize the potential of federated learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' In this work, we are approaching federated learning from the iteration perspective (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Given the distributed nature of the data over millions of user devices, the question is how can we evaluate the performance of our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' One idea is to have the users of our models give star ratings (perhaps out of five) to our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' We can then aggregate these stars as a feedback signal to guide the iteration phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Given that such a performance metric is discrete and non-differentiable we can use methods based on Reinforcement Learning to perform hyper-parameter selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Semi-Supervised & Self-Supervised Learning: Let us now move to- wards the cases where we have access to a lot of unlabeled data and very few labeled data, if any.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' We faced a similar situation in natural language pro- cessing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' However, with language we are working with discrete tokens which makes it easier to perform self-supervision by defining next token predic- tion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', GPT style models) or masked token prediction tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', BERT style models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Discrete tokens make it possible to use the softmax function as the last layer of a neural network (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', a Transformer architecture) and turn the self-supervision problem into a classification one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' However, image and speech type of data are continuous signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Normal, Laplace, or even mixture of Gaussians for modeling the distribution of continuous random variables are not as flexible as the softmax function is in modeling the dis- tribution of discrete random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' It is therefore required to rethink the semi-supervised and self-supervised learning paradigms when it comes to continuous signals (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', images and speech).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' In fact, this field has been growing at an exponential rate over the past two or three years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' One com- mon theme emerging in the literature is to take a single image and augment it into two different views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' These two views should then give consistent 18 representations once processed by the same neural network (or two similar ones).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Here, a central challenge that needs to be overcome is avoiding the trivial solutions, either implicitly or explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' A network being supervised by itself or by another similar network is prone to converging to a trivial solution (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', a constant function ignoring its inputs altogether).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' We face a similar problem with physics-informed neural networks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', any constant function is a solution to the Navier-Stokes equations) [17–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' We are going to approach this problem from the perspective of the iteration stage (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' We need to either explicitly include a term in our training objective function that encourages non-trivial solutions, or design our search space in such a way that it includes mechanisms that have shown empirical success in avoiding trivial solutions such as stop-gradients, predictor heads, model averaging, contrastive losses, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Here, balancing the trade-off between the consistency loss and avoiding the trivial solution is very delicate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Fortunately, with semi-supervised learning [12, 41, 255–258] we have some labeled data and performance metrics that we can leverage to guide the iteration phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' However, with self-supervised learning [13, 14, 259–271], neither such labeled data exists nor are there any universally accepted performance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' One idea is to define a set of downstream tasks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', object recognition, detec- tion, and segmentation for images) to judge the transferablility of the learned features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Here, how much importance we should give to each downstream task is an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Speech: Similar to the language modeling paradigm for text, there is an emerging trend over the past few years to model speech [272–276].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' The idea is that before we (as human beings) learn to read and write, we learn to listen and speak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' This self-supervised learning paradigm for speech is sometimes also called learning by listening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Speech being a continuous signal inherits many of the challenges that we went over in the previous paragraph on self-supervised learning for images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' In addition, as of this writing, there are only two well-defined downstream tasks, namely translating speech to text [277–285] and vice versa [286–289], to guide the iteration stage (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1) using their respective performance metrics (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', label error rate for speech recognition and subjective 5-scale mean opinion score in naturalness for speech synthesis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' The performance metric for speech synthesis, however, requires human in the loop evaluators and leaves Reinforcement Learning or evolutionary algorithms as the only options to guide the iteration phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Fortunately, there are some researchers in both academia and industry who 19 are trying to come up with more downstream tasks to judge the quality of speech models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Even in the presence of such downstream tasks, we will need to solve multi-objective bi-level optimisation problems of a form that gener- alizes the one given in equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' This is still an open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Reinforcement Learning: If we take a closer at the literature, many of the success stories of Reinforcement Learning are for Games (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', Atari, Chess, Shogi, Go, and StarCraft II) or in simulated environments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', Ope- nAI Gym, MuJoCo, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' For such cases, we have well-defined reward func- tions and are able to interact with the environment as many times as we like to collect enough experiences (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' This is not the case in the real world due the well-known physical constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' What makes Reinforce- ment Learning difficult is 1) the need to collect plenty of experiences (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', inefficient use of data), 2) lack of well-defined rewards signals to not only guide the training process but also the iteration process (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1) and 3) the sheer number hyper-parameters (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', degrees of freedom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' We will there- fore approach Reinforcement Learning from the perspective of the iteration process (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' A central question here whose answer can address (at least partially) all three of the aforementioned challenges is how to properly balance the trade-off between exploration and exploitation in the absence of well-denied reward functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Here, we will investigate the possibility of using human-in-the-loop reward signals [95] (see also ChatGPT);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' A human can easily take a look at the performance of a robot in the real world and give it feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' It is worth noting that this is different from imitation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' To deal with the data-inefficiency issue, we can reformulate Reinforcement Learning as a multi-task learning problem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' One task is to make the human evaluator happy and the other is to encourage exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Balancing the weights given to each objective is an open problem and has a direct impact on becoming more data-efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Physics-Informed Learning: So far, we have been working on the brain of our artificial intelligent agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' If we take this brain, mount it on a robot (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', a drone) and ask it to operate in the real physical world (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', in a fluid), it will most definitely fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' This is because it has never learned to respect the laws of physics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', conservation of mass, momentum and energy, gravity, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' If anything, it has learned to find loopholes of the simulated environment and bypass such laws (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', go faster than the speed of light), simply because it is trained only to maximize a reward signal or fit 20 the corresponding data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' This motivated research in Physics-Informed Neural Networks (PINNs) [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' The field has been growing at an exponential rate ever since its advent in 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' PINNs can be used to solve a wide range of problems involving (partial) differential equations, namely forward, inverse, model discovery, surrogate modeling and uncertainty quantification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' How- ever, PINNs have an Achilles heel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Namely, how to balance the trade-off between fitting the data and respecting the laws of physics in the absence of validation data?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' We are therefore dealing with a multi-objective optimiza- tion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Here, ideas such as the ones proposed in [50] for multi-task learning using uncertainty to weigh different objectives could be extended to solve our multi-objective optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Here, another challenge that we need to overcome is avoiding trivial solutions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', any constant function is a solution to the Navier Stokes equations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' A feasible strategy is to explicitly include a term in our training objective function that encourages non-trivial solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Concluding Remarks Artificial intelligence (AI) evangelizes the idea of automation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' On the sur- face, AI algorithms take the data, develop their own understanding of it, and generate valuable insights and predictions – all without human intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' In truth, AI involves an enormous amount of repetitive manual operations, all hidden behind the scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' This is what we call the “iteration process”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' Among many other degrees of freedom, this process entails model engineer- ing (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=', neural network architecture design) and management, experiment tracking, dataset versioning and augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' The iteration process is typ- ically carried out by data engineers, data scientists, machine learning engi- neers, and other highly-trained (and highly-paid) specialists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' However, at least part of their work can be streamlined by AutoML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' In recent years, Au- toML has demonstrated some promise in solving simple supervised learning problems, in particular (image) classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dNFIT4oBgHgl3EQfnyuw/content/2301.11316v1.pdf'} +page_content=' However, this does not mean that AutoML will be successful in the face of more 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0000000000000000000000000000000000000000..9d2d7a993afc3f4bf53ab16527e0e8831cfd677e --- /dev/null +++ b/eNE0T4oBgHgl3EQfogEC/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:befb05e117d5bc216e44e031a69ef0afe6547482b0703dccf1249e8349f209ef +size 119838 diff --git a/eNFKT4oBgHgl3EQfAC2F/content/tmp_files/2301.11697v1.pdf.txt b/eNFKT4oBgHgl3EQfAC2F/content/tmp_files/2301.11697v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..10eae8911b2766bf17663a824acb88d7726f1467 --- /dev/null +++ b/eNFKT4oBgHgl3EQfAC2F/content/tmp_files/2301.11697v1.pdf.txt @@ -0,0 +1,2386 @@ +BIG PORTFOLIO SELECTION BY GRAPH-BASED CONDITIONAL +MOMENTS METHOD +Zhoufan Zhu 1, Ningning Zhang 2 and Ke Zhu∗ 2 +1Shanghai University of Finance and Economics +and +2University of Hong Kong +How to do big portfolio selection is very important but challenging for both researchers +and practitioners. In this paper, we propose a new graph-based conditional moments (GRACE) +method to do portfolio selection based on thousands of stocks or more. The GRACE method +first learns the conditional quantiles and mean of stock returns via a factor-augmented tem- +poral graph convolutional network, which guides the learning procedure through a factor- +hypergraph built by the set of stock-to-stock relations from the domain knowledge as well +as the set of factor-to-stock relations from the asset pricing knowledge. Next, the GRACE +method learns the conditional variance, skewness, and kurtosis of stock returns from the +learned conditional quantiles by using the quantiled conditional moment (QCM) method. The +QCM method is a supervised learning procedure to learn these conditional higher-order mo- +ments, so it largely overcomes the computational difficulty from the classical high-dimensional +GARCH-type methods. Moreover, the QCM method allows the mis-specification in modeling +conditional quantiles to some extent, due to its regression-based nature. Finally, the GRACE +method uses the learned conditional mean, variance, skewness, and kurtosis to construct +several performance measures, which are criteria to sort the stocks to proceed the portfo- +lio selection in the well-known 10-decile framework. An application to NASDAQ and NYSE +stock markets shows that the GRACE method performs much better than its competitors, +particularly when the performance measures are comprised of conditional variance, skewness, +and kurtosis. +∗Address correspondence to Ke Zhu: Department of Statistics and Actuarial Science, University of Hong Kong, Hong +Kong. E-mail: mazhuke@hku.hk +Keywords: Asset pricing knowledge; Big data; Big portfolio selection; Domain knowledge; High-dimensional time +series; Machine learning; Quantiled conditional moments. +1 +arXiv:2301.11697v1 [stat.ML] 27 Jan 2023 + +2 +1. Introduction. +In conjunction with the huge growth of stock market capitalization, the num- +ber of existing stocks in the financial market is increasing rapidly nowadays, raising a big challenge to +researchers and practitioners on how to do portfolio selection based on thousands of stocks or more. +Suppose there are N different stocks with their prices over T timepoints. Let ri,t denote the return +of individual stock i at time t with the conditional mean µi,t ≡ E (ri,t | Ft−1), where i = 1, ..., N, +t = 1, ..., T, and Ft ≡ σ (ri,s; i = 1, ..., N, s ≤ t) is the available information set up to time t. Conven- +tionally, all considered N stocks are sorted into 10 deciles according to their predicted values of µi,t +(from the smallest to the largest), and then the long-short portfolio is constructed at time t − 1 by +buying the 10% highest ranking stocks (decile 10) and selling the 10% lowest (decile 1); see, for ex- +ample, Gu, Kelly and Xiu (2020, 2021). However, this conventional way using µi,t as the performance +measure to sort the stocks has two major shortcomings: First, it ignores the impact of conditional +variance hi,t ≡ Var (ri,t | Ft−1), which is the risk of uncertainty for guiding portfolio selection under +the mean-variance criterion (Markowitz, 1952) or Sharpe ratio criterion (Sharpe, 1994); Second, it +does not accommodate the observation that rational investors prefer assets with higher skewness and +lower kurtosis in the market (Scott and Horvath, 1980; Dittmar, 2002), implying the necessity of +considering the conditional skewness si,t ≡ Skew (ri,t | Ft−1) for the asymmetry risk and conditional +kurtosis ki,t ≡ Kurt (ri,t | Ft−1) for the tail risk to proceed the portfolio selection. +To overcome the two shortcomings aforementioned, we aim to sort the stocks for portfolio selection +by using the performance measures below: +Mean–variance (MV): µi,t − λ1hi,t; +(1) +Mean–variance with skewness and kurtosis (MVSK): µi,t − λ1hi,t + λ2si,t − λ3ki,t; +(2) +Sharpe ratio (SR): µi,t/ +� +hi,t; +(3) +Sharpe ratio with skewness and kurtosis (SRSK): µi,t/ +� +hi,t + λ2si,t − λ3ki,t, +(4) +where λi, i = 1, 2, 3, are positive hyperparameters, and they determine how much penalty one needs +to pay for the large values of hi,t and ki,t or how much reward one can gain for the large values of +si,t. To implement these four performance measures, we need to learn hi,t, si,t, and ki,t dynamically +for N stocks. When N = 1, these three higher-order conditional moments are studied by using +some variants of univariate generalized autoregressive conditional heteroskedasticity (GARCH) model +(Engle, 1982; Bollerslev, 1986); see, for example, Jondeau and Rockinger (2003), Le´on, Rubio and +Serna (2005), Le´on and ˜N´ıguez (2020), and references therein. However, those univariate GARCH-type + +3 +methods have the risk of model mis-specification and the instability of model estimation particularly +when the dynamics of si,t and ki,t are considered. When N is large (say, e.g., N = 1000), no clear +feasible manner so far has been offered in the literature to estimate hi,t, si,t, and ki,t by using high- +dimensional GARCH-type models, which are formed to study the dynamics of rt ≡ (r1,t, ..., rN,t)′. +The cause of this dilemma is that the high-dimensional GARCH-type models have to assume a +certain distribution of rt to estimate hi,t, si,t, and ki,t, however, the resulting log-likelihood function +is too complex to be optimized. For example, the optimization of the commonly used Gaussian log- +likelihood function needs to invert many N × N-dimensional variance-covariance matrices, and this +task becomes computationally infeasible for large N cases. Pakel et al. (2021) propose a composite +likelihood estimation (CLE) method for the parsimonious scalar BEKK model to estimate hi,t. Since +the CLE method is based on all pairwise Gaussian log-likelihood functions with the order O(N2), it +becomes computationally burdensome when N is in thousands, and meanwhile, it may not adequately +capture the dependence among stock returns at the price of pairwise technique. One way to further +reduce the computational burden is to estimate hi,t by the equation-by-equation (EbE) method, as +done for the parsimonious scalar DCC model (Francq and Zako¨ıan, 2016; Engle, Ledoit and Wolf, +2019). The EbE method is feasible in terms of computation but inefficient in terms of prediction, +since it totally ignores the dependence among stock returns. +This paper contributes to the literature by proposing a new graph-based conditional moments +(GRACE) method for portfolio selection under four performance measures in (1)–(4). The GRACE +method has two core engines. Its first engine is to study the conditional quantiles of ri,t for i = 1, ..., N +and t = 1, ..., T by a graph-based quantile model, which can be directly estimated via the quantile +loss function (Koenker and Bassett, 1978). Our graph-based quantile model is based on a new factor- +augmented temporal graph convolutional network (FTGCN), and thus it is called the FTGCN-based +quantile model. This FTGCN-based quantile model uses the stock and factor features to extract +both temporal and spatial information for all stocks, and then takes the extracted information to +learn the conditional quantiles under the guidance of a factor-augmented hypergraph. The factor- +augmented hypergraph is neither random nor time-variant, and it combines the domain knowledge of +the multiple types of relation between any two stocks and the asset pricing knowledge of the impact +of common factors on all stocks. Our factor-augmented hypergraph has a linkage with the hypergraph +in TGCN (Feng et al., 2019) that also exploits the domain knowledge to build the graph structure +among stocks, where the domain knowledge comes from the public information of the stocks (e.g., + +4 +the industrial background, financial statement, and shareholder information), and its usefulness has +been well documented by Livingston (1977), Cohen and Frazzini (2008), Lee et al. (2019), Burt and +Hrdlicka (2021), and many others. However, the hypergraph in TGCN overlooks an important fact +from the asset pricing literature that some common factors can globally affect all stocks in the market +(Fama and French, 1993, 2015, 2018; Griffin, 2002; Hou, Karolyi and Kho, 2011). This asset pricing +knowledge is obviously as informative as the domain knowledge, and it motivates the design of our +factor-augmented hypergraph. Using the similar idea above, our GRACE method further proposes a +FTGCN-based mean model to estimate µi,t. +Based on the estimated conditional quantiles of ri,t at K different quantile levels from our FTGCN- +based quantile model, the second engine of our GRACE method is to estimate hi,t, si,t, and ki,t via +their corresponding quantiled conditional moments (QCMs) in Zhang and Zhu (2022). The QCM +method estimates hi,t, si,t, and ki,t through the ordinary least squares (OLS) estimator of a linear +regression model, which is constructed by those estimated conditional quantiles. The formulation of +this linear regression model stems naturally from the Cornish-Fisher expansion (Cornish and Fisher, +1938), which exhibits a fundamental relationship between conditional quantiles and conditional mo- +ments. In principle, the QCM method transforms the estimation of hi,t, si,t, and ki,t to that of +conditional quantiles, and this brings us two substantial advantages over the GARCH-type method. +First, the QCM method is easy-to-implement as long as the estimated conditional quantiles of ri,t are +provided. Note that our FTGCN-based quantile model can estimate conditional quantiles of ri,t for +large N and T cases by a supervised learning through the use of quantile loss function. Therefore, +unlike the estimation of high-dimensional GARCH-type models, no assumption on the distribution of +rt is needed to estimate our FTGCN-based quantile model. This is the reason why the QCM method +can make the estimation of higher-order moments feasible for large N cases, although it needs to +estimate the quantile model K different times. Second, the QCM method largely alleviates the risk of +model mis-specification, since the QCMs of hi,t, si,t, and ki,t are proposed without any estimator of +µi,t, and more importantly, they are consistent even when the conditional quantile estimators of ri,t +are biased to some extent. In this sense, our FTGCN-based quantile model could generate consistent +QCMs, as long as its specification does not largely deviate from the true specification of conditional +quantile of ri,t. +We apply our GRACE method to construct long-short portfolios based on 1026 and 1737 stocks +in NASDAQ and NYSE, respectively. To build the factor-augmented hypergraph, we use the Wiki + +5 +company-based relations (Feng et al., 2019) as the domain knowledge to specify the multiple types of +relation between any two stocks, and at the same time, we take the Fama-French five factors (Fama +and French, 2015) as the asset pricing knowledge to capture the impact of common factors on all +stocks. From an economic viewpoint, our empirical results are encouraging in four aspects. First, all of +the MV, MVSK, SR, and SRSK portfolios have larger values of out-of-sample annualized SR than the +M portfolio in the GRACE method. Second, the SRSK portfolio from the GRACE method performs +the best, and its values of out-of-sample annualized SR are 4.81 and 3.48 in NASDAQ and NYSE, re- +spectively, which are 236% and 21% higher than those of the M portfolio from the benchmark method +in Feng et al. (2019). Third, the GRACE method always dominates the simple GRACE method in +portfolio selection by a wide margin, where the simple GRACE method adopts the linear structure +(Zhu et al., 2017, 2019) to extract the information from stock and factor features to learn the condi- +tional quantiles and mean of stock returns. Fourth, regardless of performance measure, the portfolios +from the GRACE method have a more robust performance than those from its competitors over +the set of stock-to-stock relations, the choice of hyperparameters, and the level of transection cost. +All of these aforementioned findings indicate the importance of using the higher-order conditional +moments to form the performance measure, the asset pricing knowledge to build the hypergraph, +and the network structure to extract the feature information. From a statistical viewpoint, the condi- +tional moments learned from the GRACE method are largely valid and better than those from other +competing methods, shedding light on the advantage of GRACE method in portfolio selection. +The remaining paper is organized as follows. Section 2 presents our entire methodology, including +the network architecture of FTGCN, the training procedure of FTGCN-based quantile and mean +models, the formal estimation procedure of the QCMs, and the implementation details of the GRACE +method. Section 3 presents our empirical studies of big portfolio selection in NASDAQ and NYSE +stock markets. Concluding remarks are offered in Section 4. +2. Methodology. +2.1. Graph-based Learning for Conditional Quantiles. +Let Qt(τ) = (Q1,t(τ), ..., QN,t(τ))′ be the +high-dimensional vector of τ-th conditional quantiles, where Qi,t(τ) is the τ-th conditional quantile +of ri,t given Ft−1. In this paper, we study Qt(τ) by a new FTGCN-based quantile model defined as +Qt(τ) = f(Xt−1; G, θτ), +(5) + +6 +where Xt−1 ∈ R(N+B)×P×S is a feature tensor built on Ft−1 including the information of N stocks +and B factors, and f(·; G, θτ) : R(N+B)×P×S → RN×1 is the FTGCN depending on a factor- +augmented hypergraph G and a vector of unknown parameters θτ. Here, Xt−1 = [X1,t−1, ..., XN,t−1, +XN+1,t−1, ..., XN+B,t−1] with Xi,t−1 ∈ RP×S having its s-th column xi,t−1−S+s ∈ RP , where Xi,t−1 +for i = 1, ..., N is the feature matrix for stock i, XN+b,t−1 for b = 1, ..., B is the feature matrix for +factor b, P with a potential high dimension is the number of stock or factor features, S is the number +of lagged values of each feature, and xi,t (or xN+b,t) is the vector of P different features of stock i +(or factor b) at time t. Below, we show the four modules to construct the FTGCN f(·; G, θτ). +2.1.1. Module I: Feature Extraction. +In the first module, we employ a one-layer long short-term +memory (LSTM) network (Hochreiter and Schmidhuber, 1997) to extract the temporal embedding +xL +i,t ∈ Rd from the feature matrix Xi,t−1 at time t − 1. Specifically, we let xs +i,t−1 = xi,t−1−S+s and +compute the hidden state vectors hs +i,t, s = 1, ..., S, recursively from the LSTM network: +zs +i,t = tanh(W1xxs +i,t−1 + W1hhs−1 +i,t ++ b1), +is +i,t = sigmoid(W2xxs +i,t−1 + W2hhs−1 +i,t ++ b2), +f s +i,t = sigmoid(W3xxs +i,t−1 + W3hhs−1 +i,t ++ b3), +cs +i,t = f s +i,t ⊙ cs−1 +i,t ++ is +i,t ⊙ zs +i,t, +os +i,t = sigmoid(W4xxs +i,t−1 + W4hhs−1 +i,t ++ b4), +hs +i,t = os +i,t ⊙ tanh(cs +i,t), +(6) +where W1x, W2x, W3x, W4x ∈ Rd×P and W1h, W2h, W3h, W4h ∈ Rd×d are matrices of weight pa- +rameters, b1, b2, b3, b4 ∈ Rd are vectors of bias parameters, d is the number of hidden units control- +ling the network complexity, tanh(·) and sigmoid(·) are two entry-wise vector-valued functions, ⊙ is +element-wise production operation, and the initial values c0 +i,t and h0 +i,t are conventionally set to the +d-dimensional vector of zeros. Then, our temporal embedding xL +i,t is taken as the last hidden state +vector hS +i,t (the output of LSTM network), that is, +xL +i,t = hS +i,t = h(Xi,t−1; θL) ∈ Rd, +(7) +where h(·; θL) is the LSTM network in (6) indexed by θL, and θL contains all the parameters in +{Wjx, Wjh, bj : j = 1, 2, 3, 4}. Clearly, the purpose of this module is to extract an expressive vector +xL +i,t, which stores the long-term temporal information of P features up to time t − 1. It is expected + +7 +that all the temporal information carried by {xL +1,t, ..., xL +N+B,t} can help us to predict the behavior of +future returns {r1,t, ..., rN,t} at time t. +2.1.2. Module II: Hypergraph Construction. +Besides the temporal information from the stock fea- +tures, the spatial information (i.e., interdependence relations) among all stocks is also important for +predictions. For example, (i) MSFT LLC and Google LLC could have an industry-specific relation, +since both of them belong to “Computer Software: Programming” industry; and (ii) Boeing Inc. and +United Airlines Inc. could have a corporate relation, in view of the fact that Boeing Inc. produces +Boeing airplanes for United Airlines Inc. Needless to say, these stock-to-stock (S2S) relations are +informative and should not be ignored. In general, we can have M different types of S2S relation +(denoted by Estock = {e1, ..., eM}) between any two stocks based on the domain knowledge. +Along with the S2S relations, the factor-to-stock (F2S) relations also exist in the market, since the +stock returns can move together driven by the common factors; see the vast evidence in the asset +pricing literature (Fama and French, 2018; Lettau and Pelger, 2020; Gu, Kelly and Xiu, 2021). The +F2S relations convey the spatial information from factors to stocks, and they are highly possible to be +factor-specific. Therefore, based on the asset pricing knowledge, we consider B different F2S relations +(denoted by Efactor = {eM+1, ..., eM+B}), where the F2S relation eM+b is induced by factor b. +To describe all of S2S and F2S relations above, we build a factor-augmented hypergraph +G = (V , A), +(8) +where V = {Vstock, Vfactor} is the set of vertices, and A = {Astock, Afactor} is the set of adjacency +matrices. Here, Vstock = {1, ..., N} is the set of stock vertices with the vertex i ∈ Vstock representing +the stock i, Vfactor = {N + 1, ..., N + B} is the set of factor vertices with the vertex N + b ∈ Vfactor +representing the factor b, Astock = {A1, ..., AM} is the set of adjacency matrices with the matrix +Am ∈ Astock representing the S2S relation em, and Afactor = {AM+1, ..., AM+B} is the set of +adjacency matrices with the matrix AM+b ∈ Afactor representing the F2S relation eM+b, where Am +has its (i, j)-th entry +ai,j,m = +� +� +� +1, +if there is an S2S relation em between vertices i ∈ Vstock and j ∈ Vstock, +0, +otherwise, + +8 +and AM+b has its (i, j)-th entry +ai,j,M+b = +� +� +� +1, +if i ∈ Vstock and j = N + b ∈ Vfactor or j ∈ Vstock and i = N + b ∈ Vfactor, +0, +otherwise. +According to the definitions of Am and AM+b, the factor-augmented hypergraph G ensures that (i) +two stock vertices in Vstock are linked when they have up to M different S2S relations; and (ii) each +factor vertex in Vfactor is linked to all of stock vertices in Vstock indicating the corresponding F2S +relation. Since our main target is to study the dynamics of stocks rather than factors, we assume that +there has no linkage between any two factor vertices in Vfactor for simplicity, and our analysis results +below do not change even when the factor vertices are allowed to have connections. +In sum, the overall relation between any two vertices i and j in G can be represented by the vector +ai,j = (ai,j,1, ..., ai,j,M, ai,j,M+1, ..., ai,j,M+B)′ ∈ RM+B, +(9) +where the first M entries and the remaining B entries carry the information of S2S relations and F2S +relations, respectively. +2.1.3. Module III: Hypergraph Learning. +Having known the relations among all stocks and factors +in G, it is natural to capture how much temporal information the stock i can receive from its linked +stocks and factors. To fulfill this goal, we define the aggregated temporal embedding for stock i as +xP +i,t = +� +j∈Vstock,j̸=i +g +� +ai,j, xL +i,t, xL +j,t; G, θP +� +dj +xL +j,t + +� +j∈Vfactor +g +� +ai,j, xL +i,t, xL +j,t; G, θP +� +N +xL +j,t ∈ Rd, +(10) +where i = 1, ..., N, ai,j in (9) represents the overall relation between vertices i and j, xL +i,t and xL +j,t are +the temporal embeddings extracted by (7), g(·) : RM+B×Rd×Rd → R is a nonlinear function indexed +by θP to measure the intensity of interplay between vertices i and j, and dj = �M +m=1 +� +i̸=j ai,j,m is +the number of stocks linked with stock j. Following Feng et al. (2019), we take +g +� +ai,j, xL +i,t, xL +j,t; G, θP +� += softmax +� +W5(xL′ +i,t, xL′ +j,t, a′ +i,j)′ + b5 +� +∈ R, +(11) +where W5 ∈ R1×(M+B+2d) is a vector of weight parameters, b5 ∈ R is a bias parameter, θP contains +all the parameters in W5 and b5, and softmax(·) is used to normalize the value of g(·) into (0, 1). The +specification of g(·) in (11) has two merits: First, it allows the intensity of interplay between any two + +9 +vertices to be stock-, factor-, and relation-specific; Second, it aims to capture some missing relations +that are not described in ai,j (i.e., ai,j ≡ 0) but presented by the similarity of xL +i,t and xL +j,t, since the +term W5(xL′ +i,t, xL′ +j,t, a′ +i,j)′ is still informative even when ai,j ≡ 0. +As a temporal graph convolution (TGC), the third module combines the temporal embedding xL +i,t +in (7) and the aggregated temporal embedding xP +i,t in (10) to form +xTGC +i,t += +� +xL′ +i,t, xP′ +i,t +�′ ∈ R2d for i = 1, ..., N. +(12) +The advantage of using xTGC +i,t +is apparent, since xTGC +i,t +captures the spatial and temporal information +of stock features simultaneously. +2.1.4. Module IV: Quantile Output. +Our last module applies a fully connected (FC) network to +revise the spatial-temporal information xTGC +i,t +in (12) to Qt(τ) in (5). Let θC contain all the parameters +in W6 and b6, where W6 ∈ R1×2d is a vector of weight parameters, and b6 ∈ R is a bias parameter. +Then, we set the form of FTGCN as +f(Xt−1; G, θ) ≡ (f1(Xt−1; G, θ), ..., fN(Xt−1; G, θ))′ +with fi(Xt−1; G, θ) = W6xTGC +i,t ++ b6 for i = 1, ..., N, +(13) +where G is the factor-augmented hypergraph in (8), and θ contains all the parameters in θL, θP , and +θC. Consequently, our FTGCN-based quantile model has the specification: +Qt(τ) = f(Xt−1; G, θτ) with Qi,t(τ) = fi(Xt−1; G, θτ) for i = 1, ..., N +(14) +(see its network architecture in Fig 1), where f(Xt−1; G, θτ) and fi(Xt−1; G, θτ) are defined as in (13). +2.1.5. Estimation of the FTGCN-based Quantile Model. +As Qi,t(τ) is the τ-th conditional quantile +of ri,t given Ft−1, we estimate θτ in (14) by the following quantile estimator: +�θτ = argmin +θτ +1 +NT +N +� +i=1 +T +� +t=1 +ρτ (ri,t − fi(Xt−1; G, θτ)) ≡ argmin +θτ +1 +T +T +� +t=1 +ℓ(rt, Xt−1; G, θτ, τ), +(15) +where ρτ(x) = x[τ −I(x < 0)] is the quantile loss function (Koenker and Bassett, 1978) with I(·) being +the indicator function. Due to the massive data volume, we adopt the adaptive moment estimation +(Adam) algorithm in Kingma and Ba (2015) to compute �θτ in (15); see Algorithm 1 for the details. +Using �θτ, we obtain +� +Qt(τ) = f(Xt−1; G, �θτ), +(16) +which is the estimator of Qt(τ) at the quantile level τ. + +10 +Fig 1: The architecture of FTGCN-based quantile model. +Algorithm 1 The training procedure of �θτ by the Adam algorithm. +Input: +The sample: {(rt, Xt−1)}; +The initial value of network parameters in the FTGCN-based quantile model: θ(0) +τ ; +The factor-augmented hypergraph: G; +Hyperparameter: learning rate γ; +1: l = 0; +2: repeat +3: +(rl, Xl−1) ← draw a random data point from {(rt, Xt−1)} (A cross-sectional minibatch); +4: +g(l) +τ +← ∇θτ +� +ℓ(rl, Xl−1; G, θ(l) +τ , τ) +� +(Gradients of minibatch estimator); +5: +θ(l+1) +τ +← update parameters using learning rate γ and gradients g(l) +τ +(Adam); +6: +l ← l + 1; +7: until convergence of parameters θ(l+1) +τ +; +Output: +The value of θ(l+1) +τ +, which is taken as the quantile estimator �θτ. +2.1.6. Comparison with the Existing Models. +Our FTGCN-based quantile model in (14) has a +linkage with the TGCN-based model in Feng et al. (2019) with regard to the network structure. As + +0.00.0 +Stock 1 +0.0 +Stock 2 +0.00.0 +O.. +Stock N +Factor 1 +0 +0.00.0 +0.0 +Factor B11 +the pioneering work, the TGCN-based model applies the domain knowledge to construct a hypergraph +for taking multiple types of S2S relation into account. The main difference between the FTGCN-based +quantile model and the TGCN-based model is two-fold. First, the FTGCN-based quantile model aims +to learn the conditional quantiles of ri,t, whereas the TGCN-based model focuses on the conditional +mean of ri,t. Second, the FTGCN-based quantile model incorporates the F2S relations to build the +factor-augmented hypergraph, but the TGCN-based model does not consider this kind of important +information in its hypergraph. +Besides our FTGCN-based quantile model, many other models are existing in the literature to study +the conditional quantile of high-dimensional data; see, for example, Koenker (2004), Kato, Galvao and +Montes-Rojas (2012), and Galvao and Kato (2016) for the quantile individual fixed effects models, +Ando and Bai (2020), Chen, Dolado and Gonzalo (2021), Ma, Linton and Gao (2021), and Yang +et al. (2022) for the quantile factor models, and H¨ardle, Wang and Yu (2016), Zhu et al. (2019), and +Xu et al. (2022) for the quantile network models. However, except for the factor-augmented dynamic +network quantile regression (FDNQR) model in Xu et al. (2022), none of the aforementioned models +takes the domain knowledge and asset pricing knowledge simultaneously into account to guide the +estimation of conditional quantile. Specifically, the FDNQR model uses the domain knowledge to +propose a weighted adjacency matrix W ∈ RN×N with the (i, j)-th entry wi,j, where wi,j = ai,j/ni, +ni = �N +j=1 ai,j, ai,j = 1 if the stock i has the connection with another stock j, and ai,j = 0 otherwise. +Based on W , the FDNQR model assumes +Qi,t(τ) = ατ + β′ +τzi + γτ +N +� +j=1 +wi,jrj,t−1 + ζτri,t−1 + +S +� +s=1 +ς′ +s,τFt−s, +(17) +where ατ ∈ R, βτ ∈ RQ×1, γτ ∈ R, and ςs,τ ∈ RB×1 are quantile regression coefficients, zi ∈ RQ×1 +is a Q-dimensional vector of time-invariant stock features, and Ft = (f1,t, ..., fB,t)′ ∈ RB×1 is a B- +dimensional vector of time-variant factors. In model (17), ατ is the constant intercept term for all +stocks, βτ is the constant intensity of the impact from stock features on stock i, γτ is the constant +intensity of spatial impact on stock i caused by its connected stocks, ζτ is the constant intensity of +temporal impact on stock i caused by its lagged term, and ςs,τ is the constant intensity of factor +impact on all stocks caused by the lagged factors. Clearly, our FTGCN-based quantile model is much +more general than model (17), since it captures multiple types of relation separately, extracts the +information of time-variant stock and factor features in a non-linear way, and allows for the time- +variant heterogenous intensity of spatial and temporal impacts on each stock caused by either its + +12 +connected stocks or factors. +Note that model (17) does not include the contemporaneous variables in the original FDNQR model +for the purpose of prediction, and it nests the network quantile autoregressive model in Zhu et al. +(2019). As the contemporaneous variables are absent, model (17) now can be consistently estimated +by using the quantile loss function as in Zhu et al. (2019). +2.2. Graph-based Learning for Conditional Mean. +So far, we have introduced the FTGCN to learn +the conditional quantile of ri,t. Following the similar idea, we can learn the conditional mean of ri,t +by an FTGCN-based mean model: +ri,t = f(Xt−1; G, θµ) + εi,t, +(18) +where f(Xt−1; G, θµ) is defined as in (13), and εi,t is the error term with zero mean. Note that model +(18) reduces to the TGCN model in Feng et al. (2019), when the factors and their features are absent. +To estimate model (18), we consider the penalized least squares (PLS) estimator of θµ given by +�θµ = argmin +θµ +1 +T +T +� +t=1 +� +1 +N +N +� +i=1 +[ri,t − fi(Xt−1; G, θµ)]2 ++ λ∗ +N2 +N +� +i=1 +N +� +j=1 +max {0, −[fi(Xt−1; G, θµ) − fj(Xt−1; G, θµ)](ri,t − rj,t)} +� +≡ argmin +θµ +1 +T +T +� +t=1 +ℓµ(rt, Xt−1; G, θµ, λ∗), +(19) +where the penalty term is utilized to ensure that the orders of ri,t and rj,t do not deviate largely from +those of their predicted values, and λ∗ is a positive hyperparameter. As for �θτ, we adopt the Adam +algorithm to compute �θµ; see Algorithm 2 for the details. Using �θµ, we then estimate the conditional +mean µi,t by +�µi,t = fi(Xt−1; G, �θµ) +for i = 1, ..., N. +(20) +It is worthwhile mentioning that the order-preserving penalty in (19) has been widely adopted in the +literature to improve the learning efficiency for conditional mean prediction (Zheng et al., 2007; Socher +et al., 2013; Feng et al., 2019). However, this penalized method is inappropriate for the conditional +quantile estimation, since Qi,t(τ) does not tend to be larger than Qj,t(τ) when ri,t is larger than rj,t. + +13 +Algorithm 2 The training procedure of �θµ by the Adam algorithm. +Input: +The sample: {(rt, Xt−1)}; +The initial value of network parameters in the FTGCN-based mean model: θ(0) +µ ; +The factor-augmented hypergraph: G; +Hyperparameters: λ∗, learning rate γ; +1: l = 0; +2: repeat +3: +(rl, Xl−1) ← draw a random data point from {(rt, Xt−1)} (A cross-sectional minibatch); +4: +g(l) ← ∇θµ +� +ℓµ(rl, Xl−1; G, θ(l) +µ , λ∗) +� +(Gradients of minibatch estimator); +5: +θ(l+1) +µ +← update parameters using learning rate γ and gradients g(l) (Adam); +6: +l ← l + 1; +7: until convergence of parameters θ(l+1) +µ +; +Output: +The value of θ(l+1) +µ +, which is taken as the PLS estimator �θµ. +2.3. The QCM Learning for Higher-order Conditional Moments. +Let � +Qt(τ1), ..., � +Qt(τK) be the +vectors of estimated conditional quantiles at K different quantile levels τ1, ..., τK, where � +Qt(τk) ≡ +( �Q1,t(τk), ..., �QN,t(τk))′ for k = 1, ..., K are computed as in (16). Below, we elaborate on how to +estimate hi,t, si,t, and ki,t by the QCM method in Zhang and Zhu (2022) for the fixed values of i and +t, based on �Qi,t(τ1), ..., �Qi,t(τK). +The QCM method is motivated by the Cornish-Fisher expansion (Cornish and Fisher, 1938), which +shows the following fundamental linkage between conditional quantiles and conditional moments: +Qi,t(τk) = µi,t + z(τk) +� +hi,t + +� +z(τk)2 − 1 +� � +hi,tsi,t +6 ++ +� +z(τk)3 − 3z(τk) +� � +hi,t(ki,t − 3) +24 ++ +� +hi,tωi,t(τk) +(21) +for k = 1, ..., K, where z(τk) is the τk-th quantile of standard normal distribution, and ωi,t(τk) is the +remainder of this expansion. Define +ε• +i,t,k = ε∗ +i,t,k + ε◦ +i,t,k with ε∗ +i,t,k = +� +hi,tωi,t(τk) and ε◦ +i,t,k = �Qi,t(τk) − Qi,t(τk), +Zk = +� +z(τk), z(τk)2 − 1, z(τk)3 − 3z(τk) +�′, +βi,t ≡ (βi,t,1, βi,t,2, βi,t,3)′ = +�� +hi,t, +� +hi,tsi,t +6 +, +� +hi,t(ki,t − 3) +24 +�′ +, +where �Qi,t(τk) is the estimator of Qi,t(τk). Then, we can rewrite (21) as follows: +�Qi,t(τk) = µi,t + Z′ +kβi,t + ε• +i,t,k +for k = 1, ..., K, +(22) + +14 +where ε• +i,t,k is the gross error containing the expansion error ε∗ +i,t,k and the quantile estimation error +ε◦ +i,t,k. Clearly, the equation (22) is a linear regression model with the response variable �Qi,t(τk), +explanatory variables Zk, parameter vector (µi,t, β′ +i,t)′, and error term ε• +i,t,k. Since ε• +i,t,k may not have +zero mean for model identification, we add an additional deterministic intercept term γi,t into the +equation (22) to form the following linear regression model: +�Qi,t(τk) = (µi,t + γi,t) + Z′ +kβi,t + εi,t,k +≡ ¯Z′ +kθi,t + εi,t,k +for k = 1, ..., K, +(23) +where εi,t,k = ε• +i,t,k − γi,t, ¯Zk = (1, Z′ +k)′, and θi,t = (βi,t,0, β′ +i,t)′ with βi,t,0 = µi,t + γi,t. +Let Yi,t be a K × 1 vector with entries �Qi,t(τk), ¯Z be a K × 4 matrix with rows ¯Z′ +k, and εi,t be a +K × 1 vector with entries εi,t,k. Then, the ordinary least squares (OLS) estimator of θi,t in (23) is +�θi,t ≡ (�βi,t,0, �β′ +i,t)′ = ( ¯Z′ ¯Z)−1 ¯Z′Yi,t, +(24) +where �βi,t = (�βi,t,1, �βi,t,2, �βi,t,3)′. Zhang and Zhu (2022) show that �θi,t −→ θi,t in probability as +K → ∞ under the following two classical conditions in the regression literature: +Condition 1. +¯Z′ ¯Z is positive definite. +Condition 2. +¯Z′εi,t/K −→ 000 in probability as K → ∞. +Consequently, by the continuous mapping theorem, we have +(25) +�hi,t ≡ �β 2 +i,t,1 −→ hi,t, �si,t ≡ 6�βi,t,2 +�βi,t,1 +−→ si,t, +and +�ki,t ≡ 24�βi,t,3 +�βi,t,1 ++ 3 −→ ki,t +in probability as K → ∞, where �hi,t, �si,t, and �ki,t are the QCMs of hi,t, si,t, and ki,t, respectively. In +order to make sure that �hi,t, �si,t, and �ki,t are moments under certain distribution of ri,t, two necessary +constraints are required: +�hi,t ≥ 0 and �ki,t ≥ �s2 +i,t + 1. +Clearly, the first constraint holds automatically, and the second constraint can be checked directly +based on the values of �ki,t and �si,t. If the second constraint does not hold, we can easily replace �θi,t in +(24) by a constrained least squares estimator, so that the resulting �ki,t and �si,t satisfy this constraint; + +15 +see more detailed discussions in Zhang and Zhu (2022). Moreover, it should be noted that we are +unable to estimate µi,t by the QCM method. The reason is that µi,t can not be estimated by �βi,t,0 in +(24) due to the presence of γi,t. Therefore, we have to estimate µi,t separately by other methods (see, +e.g., the graph-based method in Section 2.2 above). +As we observed, the core idea of QCM method is to transform the estimation of conditional moments +to that of conditional quantiles, giving us two remarkable advantages particularly in the realm of high- +dimensional data analysis. +First, the QCM method is easy-to-implement, since it only requires the estimated conditional +quantiles as the input to compute the OLS estimator �θi,t. When N is large, a direct estimation for +the higher-order conditional moments hi,t, si,t, and ki,t via high-dimensional GARCH-type models is +computationally infeasible. The reason is that the high-dimensional GARCH-type models are fitted +by the QML estimation method, which relies on a certain distribution of rt to write down the log- +likelihood function. However, the log-likelihood function is too complex to be optimized for large N +cases. For example, the often used Gaussian log-likelihood function depends on the inverse of many +N × N-dimensional variance-covariance matrices, making its optimization infeasible. Transforming +the estimation of higher-order moments to that of quantiles circumvents this annoying difficulty, since +the estimation of quantiles is a classical supervised learning but that of higher-order moments is not. +To be more specific, the supervised learning is a machine learning paradigm, and it aims to learn +a function f0 that maps features (say, x) to labels (say, y) supervised by a certain loss function +without assuming the distribution of x or y. For example, the τ-th quantile of y can be learned by +f0(x) using the quantile loss function ρτ(y − f0(x)) as the supervisor. However, it is unclear how to +design an appropriate loss function as the supervisor for learning the variance, skewness, or kurtosis +of y by f0(x), unless certain distributional assumption is made for x or y. This indicates that the +direct estimation of hi,t, si,t, and ki,t has to rely on a certain distribution of ri,t, as done by the +QML estimation in the high-dimensional GARCH-type models. Owing to the supervised learning +feature of quantiles, our indirect estimation of hi,t, si,t, and ki,t from the QCM method does not need +any distributional assumption of ri,t, so it bypasses the computational difficulty raised in the direct +estimation method to deal with large N cases. +Second, the QCM method can largely reduce the risk of model mis-specification, since �hi,t, �si,t, and +�ki,t are simultaneously computed without any prior estimation of µi,t, and their consistency holds even +when the specification of Qi,t(τ) is mis-specified. This advantage is far beyond our expectations, since + +16 +normally we have to first estimate µi,t and then hi,t, si,t, and ki,t using some parametric models that are +needed to be correctly specified to generate consistent estimators. The reason leading to this advantage +is that the QCM method is regression-based, so that the impact of µi,t is eliminated by absorbing it +into the intercept term and the consistency of �hi,t, �si,t, and �ki,t is ensured by Conditions 1–2. Note +that Condition 1 holds for the often choices of quantile sequence {τ1, ..., τK}, and Condition 2 allows +each model error εi,t,k (including the quantile estimation error ε◦ +i,t,k) to deviate from zero to some +extent, as long as the averages of {εi,t,k}, {εi,t,kz(τk)}, {εi,t,k[z(τk)2 − 1]}, and {εi,t,k[z(τk)3 − 3z(τk)]} +across k are close to zero; see Zhang and Zhu (2022) for more discussions on this aspect. +2.4. Implementation Details of the GRACE Method. +Due to the use of FTGCN, the GRACE +method first needs to alleviate the risk of overfitting, a prevalent deficiency of the neural network. +Following the standard approach to circumvent overfitting, we chronologically partition the full data +into three disjoint parts: training sample, validation sample, and testing sample. The training and +validation samples are taken to do parameter estimation, and the testing sample is used to evaluate +the truly out-of-sample performance of the GRACE method. To be more specific, we compute θ(l) +τ +at l-th iteration in Algorithm 1 based on the training sample, and then calculate its corresponding +validation sample error. Here, the validation sample error is the value of the objective function in (15) +based on the validation sample and θτ = θ(l) +τ . To regularize against overfitting, we utilize the early +stopping method to terminate the iteration process early in Algorithm 1 when the validation sample +error increases for several iterations, and select the estimator θ(l) +τ +having the smallest validation +sample error as the quantile estimator �θτ. Similarly, the PLS estimator �θµ is computed from the +training and validation samples under Algorithm 2. +Next, the GRACE method uses �θτk and �θµ to predict the values of �Qi,t(τk) and µi,t on the test- +ing sample, respectively, where τk = k/(K + 1), k = 1, ..., K, for simplicity. In the large pool of +�Qi,t(τk), some of �Qi,t(τk) are inevitable to be invalid. Intuitively, it is reasonable to exclude those +invalid �Qi,t(τk) for the computation of QCMs. To achieve this goal, we make use of the uncondi- +tional coverage test LRuc in Kupiec (1995) and conditional coverage test LRcc in Christoffersen +(1998). Specifically, we compute �Qi,t(τk) on the training and validation samples, and apply LRuc +and LRcc to detect whether the sequence of estimated conditional quantiles Qi(τk) ≡ { �Qi,t(τk) : t ∈ +training and validation samples} is valid at the significance level α for each stock i and quantile level + +17 +τk. Then, we build a valid quantile level set for stock i: +(26) +Ωi = {τk : the validity of Qi(τk) is accepted by both LRuc and LRcc at the level α}. +That is, Ωi groups all of those quantile levels τk, for which the sequence Qi(τk) is valid. Clearly, Ωi +depends on α and K jointly, where its size (denoted by |Ωi|) is decreasing with the value of α while +increasing with the value of K. In particular, we know that |Ωi| = K when α = 0. After having Ωi, +we use the predicted values { �Qi,t(τk) : τk ∈ Ωi} to predict the values of hi,t, si,t, and ki,t for stock +i via the related QCMs on the testing sample. As K is essentially replaced by |Ωi| from the above +manipulation, we need a large value of |Ωi| to ensure the consistency of the QCMs. This motivates +us to discard those stocks having the value of |Ωi| less than a predetermined tolerance K0 (say, e.g., +K0 = 30). +Finally, the GRACE method employs different performance measures from the predicted values of +µi,t, hi,t, si,t, and ki,t to construct portfolios, based on all of remaining stocks. +3. Empirical Analysis. +3.1. Data. +We apply our GRACE method to construct portfolios based on the stocks in two +major exchanges: NASDAQ and NYSE. The stock data we consider are the same as those in Feng +et al. (2019), and they contain daily prices from January 2, 2013 to December 8, 2017 for 1026 and +1737 stocks in NASDAQ and NYSE, respectively. Alone with the stock price data, we also take the +S2S relation data in Feng et al. (2019) to describe the multiple types of S2S relation. Based on the +domain knowledge, the S2S relations can be divided into two groups: Sector-industry relations Esi +stock +and Wiki company-based relations Ewiki +stock (see the Appendix A of Feng et al. (2019) for their detailed +definitions). Specifically, two stocks (say, stock i and stock j) have a sector-industry relation if they +belong to the same industry, where the industries are classified by the GICS standard. For example, +all 1026 stocks in NASDAQ are divided into 13 different sectors, where each sector contains several +industries; see Fig 2(a) for the sector-industry hierarchy of all 1026 stocks in this market. From this +figure, we know that MSFT LLC and Google LLC have an S2S relation since they belong to the same +industry “Computer Software: Programming”. +Meanwhile, two stocks can also have a Wiki company-based relation if they have an either first-order +or second-order relation. The first-order and second-order relations have the format of “company i +R +−→ +company j” and “company i +R1 +−→ entity k +R2 +←− company j”, respectively, where the companies i and j + +18 +Fig 2: Examples of the sector-industry, first-order and second-order relations. +bridged by an entity k are corresponding to the stocks i and j, respectively, and the relations R, R1, +and R2 are defined in Wikidata (https://www.wikidata.org/wiki/Wikidata:List of properties/all). It +turns out that there are 5 and 53 different types of first-order and second-order relations, respectively. +Fig 2(b) and 2(c) give some illustrating examples on the first-order and second-order relations. We +see from this figure that BlackRock Inc. has a first-order relation with Netflix Inc. since BlackRock +Inc. owns Netflix Inc., and United Airlines Inc. and Boeing Inc. have a second-order relation since +Boeing Inc. produces Boeing 747 that is sold to United Airlines Inc. In sum, Table 1 lists the number +of S2S relation types and the ratio of S2S relations to all possible stock pairs in NASDAQ and NYSE. +Since the ratio of S2S relations is always less than 10%, it indicates that the S2S relations in both +markets are sparse. +Table 1 +Summary of S2S relations +Sector-industry relation +Wiki company-based relation +Market +Relation types +Relation ratio (pairwise) +Relation types +Relation ratio (pairwise) +NASDAQ +112 +5.00% +42 +0.21% +NYSE +130 +9.37% +32 +0.30% + +Technology +Finance +Computer +Software: +Data Processing +Programming19 +In addition, the F2S relation data are also needed to facilitate our GRACE method. Among a great +variety of factors, we use the prevalent daily Fama-French five factors (Fama and French, 2015) to spec- +ify the related F2S relations Eff5 +factor, based on the asset-pricing knowledge. These five factors are excess +market return, RMW, HML, SMB, and CMA, and their daily data can be downloaded from the home- +page of Kenneth French (https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data library.html). +For stock i, we let ri,t be its 1-day return and r(k∗) +i,t += +1 +k∗ +�k∗ +s=1 ri,t+1−s be its k∗-day moving average +of returns at day t, and use the OLS method to calculate its 5-dimensional vector of factor exposures +(denoted by λi,t = (λ(1) +i,t , ..., λ(5) +i,t )′) based on the sample of 1-day returns in a half-year rolling window +up to day t. For factor b, we similarly let fb,t be its value, f(k∗) +b,t +be its k∗-day moving average, and ¯λ(b∗) +b,t +be its exposure on factor b∗ at day t, where we assume that factor b has one exposure on itself and zero +exposure on other factors (w.r.t., b∗ ̸= b). Now, based on ten stock features r(k∗) +i,t +and λ(b∗) +i,t +together +with ten factor features f(k∗) +b,t +and ¯λ(b∗) +b,t +for k∗ = 1, 5, 10, 20, 30 and b∗ = 1, 2, 3, 4, 5, our feature tensor +Xt−1 = [X1,t−1, ..., XN,t−1, XN+1,t−1, ..., XN+B,t−1] in (5) is taken as +(27) +Xi,t−1 = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +r(1) +i,t−S +· · · +r(1) +i,t−1 +· · · +· · · +· · · +r(30) +i,t−S +· · · +r(30) +i,t−1 +λ(1) +i,t−S +· · · +λ(1) +i,t−1 +· · · +· · · +· · · +λ(5) +i,t−S +· · · +λ(5) +i,t−1 +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +and XN+b,t−1 = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +f(1) +b,t−S +· · · +f(1) +b,t−1 +· · · +· · · +· · · +f(30) +b,t−S +· · · +f(30) +b,t−1 +¯λ(1) +b,t−S +· · · +¯λ(1) +b,t−1 +· · · +· · · +· · · +¯λ(5) +b,t−S +· · · +¯λ(5) +b,t−1 +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +, +for i = 1, ..., N and b = 1, ..., B, where N = 1026 (or 1737) for the NASDAQ (or the NYSE) market, +B = 5, Xi,t−1 ∈ RP×S, and XN+b,t−1 ∈ RP×S with P = 10. It is worthy noting that each entry in +Xi,t−1 and XN+b,t−1 is normalized by its range in the training sample to reduce its skewness and +leptokurtosis; see the similar implementation in Feng et al. (2019). +With the full data sample {rt, Xt−1} in hand, we divide it into three disjoint parts in the same +way as Feng et al. (2019): The training sample from January 2, 2013 to December 31, 2015 has 756 +trading days, the validation sample follows and ends on December 30, 2016 with 252 trading days, +and the testing sample covers the remaining 237 trading days from January 3, 2017 to December 8, +2017 (i.e., the out-of-sample period). Now, based on the values of hyperparameters listed in Table 2, + +20 +our GRACE method is implemented as the details specified in Section 2.4 above. Here, the selection +of λ∗, γ, S, and d for the FTGCN-based models is suggested by the tuning results in Feng et al. +(2019), and that of α and K for the QCM method will be examined in the sequel. +Table 2 +List of hyperparameters. +Hyperparameter +Description +Value +λ∗ +tuning parameter in the computation of PLS estimator +0.1 +γ +learning rate in the Adam algorithm +10−3 +S +length of lagged features in the LSTM +16 +d +number of hidden units in the LSTM +64 +α +significance level of LRuc and LRcc tests +0.01 +K +number of predetermined quantile levels +199 +3.2. Comparison Methods. +Besides our GRACE method, other graph-based methods can also be +adopted to select portfolios using the same idea, except for different models to predict the conditional +quantiles and mean of ri,t. Below, we introduce two alternative graph-based methods for the purpose +of comparison. +The first competitor is labeled as GRACE1, which replaces the factor-augmented hypergraph in the +GRACE method with the hypergraph in Feng et al. (2019), and leaves other mechanisms (including +the input features and the selection of tuning hyperparameters) unchanged. The comparison between +the GRACE and GRACE1 methods is to verify whether incorporating the asset pricing knowledge +into the hypergraph is informative for portfolio selection. +The second competitor is the simple GRACE (denoted by GRACE2) method, which is motivated +by the network autoregression model in Zhu et al. (2017) and the FDNQR model in Xu et al. (2022). +Specifically, the GRACE2 method predicts the conditional quantiles of ri,t based on the following +specification: +Qi,t(τ) = ατ + γτ +N +� +j=1 +wi,jrj,t−1 + ζ′ +τxi,t−1 + ς′ +1,τFt−1, +(28) +where xi,t−1 is the last column of Xi,t−1 in (27), Ft−1 is the 5-dimensional vector containing the values +of Fama-French five factors at t − 1, and other notations are inherited from (17). By construction, + +21 +model (28) uses the term ζ′ +τxi,t−1 (replacing the term βτzi + ζri,t−1 in (17)) to account for the lag- +1 stock features, and it takes the term ς′ +1,τFt−1 to include the lag-1 factor features. Similarly, the +GRACE2 method predicts the conditional mean of ri,t by the following factor-augmented network +autoregressive specification: +ri,t = αµ + γµ +N +� +j=1 +wi,jrj,t−1 + ζ′ +µxi,t−1 + ς′ +1,µFt−1 + ε∗ +i,t, +(29) +where ε∗ +i,t is the error term with zero mean, and αµ, γµ, ζµ, and ς1,µ are unknown regression coefficients. +Here, wi,j = ai,j/ni in models (28) and (29) is determined by Estock in the GRACE method, such that +ai,j = 1 if there is any S2S relation between stock i and stock j, and ai,j = 0 otherwise. We estimate +models (28) and (29) respectively via the the quantile loss and L2 loss functions using the data from +the combination of training and validation samples, and then proceed the portfolio selection on the +testing sample. One may extend both models to contain the stock and factor features up to lag- +S. However, our unreported analysis shows that this extension makes model estimation less stable, +leading to the worse performance in portfolio selection. Clearly, the comparison between the GRACE +and GRACE2 methods aims to exhibit how inadequate are those simple settings in models (28) and +(29) for portfolio selection. +3.3. Economic Performance Evaluation. +This subsection evaluates the out-of-sample performance +of the long-short portfolios selected by the GRACE method. After sorting all stocks via a certain +performance measure into 10 deciles, the long-short portfolio is re-balanced on every trading day via +buying the 10% highest ranking stocks (decile 10) and selling the 10% lowest ranking stocks (decile 1) +with equal weights. The performance measures to sort all stocks include the M (mean), MV (Mean– +variance), MVSK (Mean–variance with skewness and kurtosis), SR (Sharpe ratio), and SRSK (Sharpe +ratio with skewness and kurtosis), where the definitions of the last four performance measures are +given in (1)–(4), and the values of all five performance measures are computed based on the predicted +values of µi,t, hi,t, si,t, and ki,t from the GRACE method with Estock = Ewiki +stock, Efactor = Eff5 +factor, +and hyperparameters taken as in Table 2. For the performance measures MV, MVSK, and SRSK, +the hyperparameters λ1, λ2, and λ3 are tuned by the grid search within the sets A1, A2, and A3, +respectively, to maximize the value of SR of each long-short portfolio re-balanced on training and + +22 +validation samples. Here, due to different ranges of hi,t and si,t (or ki,t), we take +A1 = {a × 10−b : a = 1, 2, ..., 9 and b = −1, 0, ..., 3}, +A2 = A3 = {a × 10−b : a = 1, 2, ..., 9 and b = 2, 3, ..., 6}. +As a comparison, the GRACE1 and GRACE2 methods are also used to select the long-short portfolios +under the similar procedure as above, where the GRACE1 method with the performance measure M is +the benchmark method proposed by Feng et al. (2019). Since the QCM method used by each method +relies on the choices of α and K, the stock pool for portfolio selection under the performance measures +MV, MVSK, SR, and SRSK varies with the choice of α or K. For the sake of consistency, the portfolio +selection under the performance measure M will use the same stock pool as for other performance +measures in the sequel, although the implementation of performance measure M is independent of +higher-order conditional moments and not affected by the choice of α or K technically. +From an economic viewpoint, we compare all of the selected portfolios in terms of their annualized +SR, which is the ratio of annualized excess return to annualized risk. To compute the annualized excess +return, we use the Treasury bill rate as a proxy for the risk-free return, and take the transaction cost +of 3‰ (i.e., 30 basic points) for buying and selling into account as done in Engle, Ferstenberg and +Russell (2012). Note that when there is no ambiguity, we omit the wording “annualized” below for +ease of presentation. +3.3.1. Impacts of Method and Performance Measure. +We first assess how the graph-based methods +and performance measures affect the out-of-sample performance of portfolios. Table 3 reports the +values of (excess) return, risk, and SR of the out-of-sample portfolios selected by three different +methods with five different performance measures in NASDAQ and NYSE. From this table, we can +have the following findings: +(i) For the GRACE method, the SRSK and M portfolios have the largest and smallest values of +SR, respectively, in both markets, implying the necessity of using three higher-order conditional +moments for portfolio selection. Particularly, the advantage of using higher-order conditional +moments is more evident in NASDAQ than NYSE by observing that the value of SR for the +SRSK portfolio is 61% (or 12%) higher than that for the M portfolio in NASDAQ (or NYSE). +Moreover, the values of return and risk indicate that the SRSK portfolio has a larger value of +SR mainly because it can generate a much larger (or smaller) value of return (or risk) than + +23 +Table 3 +Out-of-sample performances of long-short portfolios across different methods and performance measures. +NASDAQ +NYSE +Method +Measure +Return (%) +Risk (%) +SR +Return (%) +Risk (%) +SR +GRACE +M +25.40 +8.53 +2.98 +29.04 +8.74 +3.10 +MV +38.81 +10.43 +3.72 +29.36 +8.62 +3.17 +MVSK +37.04 +8.43 +4.39 +30.01 +8.62 +3.25 +SR +37.48 +9.03 +4.15 +13.90 +3.43 +3.47 +SRSK +39.21 +8.15 +4.81 +13.95 +3.43 +3.48 +GRACE1 +M +12.08 +8.45 +1.43 +25.74 +8.97 +2.87 +MV +13.55 +7.83 +1.73 +25.82 +8.97 +2.88 +MVSK +33.45 +8.24 +4.06 +25.82 +8.97 +2.88 +SR +26.57 +6.48 +4.10 +16.91 +5.10 +3.32 +SRSK +42.09 +8.86 +4.75 +16.97 +5.10 +3.33 +GRACE2 +M +0.06 +8.59 +0.01 +-2.09 +8.94 +-0.46 +MV +4.55 +10.57 +0.43 +0.45 +8.80 +-0.18 +MVSK +6.66 +8.50 +0.78 +2.26 +8.79 +0.03 +SR +4.53 +9.11 +0.50 +3.00 +3.51 +0.28 +SRSK +5.90 +8.19 +0.72 +3.56 +3.51 +0.45 +the M portfolio in NASDAQ (or NYSE). Another difference between NASDAQ and NYSE is +the influence of conditional skewness and kurtosis. Specifically, using the conditional skewness +and kurtosis can well decrease the portfolio risk in NASDAQ while only marginally increase +the portfolio return in NYSE, according to the comparison between MV and MVSK (or SR +and SRSK) portfolios. The aforementioned distinction in two markets may attribute to the +fact that the NYSE has a relatively more “normal” environment than the NASDAQ, so that +the function of higher-order conditional moments (particularly the conditional skewness and +kurtosis) is relatively weaker for portfolio selection. +(ii) For the GRACE1 and GRACE2 methods, both of them perform worse than the GRACE +method, regardless of the choice of performance measure. The advantage of GRACE method +over GRACE1 method is exceptionally significant for M and MV portfolios in NASDAQ, since + +24 +the M and MV portfolios selected by GRACE method have 108% and 115% higher value of +SR than those selected by GRACE1 method, respectively. This finding shows that incorporat- +ing asset price information is more important for portfolio selection in NASDAQ than NYSE, +especially when the conditional skewness and kurtosis are not taken into account. Moreover, +the value of SR for the best portfolio selected by the GRACE method is 236% and 21% higher +than that selected by the benchmark method (i.e., the GRACE1 method with the performance +measure M) in NASDAQ and NYSE, respectively. In all cases, the GRACE2 method has a +much worse performance than other two methods. This is not unexpected, because the simple +model settings in the GRACE2 method can not capture the effects of features on stock returns +adequately. +Overall, the above findings clearly demonstrate the importance of higher-order conditional moments +as well as the asset pricing knowledge in portfolio selection through the GRACE method. +3.3.2. Impact of EEEstock. +Since all considered three graph-based methods depend on the S2S rela- +tion set Estock, a natural question is what kind of S2S relation set is more informative for portfolio +selection. To answer this question, we alter all three methods by choosing Estock = Esi +stock or Eall +stock +while keeping other settings as for Table 3 unchanged, where Eall +stock is the union set of Ewiki +stock and +Esi +stock. Table 4 reports the values of out-of-sample SR for portfolios selected from three different +choices of Estock. From Table 4, we find that the value of SR for the Esi +stock-based portfolio is smaller +than that for the corresponding Ewiki +stock-based portfolio in all cases, except for the M portfolios se- +lected by the GRACE1 method in NASDAQ. Particularly, the advantage of Ewiki +stock-based portfolio +over Esi +stock-based portfolio is much more substantial for the GRACE1 method with the performance +measures MVSK, SR, and SRSK in NASDAQ. This finding indicates that the S2S relations in Esi +stock +could be less informative than those in Ewiki +stock to learn higher-order conditional moments, especially +when the asset pricing knowledge is absent. +Moreover, we find from Table 4 that using a richer S2S relation set Eall +stock to replace the single S2S +relation set Ewiki +stock gives no change or little change to the values of SR in the GRACE and GRACE1 +methods, and this replacement even makes the portfolios have smaller values of SR for many cases +in the GRACE1 method. The reason is probably that the long-term correlations between stocks are +largely driven by the factors through the F2S relations in Eff5 +factor, and they could be wrongly captured +by the S2S relations in Esi +stock when the asset pricing knowledge is absent. For example, the stocks + +25 +belonging to the sector “Basic Industries” tend to have large market capitalization, while the SMB +factor in Fama and French (2015) represents the outperformance of small-cap stocks over large-cap +ones during a long-term. Hence, the comovement of stocks in the sector “Basic Industries” is more +properly captured by the F2S relations with respect to the SMB factor in Eff5 +factor rather than the +sector-industry relations in Esi +stock. The unsatisfactory performance from the use of Eall +stock becomes +more evident in the GRACE2 method. This conveys the information that it is inappropriate to ignore +the type of S2S relation as done by models (28)–(29), when the domain knowledge on multiple types +of S2S relation is available. +In sum, we could reach a general conclusion that Esi +stock is less informative than Ewiki +stock for portfolio +selection. Hence, if Ewiki +stock is accessible, we recommend it for practical use. +Table 4 +Out-of-sample SRs of long-short portfolios across different choices of Estock. +NASDAQ +NYSE +Method +Measure +Esi +stock +Ewiki +stock +Eall +stock +Esi +stock +Ewiki +stock +Eall +stock +GRACE +M +2.74 +2.98 +2.98 +3.04 +3.10 +3.12 +MV +2.86 +3.72 +3.74 +3.04 +3.17 +3.17 +MVSK +3.98 +4.39 +4.40 +3.04 +3.25 +3.25 +SR +3.81 +4.15 +4.18 +3.04 +3.47 +3.47 +SRSK +4.52 +4.81 +4.81 +3.13 +3.48 +3.48 +GRACE1 +M +1.58 +1.43 +1.52 +2.76 +2.88 +2.87 +MV +1.60 +1.73 +1.55 +2.77 +2.88 +2.87 +MVSK +2.94 +4.06 +4.02 +2.77 +2.88 +2.88 +SR +2.14 +4.10 +4.01 +3.19 +3.32 +3.32 +SRSK +3.19 +4.75 +4.65 +3.19 +3.32 +3.32 +GRACE2 +M +-0.17 +0.01 +-1.24 +-1.88 +-0.46 +-2.81 +MV +0.12 +0.43 +-1.02 +-1.77 +-0.18 +-2.80 +MVSK +0.66 +0.78 +-0.55 +-1.71 +0.03 +-2.78 +SR +0.43 +0.50 +-1.34 +-1.56 +0.28 +-2.63 +SRSK +0.61 +0.72 +-1.29 +-1.33 +0.45 +-2.58 + +26 +3.3.3. Impact of α and K. +Before implementing the QCM method, a valid quantile level set Ωi in +(26) is defined for each stock i. As discussed in Section 2.4 above, we only use the predicted conditional +quantiles at those quantile levels in Ωi to predict higher-order conditional moments. Intuitively, when +the value of K is fixed (i.e., the quantile sequence {τ1, ..., τK} is given), a large value of α enhances +the reliability of the conditional quantile prediction, but at the same time, it reduces the learning +efficiency of the higher-order conditional moments as the size of Ωi becomes small. Hence, there is a +trade-off between reliability and learning efficiency, in terms of the choice of α. To address this issue, +we examine the impact of α on the performance of portfolio by changing the value of α while keeping +other settings in the same way as for Table 3. +Specifically, we select the long-short portfolios for each method when α ∈ {0%, 1%, 5%, 10%}, and +report the related results of out-of-sample SR in Table 5. From this table, we find that except for +the M and SR portfolios from the GRACE1 method in NASDAQ, all of other portfolios achieve the +maximum value of SR at α = 1%. When the value of α decreases from 1% to 0% (i.e., |Ωi| = K), the +performance of most portfolios becomes worse, particularly for those selected by the GRACE1 method +in NYSE. This finding indicates that the tests LRuc and LRcc can effectively exclude those extremely +ill-behaved predictions of conditional quantiles to improve the performance of QCM method. When +the value of α increases from 1% to 5% or 10%, the performance of most portfolios also becomes +worse but with different tendencies in two markets. For example, the reduction in SR under GRACE1 +method is much larger than that under the GRACE method in NASDAQ, whereas the reduction in +SR has similar patterns in NYSE based on both methods. This observation shows the advantage of +the factor-augmented hypergraph, which stabilizes the performance of the GRACE method across +the choice of α. +Besides α, K is also related to the construction of Ωi for all three methods. A large value of K +is needed to ensure the consistency of the QCMs in theory, however, it is inevitable to increase the +cost in computation as the conditional quantile model has to be trained K different times. Clearly, +the choice of K reflects the trade-off between the computational cost and learning efficiency. To +study the impact of K, we select the long-short portfolios as for Table 3 but with different values +of K, and report the values of out-of-sample SR for those portfolios in Table 6. The findings from +this table generally match our expectation. First, the value of SR increases with the value of K for +each portfolio. This is because a larger value of K can grab more information on the conditional +distribution of ri,t by learning its conditional quantiles at more refined quantile levels. Second, the + +27 +performance of portfolios from the GRACE method is more stable across the choice of K than that +from the GRACE1 method. This finding is consistent to that in Table 5. Third, the values of SR +under the GRACE2 method are always less than one or even below zero in many cases, implying the +inadequacy of models (28)–(29) for learning conditional quantiles and mean of ri,t. +Overall, the portfolios from the GRACE method show a robust performance over the choice of α +and K, and they outperform those from the GRACE1 and GRACE2 methods for all examined choices +of α and K. In practice, we recommend to take α = 1% and K = 199 for the GRACE method, since +this seems a desirable choice to balance the reliability (or computational cost) and learning efficiency, +as demonstrated by the aforementioned results. +Table 5 +Out-of-sample SRs of long-short portfolios across different choices of α. +NASDAQ +NYSE +Method +Measure +0% +1% +5% +10% +0% +1% +5% +10% +GRACE +M +2.35 +2.98 +2.83 +2.81 +2.88 +3.10 +3.06 +3.05 +MV +3.58 +3.72 +3.67 +3.61 +3.01 +3.17 +3.11 +3.09 +MVSK +4.17 +4.39 +4.21 +4.09 +3.03 +3.25 +3.22 +3.21 +SR +3.99 +4.15 +3.94 +3.87 +3.11 +3.47 +3.45 +3.42 +SRSK +4.78 +4.81 +4.76 +4.62 +3.15 +3.48 +3.43 +3.43 +GRACE1 +M +1.42 +1.43 +1.34 +1.46 +1.46 +2.88 +2.81 +2.83 +MV +1.48 +1.73 +1.62 +1.59 +1.53 +2.88 +2.81 +2.83 +MVSK +3.77 +4.06 +4.03 +2.98 +1.67 +2.88 +2.81 +2.83 +SR +4.61 +4.10 +3.81 +2.11 +2.11 +3.32 +3.24 +3.22 +SRSK +4.73 +4.75 +4.68 +3.06 +2.76 +3.32 +3.25 +3.26 +GRACE2 +M +0.01 +0.01 +-0.02 +-0.04 +-0.63 +-0.46 +-0.49 +-0.48 +MV +0.22 +0.43 +0.31 +0.26 +-0.41 +-0.18 +-0.22 +-0.26 +MVSK +0.68 +0.78 +0.67 +0.65 +-0.14 +0.03 +-0.08 +-0.12 +SR +0.31 +0.50 +0.41 +0.38 +0.17 +0.28 +0.21 +0.18 +SRSK +0.64 +0.72 +0.59 +0.53 +0.25 +0.45 +0.31 +0.27 + +28 +Table 6 +Out-of-sample SRs of long-short portfolios across different choices of K. +NASDAQ +NYSE +Method +Measure +49 +99 +199 +49 +99 +199 +GRACE +M +2.88 +2.91 +2.98 +3.07 +3.08 +3.10 +MV +3.16 +3.51 +3.72 +3.07 +3.12 +3.17 +MVSK +3.22 +3.64 +4.39 +3.19 +3.21 +3.25 +SR +3.59 +3.77 +4.15 +3.37 +3.42 +3.47 +SRSK +3.63 +3.92 +4.81 +3.43 +3.44 +3.48 +GRACE1 +M +1.01 +1.31 +1.43 +1.73 +2.44 +2.88 +MV +1.13 +1.52 +1.73 +1.78 +2.51 +2.88 +MVSK +3.18 +3.44 +4.06 +2.61 +2.71 +2.88 +SR +3.09 +3.59 +4.10 +2.43 +3.04 +3.32 +SRSK +3.47 +3.76 +4.75 +2.87 +3.04 +3.32 +GRACE2 +M +-0.12 +-0.03 +0.01 +-0.50 +-0.49 +-0.46 +MV +0.30 +0.38 +0.43 +-0.24 +-0.20 +-0.18 +MVSK +0.47 +0.61 +0.78 +-0.22 +-0.19 +0.03 +SR +0.29 +0.42 +0.50 +-0.08 +0.05 +0.28 +SRSK +0.49 +0.62 +0.72 +-0.02 +0.09 +0.45 +3.3.4. Impact of Transaction Cost. +The transaction cost is a non-negligible important factor for +portfolio selection. In NASDAQ and NYSE, the cost of a transaction mainly consists of three compo- +nents: Commission, stamp tax, and slippage. While the rates of commission and stamp tax are easily +accessible, the slippage is hard to be measured quantitatively as the liquidity issue could make it +difficult to execute transactions at a pre-specified price without affecting the market price, especially +for small-cap stocks. Normally, the transaction cost of 30 basis points is appropriate to capture the +total effect of commission, stamp tax, and slippage. However, the slippage could be larger in some +extreme circumstances. Therefore, it is necessary to conservatively consider some higher transaction +costs than 30 basis points to investigate their impact on portfolio selection. +To achieve this goal, Table 7 reports the values of out-of-sample SR for long-short portfolios selected +from each method, when the transaction costs are 30, 50, 75, and 100 basis points. From this table, we + +29 +can obtain some interesting findings. First, as expected, the value of SR for each portfolio decreases +with the transaction cost. Second, regardless of transaction cost, the SRSK portfolio is the best one +followed successively by the SR, MVSK, MV, and M portfolios in the GRACE and GRACE1 methods. +Moreover, the SRSK portfolio from the GRACE method has a more stable performance to the level +of transaction cost than other portfolios in NASDAQ. Third, the GRACE method outperforms the +GRACE1 method in all considered cases, except that the SR and SRSK portfolios from the GRACE1 +method perform better than those from the GRACE method when the transaction cost is 100 basis +points in NYSE. Fourth, the GRACE2 method performs much worse than the GRACE method, and +its performance becomes more unsatisfactory when the transaction cost is higher. +Overall, we find that the advantage of using the higher-order conditional moments together with +the asset pricing knowledge from the GRACE method for portfolio selection is unchanged with the +setting of transaction cost. +3.4. Statistical Performance Evaluation. +From an economic viewpoint, the good performance of +our GRACE method in portfolio selection has been demonstrated above. From a statistical viewpoint, +it is worthwhile to check whether the GRACE method can well estimate and predict the conditional +moments of ri,t, shedding light on its success in portfolio selection. +3.4.1. The Comovement of Conditional Moments. +Intuitively, the conditional moments of two +linked stocks should have a tendency of comovement. Since the number of stocks is large, there has +no easy way to exhibit whether most pairs of linked stocks have this tendency. For ease of illustration, +we only plot ri,t, �µi,t, �hi,t, �si,t, and �ki,t of MSFT LLC and Google LLC based on the GRACE method +in Fig 3. Here, �µi,t, �hi,t, �si,t, and �ki,t are estimated values during the in-sample period (i.e., the period +of the training and validation samples), and they are predicted values during the out-of-sample period. +From Fig 3, we find that not only the stock returns but also all estimated and predicted conditional +moments have similar trends during the entire period. +3.4.2. The Validity of Conditional Moments. +Since the true values of conditional moments are +unobserved, no explicit quantity can measure the precision of �hi,t, �si,t, and �ki,t. To circumvent this +deficiency, we propose some hypothesis tests to detect whether �µi,t, �hi,t, �si,t, and �ki,t are valid by +using a similar idea as in Gu, Kelly and Xiu (2020). Specifically, we define αµ +i = E(eµ +i,t), αh +i = E(eh +i,t), +αs +i = E(es +i,t), and αk +i = E(ek +i,t), where eµ +i,t = ri,t −µi,t, eh +i,t = (ri,t −µi,t)2 −hi,t, es +i,t = +� +ri,t−µi,t +√ +hi,t +�3 +−si,t, + +30 +Table 7 +Out-of-sample SRs of long-short portfolios across different transaction costs (in basis points). +NASDAQ +NYSE +Method +Measure +30 +50 +75 +100 +30 +50 +75 +100 +GRACE +M +2.98 +1.48 +0.54 +-0.40 +3.10 +1.79 +1.25 +0.83 +MV +3.72 +2.88 +1.97 +0.71 +3.17 +1.80 +1.27 +0.83 +MVSK +4.39 +3.12 +2.04 +0.87 +3.25 +1.84 +1.27 +0.85 +SR +4.15 +3.46 +2.69 +1.95 +3.47 +2.45 +1.38 +0.90 +SRSK +4.81 +4.01 +3.13 +2.30 +3.48 +2.51 +1.39 +0.92 +GRACE1 +M +1.43 +1.14 +0.29 +-0.77 +2.88 +1.44 +1.01 +0.45 +MV +1.73 +1.60 +0.78 +0.44 +2.88 +1.50 +1.16 +0.46 +MVSK +4.06 +2.71 +1.63 +0.65 +2.88 +1.52 +1.19 +0.48 +SR +4.10 +3.11 +2.48 +1.23 +3.32 +2.11 +1.22 +0.92 +SRSK +4.75 +3.82 +2.53 +1.49 +3.32 +2.33 +1.24 +0.94 +GRACE2 +M +0.01 +-1.28 +-2.33 +-2.89 +-0.46 +-0.78 +-1.33 +-1.87 +MV +0.43 +-0.27 +-1.85 +-2.19 +-0.18 +-0.73 +-0.98 +-1.47 +MVSK +0.78 +-0.21 +-1.71 +-1.97 +0.03 +-0.68 +-0.90 +-1.32 +SR +0.50 +0.04 +-0.36 +-0.75 +0.28 +-0.40 +-0.74 +-1.29 +SRSK +0.72 +0.26 +-0.31 +-0.48 +0.45 +-0.37 +-0.72 +-1.22 +and ek +i,t = +� +ri,t−µi,t +√ +hi,t +�4 +− ki,t for i = 1, ..., N. Based on the estimates {�eµ +i,t}T +t=1 with �eµ +i,t = ri,t − �µi,t, we +adopt the classical Student’s t test Tµ +i to detect the null hypothesis Hµ +i : αµ +i = 0. If Hµ +i is not rejected +by Tµ +i at the significance level α∗, then we regard that {�µi,t}T +t=1 is valid. Similarly, the Student’s t +tests Th +i , Ts +i, and Tk +i to detect the null hypotheses Hh +i : αh +i = 0, Hs +i : αs +i = 0, and Hk +i : αk +i = 0, +respectively, can be used to examine the validity of {�µi,t}T +t=1, {�hi,t}T +t=1, {�si,t}T +t=1, and {�ki,t}T +t=1. +We apply each Student’s t test to check the validity of conditional moments of stock i at the +significance level α∗ ∈ {1%, 5%, 10%}, and then report the percentage of stocks having valid con- +ditional moments in Table 8. From this table, we first find that the percentage of stocks having +valid conditional moments during the out-of-sample period is higher than that during the in-sample +period in most cases. This may indicate that none of the methods has the problem of overfitting. + +31 +Date +Return +MSFT +Google +Date +Conditional Mean +Date +Conditional Variance +Date +Conditional Skewness +Date +Conditional Kurtosis +Fig 3: The plots of ri,t, �µi,t, �hi,t, �si,t and �ki,t for MSFT LLC and Google LLC from January 2, 2013 +to December 8, 2017. +Next, except for the out-of-sample results of Ts +i in NYSE, all of the testing results show that both +GRACE and GRACE1 methods deliver much better estimated and predicted conditional moments +than the GRACE2 method. This statistically explains why both GRACE and GRACE1 methods can +select better portfolios than the GRACE2 method. Moreover, we observe that the GRACE method +performs better than GARCE1 method according to the results at the significance level of 5% and +10%, although both methods have the same results at the significance level of 1%. This advantage of +GRACE method over GARCE1 method shows the necessity of using factor-augmented hypergraph +for learning conditional moments. +4. Concluding Remarks. +This paper proposes a new GRACE method for big portfolio selection +under different performance measures that are defined by four conditional moments (with respect to +mean, variance, skewness, and kurtosis) of stock returns. The GRACE method builds on the FTGCN +and the QCM method: The former embeds the factor-augmented hypergraph within a graph neural +network to obtain the estimates of mean and quantiles, and the latter transforms the estimates of + +32 +Table 8 +Percentages of stocks having valid conditional moments. +NASDAQ +NYSE +GRACE +GRACE1 +GRACE2 +GRACE +GRACE1 +GRACE2 +Test +1% +5% 10% +1% +5% 10% +1% +5% 10% +1% +5% 10% +1% +5% 10% +1% +5% 10% +Panel A: In-sample period +Tµ +i +95.0 71.8 +57.6 +95.0 71.7 +56.6 +87.2 65.3 +49.0 +96.4 83.4 +75.4 +96.4 82.8 +71.9 +78.1 75.9 +70.0 +Th +i +79.4 72.0 +69.0 +79.4 71.4 +67.0 +71.8 64.7 +61.0 +82.9 77.5 +74.3 +82.9 76.8 +73.2 +73.8 67.9 +65.5 +Ts +i +86.7 82.1 +73.4 +86.7 80.6 +71.5 +77.2 73.3 +68.0 +86.6 82.7 +76.8 +86.6 81.4 +75.1 +75.6 63.6 +61.2 +Tk +i +90.3 73.9 +61.2 +90.3 73.6 +59.6 +84.2 72.3 +60.5 +94.7 83.2 +76.6 +94.7 82.6 +71.0 +72.2 71.3 +65.8 +Panel B: Out-of-sample period +Tµ +i +94.0 85.7 +78.3 +94.0 83.4 +74.9 +79.3 77.1 +65.8 +93.2 80.9 +73.9 +93.2 78.8 +66.8 +86.8 71.3 +50.8 +Th +i +94.1 86.4 +82.4 +94.1 86.4 +80.8 +82.6 78.7 +72.1 +93.4 89.4 +86.2 +93.4 87.3 +82.8 +88.5 81.4 +66.4 +Ts +i +89.6 82.3 +74.8 +89.6 82.0 +73.2 +92.8 78.2 +76.5 +71.5 66.8 +61.4 +71.5 63.7 +56.1 +85.6 78.9 +72.6 +Tk +i +99.3 94.9 +86.1 +99.3 94.9 +84.5 +98.5 92.1 +71.1 +99.0 94.6 +85.9 +99.0 94.6 +85.9 +71.3 72.1 +62.4 +quantiles into those of higher-order moments. +The most attractive feature of the GRACE method is its capacity to estimate conditional variance, +skewness, and kurtosis of high-dimensional stock returns, so the big portfolios under four performance +measures MV, MVSK, SR, and SRSK can be constructed from thousands of stocks or even more. +There are two reasons leading to this feature. First, the FTGCN takes the domain knowledge on S2S +relations and the asset pricing knowledge on F2S relations to form the factor-augmented hypergraph. +The knowledge of S2S and F2S relations is sparse to make estimation tractable in high-dimension, +and at the same time, it is informative to capture the interplay between two stocks as well as the +driving force from common factors in all stocks. Second, the QCM method only needs the estimates +of quantiles to obtain those of higher-order moments, and the quantiles can be estimated by a super- +vised learning without assuming any distribution of stock returns. If we intend to estimate variance, +skewness, and kurtosis directly, a specific distribution of stock returns is inevitably needed, as done +by the traditional method based on high-dimensional GARCH models. However, due to the pres- +ence of distribution of stock returns, the estimation of high-dimensional GARCH models becomes +computationally infeasible when the dimension of stocks is large. + +33 +The importance of our GRACE method is further demonstrated by the empirical studies in NAS- +DAQ and NYSE. From an economic viewpoint, we find that the SRSK portfolio selected by the +GRACE method earns the out-of-sample SR of 4.81 and 3.48 in NASDAQ and NYSE, respectively, +and it beats the M portfolio (with the out-of-sample SR of 1.43 in NASDAQ and 2.87 in NYSE) +selected by the benchmark method in Feng et al. (2019) by a wide margin. Moreover, we find that +regardless of the performance measure, the portfolios from the GRACE method have more stable and +larger values of out-of-sample SR than those from the competing methods, across different settings of +hyperparameter and transaction cost. From a statistical viewpoint, we find that the predicted condi- +tional moments from the GRACE method are largely valid and capable of reflecting the comovement +between linked stocks. +In future, our GRACE method can be extended in several directions. First, it is interesting to +design some new hypergraphs to incorporate subjective information from the experiences and beliefs +of investors) or unstructured data information from financial news and social media contents (Ke, +Kelly and Xiu, 2019; Fan, Xue and Zhou, 2021). Second, it is intriguing to examine whether other +observed factors or data-driven factors (Giglio, Kelly and Xiu, 2022) are useful for specifying the +F2S relations. 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A regression framework for learning ranking functions using +relative relevance judgments. In Proceedings of the 30th annual international ACM SIGIR conference on Research +and development in information retrieval 287–294. +Zhu, X., Pan, R., Li, G., Liu, Y. and Wang, H. (2017). Network vector autoregression. Annals of Statistics 45 +1096–1123. +Zhu, X., Wang, W., Wang, H. and H¨ardle, W. K. (2019). Network quantile autoregression. Journal of Econometrics +212 345–358. + diff --git a/eNFKT4oBgHgl3EQfAC2F/content/tmp_files/load_file.txt b/eNFKT4oBgHgl3EQfAC2F/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f55aff7a740287087ffc576da5e841cb34e71d05 --- /dev/null +++ b/eNFKT4oBgHgl3EQfAC2F/content/tmp_files/load_file.txt @@ -0,0 +1,1752 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf,len=1751 +page_content='BIG PORTFOLIO SELECTION BY GRAPH-BASED CONDITIONAL MOMENTS METHOD Zhoufan Zhu 1, Ningning Zhang 2 and Ke Zhu∗ 2 1Shanghai University of Finance and Economics and 2University of Hong Kong How to do big portfolio selection is very important but challenging for both researchers and practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' In this paper, we propose a new graph-based conditional moments (GRACE) method to do portfolio selection based on thousands of stocks or more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The GRACE method first learns the conditional quantiles and mean of stock returns via a factor-augmented tem- poral graph convolutional network, which guides the learning procedure through a factor- hypergraph built by the set of stock-to-stock relations from the domain knowledge as well as the set of factor-to-stock relations from the asset pricing knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Next, the GRACE method learns the conditional variance, skewness, and kurtosis of stock returns from the learned conditional quantiles by using the quantiled conditional moment (QCM) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The QCM method is a supervised learning procedure to learn these conditional higher-order mo- ments, so it largely overcomes the computational difficulty from the classical high-dimensional GARCH-type methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Moreover, the QCM method allows the mis-specification in modeling conditional quantiles to some extent, due to its regression-based nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Finally, the GRACE method uses the learned conditional mean, variance, skewness, and kurtosis to construct several performance measures, which are criteria to sort the stocks to proceed the portfo- lio selection in the well-known 10-decile framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' An application to NASDAQ and NYSE stock markets shows that the GRACE method performs much better than its competitors, particularly when the performance measures are comprised of conditional variance, skewness, and kurtosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' ∗Address correspondence to Ke Zhu: Department of Statistics and Actuarial Science, University of Hong Kong, Hong Kong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' E-mail: mazhuke@hku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='hk Keywords: Asset pricing knowledge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Big data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Big portfolio selection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Domain knowledge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' High-dimensional time series;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Machine learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Quantiled conditional moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='11697v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='ML] 27 Jan 2023 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' In conjunction with the huge growth of stock market capitalization, the num- ber of existing stocks in the financial market is increasing rapidly nowadays, raising a big challenge to researchers and practitioners on how to do portfolio selection based on thousands of stocks or more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Suppose there are N different stocks with their prices over T timepoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Let ri,t denote the return of individual stock i at time t with the conditional mean µi,t ≡ E (ri,t | Ft−1), where i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', N, t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', T, and Ft ≡ σ (ri,s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', N, s ≤ t) is the available information set up to time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Conven- tionally, all considered N stocks are sorted into 10 deciles according to their predicted values of µi,t (from the smallest to the largest), and then the long-short portfolio is constructed at time t − 1 by buying the 10% highest ranking stocks (decile 10) and selling the 10% lowest (decile 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' see, for ex- ample, Gu, Kelly and Xiu (2020, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' However, this conventional way using µi,t as the performance measure to sort the stocks has two major shortcomings: First, it ignores the impact of conditional variance hi,t ≡ Var (ri,t | Ft−1), which is the risk of uncertainty for guiding portfolio selection under the mean-variance criterion (Markowitz, 1952) or Sharpe ratio criterion (Sharpe, 1994);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Second, it does not accommodate the observation that rational investors prefer assets with higher skewness and lower kurtosis in the market (Scott and Horvath, 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Dittmar, 2002), implying the necessity of considering the conditional skewness si,t ≡ Skew (ri,t | Ft−1) for the asymmetry risk and conditional kurtosis ki,t ≡ Kurt (ri,t | Ft−1) for the tail risk to proceed the portfolio selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' To overcome the two shortcomings aforementioned, we aim to sort the stocks for portfolio selection by using the performance measures below: Mean–variance (MV): µi,t − λ1hi,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' (1) Mean–variance with skewness and kurtosis (MVSK): µi,t − λ1hi,t + λ2si,t − λ3ki,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' (2) Sharpe ratio (SR): µi,t/ � hi,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' (3) Sharpe ratio with skewness and kurtosis (SRSK): µi,t/ � hi,t + λ2si,t − λ3ki,t, (4) where λi, i = 1, 2, 3, are positive hyperparameters, and they determine how much penalty one needs to pay for the large values of hi,t and ki,t or how much reward one can gain for the large values of si,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' To implement these four performance measures, we need to learn hi,t, si,t, and ki,t dynamically for N stocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' When N = 1, these three higher-order conditional moments are studied by using some variants of univariate generalized autoregressive conditional heteroskedasticity (GARCH) model (Engle, 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Bollerslev, 1986);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' see, for example, Jondeau and Rockinger (2003), Le´on, Rubio and Serna (2005), Le´on and ˜N´ıguez (2020), and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' However, those univariate GARCH-type 3 methods have the risk of model mis-specification and the instability of model estimation particularly when the dynamics of si,t and ki,t are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' When N is large (say, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', N = 1000), no clear feasible manner so far has been offered in the literature to estimate hi,t, si,t, and ki,t by using high- dimensional GARCH-type models, which are formed to study the dynamics of rt ≡ (r1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', rN,t)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The cause of this dilemma is that the high-dimensional GARCH-type models have to assume a certain distribution of rt to estimate hi,t, si,t, and ki,t, however, the resulting log-likelihood function is too complex to be optimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' For example, the optimization of the commonly used Gaussian log- likelihood function needs to invert many N × N-dimensional variance-covariance matrices, and this task becomes computationally infeasible for large N cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Pakel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' (2021) propose a composite likelihood estimation (CLE) method for the parsimonious scalar BEKK model to estimate hi,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Since the CLE method is based on all pairwise Gaussian log-likelihood functions with the order O(N2), it becomes computationally burdensome when N is in thousands, and meanwhile, it may not adequately capture the dependence among stock returns at the price of pairwise technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' One way to further reduce the computational burden is to estimate hi,t by the equation-by-equation (EbE) method, as done for the parsimonious scalar DCC model (Francq and Zako¨ıan, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Engle, Ledoit and Wolf, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The EbE method is feasible in terms of computation but inefficient in terms of prediction, since it totally ignores the dependence among stock returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' This paper contributes to the literature by proposing a new graph-based conditional moments (GRACE) method for portfolio selection under four performance measures in (1)–(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The GRACE method has two core engines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Its first engine is to study the conditional quantiles of ri,t for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', N and t = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', T by a graph-based quantile model, which can be directly estimated via the quantile loss function (Koenker and Bassett, 1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Our graph-based quantile model is based on a new factor- augmented temporal graph convolutional network (FTGCN), and thus it is called the FTGCN-based quantile model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' This FTGCN-based quantile model uses the stock and factor features to extract both temporal and spatial information for all stocks, and then takes the extracted information to learn the conditional quantiles under the guidance of a factor-augmented hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The factor- augmented hypergraph is neither random nor time-variant, and it combines the domain knowledge of the multiple types of relation between any two stocks and the asset pricing knowledge of the impact of common factors on all stocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Our factor-augmented hypergraph has a linkage with the hypergraph in TGCN (Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', 2019) that also exploits the domain knowledge to build the graph structure among stocks, where the domain knowledge comes from the public information of the stocks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', 4 the industrial background, financial statement, and shareholder information), and its usefulness has been well documented by Livingston (1977), Cohen and Frazzini (2008), Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' (2019), Burt and Hrdlicka (2021), and many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' However, the hypergraph in TGCN overlooks an important fact from the asset pricing literature that some common factors can globally affect all stocks in the market (Fama and French, 1993, 2015, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Griffin, 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Hou, Karolyi and Kho, 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' This asset pricing knowledge is obviously as informative as the domain knowledge, and it motivates the design of our factor-augmented hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Using the similar idea above, our GRACE method further proposes a FTGCN-based mean model to estimate µi,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Based on the estimated conditional quantiles of ri,t at K different quantile levels from our FTGCN- based quantile model, the second engine of our GRACE method is to estimate hi,t, si,t, and ki,t via their corresponding quantiled conditional moments (QCMs) in Zhang and Zhu (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The QCM method estimates hi,t, si,t, and ki,t through the ordinary least squares (OLS) estimator of a linear regression model, which is constructed by those estimated conditional quantiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The formulation of this linear regression model stems naturally from the Cornish-Fisher expansion (Cornish and Fisher, 1938), which exhibits a fundamental relationship between conditional quantiles and conditional mo- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' In principle, the QCM method transforms the estimation of hi,t, si,t, and ki,t to that of conditional quantiles, and this brings us two substantial advantages over the GARCH-type method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' First, the QCM method is easy-to-implement as long as the estimated conditional quantiles of ri,t are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Note that our FTGCN-based quantile model can estimate conditional quantiles of ri,t for large N and T cases by a supervised learning through the use of quantile loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Therefore, unlike the estimation of high-dimensional GARCH-type models, no assumption on the distribution of rt is needed to estimate our FTGCN-based quantile model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' This is the reason why the QCM method can make the estimation of higher-order moments feasible for large N cases, although it needs to estimate the quantile model K different times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Second, the QCM method largely alleviates the risk of model mis-specification, since the QCMs of hi,t, si,t, and ki,t are proposed without any estimator of µi,t, and more importantly, they are consistent even when the conditional quantile estimators of ri,t are biased to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' In this sense, our FTGCN-based quantile model could generate consistent QCMs, as long as its specification does not largely deviate from the true specification of conditional quantile of ri,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' We apply our GRACE method to construct long-short portfolios based on 1026 and 1737 stocks in NASDAQ and NYSE, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' To build the factor-augmented hypergraph, we use the Wiki 5 company-based relations (Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', 2019) as the domain knowledge to specify the multiple types of relation between any two stocks, and at the same time, we take the Fama-French five factors (Fama and French, 2015) as the asset pricing knowledge to capture the impact of common factors on all stocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' From an economic viewpoint, our empirical results are encouraging in four aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' First, all of the MV, MVSK, SR, and SRSK portfolios have larger values of out-of-sample annualized SR than the M portfolio in the GRACE method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Second, the SRSK portfolio from the GRACE method performs the best, and its values of out-of-sample annualized SR are 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='81 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='48 in NASDAQ and NYSE, re- spectively, which are 236% and 21% higher than those of the M portfolio from the benchmark method in Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Third, the GRACE method always dominates the simple GRACE method in portfolio selection by a wide margin, where the simple GRACE method adopts the linear structure (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', 2017, 2019) to extract the information from stock and factor features to learn the condi- tional quantiles and mean of stock returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Fourth, regardless of performance measure, the portfolios from the GRACE method have a more robust performance than those from its competitors over the set of stock-to-stock relations, the choice of hyperparameters, and the level of transection cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' All of these aforementioned findings indicate the importance of using the higher-order conditional moments to form the performance measure, the asset pricing knowledge to build the hypergraph, and the network structure to extract the feature information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' From a statistical viewpoint, the condi- tional moments learned from the GRACE method are largely valid and better than those from other competing methods, shedding light on the advantage of GRACE method in portfolio selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The remaining paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Section 2 presents our entire methodology, including the network architecture of FTGCN, the training procedure of FTGCN-based quantile and mean models, the formal estimation procedure of the QCMs, and the implementation details of the GRACE method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Section 3 presents our empirical studies of big portfolio selection in NASDAQ and NYSE stock markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Concluding remarks are offered in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Graph-based Learning for Conditional Quantiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Let Qt(τ) = (Q1,t(τ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', QN,t(τ))′ be the high-dimensional vector of τ-th conditional quantiles, where Qi,t(τ) is the τ-th conditional quantile of ri,t given Ft−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' In this paper, we study Qt(τ) by a new FTGCN-based quantile model defined as Qt(τ) = f(Xt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' G, θτ), (5) 6 where Xt−1 ∈ R(N+B)×P×S is a feature tensor built on Ft−1 including the information of N stocks and B factors, and f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' G, θτ) : R(N+B)×P×S → RN×1 is the FTGCN depending on a factor- augmented hypergraph G and a vector of unknown parameters θτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Here, Xt−1 = [X1,t−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', XN,t−1, XN+1,t−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', XN+B,t−1] with Xi,t−1 ∈ RP×S having its s-th column xi,t−1−S+s ∈ RP , where Xi,t−1 for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', N is the feature matrix for stock i, XN+b,t−1 for b = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', B is the feature matrix for factor b, P with a potential high dimension is the number of stock or factor features, S is the number of lagged values of each feature, and xi,t (or xN+b,t) is the vector of P different features of stock i (or factor b) at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Below, we show the four modules to construct the FTGCN f(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' G, θτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Module I: Feature Extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' In the first module, we employ a one-layer long short-term memory (LSTM) network (Hochreiter and Schmidhuber, 1997) to extract the temporal embedding xL i,t ∈ Rd from the feature matrix Xi,t−1 at time t − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Specifically, we let xs i,t−1 = xi,t−1−S+s and compute the hidden state vectors hs i,t, s = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' recursively from the LSTM network: zs i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t = tanh(W1xxs i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t−1 + W1hhs−1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t + b1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' is i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t = sigmoid(W2xxs i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t−1 + W2hhs−1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t + b2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' f s i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t = sigmoid(W3xxs i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t−1 + W3hhs−1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t + b3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' cs i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t = f s i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t ⊙ cs−1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t + is i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t ⊙ zs i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' os i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t = sigmoid(W4xxs i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t−1 + W4hhs−1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t + b4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' hs i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t = os i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t ⊙ tanh(cs i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' (6) where W1x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' W2x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' W3x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' W4x ∈ Rd×P and W1h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' W2h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' W3h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' W4h ∈ Rd×d are matrices of weight pa- rameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' b1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' b2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' b3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' b4 ∈ Rd are vectors of bias parameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' d is the number of hidden units control- ling the network complexity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' tanh(·) and sigmoid(·) are two entry-wise vector-valued functions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' ⊙ is element-wise production operation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' and the initial values c0 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t and h0 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t are conventionally set to the d-dimensional vector of zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Then, our temporal embedding xL i,t is taken as the last hidden state vector hS i,t (the output of LSTM network), that is, xL i,t = hS i,t = h(Xi,t−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' θL) ∈ Rd, (7) where h(·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' θL) is the LSTM network in (6) indexed by θL, and θL contains all the parameters in {Wjx, Wjh, bj : j = 1, 2, 3, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Clearly, the purpose of this module is to extract an expressive vector xL i,t, which stores the long-term temporal information of P features up to time t − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' It is expected 7 that all the temporal information carried by {xL 1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', xL N+B,t} can help us to predict the behavior of future returns {r1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', rN,t} at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Module II: Hypergraph Construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Besides the temporal information from the stock fea- tures, the spatial information (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', interdependence relations) among all stocks is also important for predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' For example, (i) MSFT LLC and Google LLC could have an industry-specific relation, since both of them belong to “Computer Software: Programming” industry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' and (ii) Boeing Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' and United Airlines Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' could have a corporate relation, in view of the fact that Boeing Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' produces Boeing airplanes for United Airlines Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Needless to say, these stock-to-stock (S2S) relations are informative and should not be ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' In general, we can have M different types of S2S relation (denoted by Estock = {e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', eM}) between any two stocks based on the domain knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Along with the S2S relations, the factor-to-stock (F2S) relations also exist in the market, since the stock returns can move together driven by the common factors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' see the vast evidence in the asset pricing literature (Fama and French, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Lettau and Pelger, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Gu, Kelly and Xiu, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The F2S relations convey the spatial information from factors to stocks, and they are highly possible to be factor-specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Therefore, based on the asset pricing knowledge, we consider B different F2S relations (denoted by Efactor = {eM+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', eM+B}), where the F2S relation eM+b is induced by factor b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' To describe all of S2S and F2S relations above, we build a factor-augmented hypergraph G = (V , A), (8) where V = {Vstock, Vfactor} is the set of vertices, and A = {Astock, Afactor} is the set of adjacency matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Here, Vstock = {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', N} is the set of stock vertices with the vertex i ∈ Vstock representing the stock i, Vfactor = {N + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', N + B} is the set of factor vertices with the vertex N + b ∈ Vfactor representing the factor b, Astock = {A1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', AM} is the set of adjacency matrices with the matrix Am ∈ Astock representing the S2S relation em, and Afactor = {AM+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', AM+B} is the set of adjacency matrices with the matrix AM+b ∈ Afactor representing the F2S relation eM+b, where Am has its (i, j)-th entry ai,j,m = � � � 1, if there is an S2S relation em between vertices i ∈ Vstock and j ∈ Vstock, 0, otherwise, 8 and AM+b has its (i, j)-th entry ai,j,M+b = � � � 1, if i ∈ Vstock and j = N + b ∈ Vfactor or j ∈ Vstock and i = N + b ∈ Vfactor, 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' According to the definitions of Am and AM+b, the factor-augmented hypergraph G ensures that (i) two stock vertices in Vstock are linked when they have up to M different S2S relations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' and (ii) each factor vertex in Vfactor is linked to all of stock vertices in Vstock indicating the corresponding F2S relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Since our main target is to study the dynamics of stocks rather than factors, we assume that there has no linkage between any two factor vertices in Vfactor for simplicity, and our analysis results below do not change even when the factor vertices are allowed to have connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' In sum, the overall relation between any two vertices i and j in G can be represented by the vector ai,j = (ai,j,1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', ai,j,M, ai,j,M+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', ai,j,M+B)′ ∈ RM+B, (9) where the first M entries and the remaining B entries carry the information of S2S relations and F2S relations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Module III: Hypergraph Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Having known the relations among all stocks and factors in G, it is natural to capture how much temporal information the stock i can receive from its linked stocks and factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' To fulfill this goal, we define the aggregated temporal embedding for stock i as xP i,t = � j∈Vstock,j̸=i g � ai,j, xL i,t, xL j,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' G, θP � dj xL j,t + � j∈Vfactor g � ai,j, xL i,t, xL j,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' G, θP � N xL j,t ∈ Rd, (10) where i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', N, ai,j in (9) represents the overall relation between vertices i and j, xL i,t and xL j,t are the temporal embeddings extracted by (7), g(·) : RM+B×Rd×Rd → R is a nonlinear function indexed by θP to measure the intensity of interplay between vertices i and j, and dj = �M m=1 � i̸=j ai,j,m is the number of stocks linked with stock j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Following Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' (2019), we take g � ai,j, xL i,t, xL j,t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' G, θP � = softmax � W5(xL′ i,t, xL′ j,t, a′ i,j)′ + b5 � ∈ R, (11) where W5 ∈ R1×(M+B+2d) is a vector of weight parameters, b5 ∈ R is a bias parameter, θP contains all the parameters in W5 and b5, and softmax(·) is used to normalize the value of g(·) into (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The specification of g(·) in (11) has two merits: First, it allows the intensity of interplay between any two 9 vertices to be stock-, factor-, and relation-specific;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Second, it aims to capture some missing relations that are not described in ai,j (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', ai,j ≡ 0) but presented by the similarity of xL i,t and xL j,t, since the term W5(xL′ i,t, xL′ j,t, a′ i,j)′ is still informative even when ai,j ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' As a temporal graph convolution (TGC), the third module combines the temporal embedding xL i,t in (7) and the aggregated temporal embedding xP i,t in (10) to form xTGC i,t = � xL′ i,t, xP′ i,t �′ ∈ R2d for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' (12) The advantage of using xTGC i,t is apparent, since xTGC i,t captures the spatial and temporal information of stock features simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Module IV: Quantile Output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Our last module applies a fully connected (FC) network to revise the spatial-temporal information xTGC i,t in (12) to Qt(τ) in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Let θC contain all the parameters in W6 and b6, where W6 ∈ R1×2d is a vector of weight parameters, and b6 ∈ R is a bias parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Then, we set the form of FTGCN as f(Xt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' G, θ) ≡ (f1(Xt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' G, θ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', fN(Xt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' G, θ))′ with fi(Xt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' G, θ) = W6xTGC i,t + b6 for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', N, (13) where G is the factor-augmented hypergraph in (8), and θ contains all the parameters in θL, θP , and θC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Consequently, our FTGCN-based quantile model has the specification: Qt(τ) = f(Xt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' G, θτ) with Qi,t(τ) = fi(Xt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' G, θτ) for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', N (14) (see its network architecture in Fig 1), where f(Xt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' G, θτ) and fi(Xt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' G, θτ) are defined as in (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Estimation of the FTGCN-based Quantile Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' As Qi,t(τ) is the τ-th conditional quantile of ri,t given Ft−1, we estimate θτ in (14) by the following quantile estimator: �θτ = argmin θτ 1 NT N � i=1 T � t=1 ρτ (ri,t − fi(Xt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' G, θτ)) ≡ argmin θτ 1 T T � t=1 ℓ(rt, Xt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' G, θτ, τ), (15) where ρτ(x) = x[τ −I(x < 0)] is the quantile loss function (Koenker and Bassett, 1978) with I(·) being the indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Due to the massive data volume, we adopt the adaptive moment estimation (Adam) algorithm in Kingma and Ba (2015) to compute �θτ in (15);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' see Algorithm 1 for the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Using �θτ, we obtain � Qt(τ) = f(Xt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' G, �θτ), (16) which is the estimator of Qt(τ) at the quantile level τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 10 Fig 1: The architecture of FTGCN-based quantile model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Algorithm 1 The training procedure of �θτ by the Adam algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Input: The sample: {(rt, Xt−1)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The initial value of network parameters in the FTGCN-based quantile model: θ(0) τ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The factor-augmented hypergraph: G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Hyperparameter: learning rate γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 1: l = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 2: repeat 3: (rl, Xl−1) ← draw a random data point from {(rt, Xt−1)} (A cross-sectional minibatch);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 4: g(l) τ ← ∇θτ � ℓ(rl, Xl−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' G, θ(l) τ , τ) � (Gradients of minibatch estimator);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 5: θ(l+1) τ ← update parameters using learning rate γ and gradients g(l) τ (Adam);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 6: l ← l + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 7: until convergence of parameters θ(l+1) τ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Output: The value of θ(l+1) τ , which is taken as the quantile estimator �θτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Comparison with the Existing Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Our FTGCN-based quantile model in (14) has a linkage with the TGCN-based model in Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' (2019) with regard to the network structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' As 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='0 Stock 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='0 Stock 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='0 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='. Stock N Factor 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='0 Factor B11 the pioneering work, the TGCN-based model applies the domain knowledge to construct a hypergraph for taking multiple types of S2S relation into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The main difference between the FTGCN-based quantile model and the TGCN-based model is two-fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' First, the FTGCN-based quantile model aims to learn the conditional quantiles of ri,t, whereas the TGCN-based model focuses on the conditional mean of ri,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Second, the FTGCN-based quantile model incorporates the F2S relations to build the factor-augmented hypergraph, but the TGCN-based model does not consider this kind of important information in its hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Besides our FTGCN-based quantile model, many other models are existing in the literature to study the conditional quantile of high-dimensional data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' see, for example, Koenker (2004), Kato, Galvao and Montes-Rojas (2012), and Galvao and Kato (2016) for the quantile individual fixed effects models, Ando and Bai (2020), Chen, Dolado and Gonzalo (2021), Ma, Linton and Gao (2021), and Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' (2022) for the quantile factor models, and H¨ardle, Wang and Yu (2016), Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' (2019), and Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' (2022) for the quantile network models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' However, except for the factor-augmented dynamic network quantile regression (FDNQR) model in Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' (2022), none of the aforementioned models takes the domain knowledge and asset pricing knowledge simultaneously into account to guide the estimation of conditional quantile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Specifically, the FDNQR model uses the domain knowledge to propose a weighted adjacency matrix W ∈ RN×N with the (i, j)-th entry wi,j, where wi,j = ai,j/ni, ni = �N j=1 ai,j, ai,j = 1 if the stock i has the connection with another stock j, and ai,j = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Based on W , the FDNQR model assumes Qi,t(τ) = ατ + β′ τzi + γτ N � j=1 wi,jrj,t−1 + ζτri,t−1 + S � s=1 ς′ s,τFt−s, (17) where ατ ∈ R, βτ ∈ RQ×1, γτ ∈ R, and ςs,τ ∈ RB×1 are quantile regression coefficients, zi ∈ RQ×1 is a Q-dimensional vector of time-invariant stock features, and Ft = (f1,t, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', fB,t)′ ∈ RB×1 is a B- dimensional vector of time-variant factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' In model (17), ατ is the constant intercept term for all stocks, βτ is the constant intensity of the impact from stock features on stock i, γτ is the constant intensity of spatial impact on stock i caused by its connected stocks, ζτ is the constant intensity of temporal impact on stock i caused by its lagged term, and ςs,τ is the constant intensity of factor impact on all stocks caused by the lagged factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Clearly, our FTGCN-based quantile model is much more general than model (17), since it captures multiple types of relation separately, extracts the information of time-variant stock and factor features in a non-linear way, and allows for the time- variant heterogenous intensity of spatial and temporal impacts on each stock caused by either its 12 connected stocks or factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Note that model (17) does not include the contemporaneous variables in the original FDNQR model for the purpose of prediction, and it nests the network quantile autoregressive model in Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' As the contemporaneous variables are absent, model (17) now can be consistently estimated by using the quantile loss function as in Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Graph-based Learning for Conditional Mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' So far, we have introduced the FTGCN to learn the conditional quantile of ri,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Following the similar idea, we can learn the conditional mean of ri,t by an FTGCN-based mean model: ri,t = f(Xt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' G, θµ) + εi,t, (18) where f(Xt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' G, θµ) is defined as in (13), and εi,t is the error term with zero mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Note that model (18) reduces to the TGCN model in Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' (2019), when the factors and their features are absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' To estimate model (18), we consider the penalized least squares (PLS) estimator of θµ given by �θµ = argmin θµ 1 T T � t=1 � 1 N N � i=1 [ri,t − fi(Xt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' G, θµ)]2 + λ∗ N2 N � i=1 N � j=1 max {0, −[fi(Xt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' G, θµ) − fj(Xt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' G, θµ)](ri,t − rj,t)} � ≡ argmin θµ 1 T T � t=1 ℓµ(rt, Xt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' G, θµ, λ∗), (19) where the penalty term is utilized to ensure that the orders of ri,t and rj,t do not deviate largely from those of their predicted values, and λ∗ is a positive hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' As for �θτ, we adopt the Adam algorithm to compute �θµ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' see Algorithm 2 for the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Using �θµ, we then estimate the conditional mean µi,t by �µi,t = fi(Xt−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' G, �θµ) for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' (20) It is worthwhile mentioning that the order-preserving penalty in (19) has been widely adopted in the literature to improve the learning efficiency for conditional mean prediction (Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Socher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' However, this penalized method is inappropriate for the conditional quantile estimation, since Qi,t(τ) does not tend to be larger than Qj,t(τ) when ri,t is larger than rj,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 13 Algorithm 2 The training procedure of �θµ by the Adam algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Input: The sample: {(rt, Xt−1)};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The initial value of network parameters in the FTGCN-based mean model: θ(0) µ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The factor-augmented hypergraph: G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Hyperparameters: λ∗, learning rate γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 1: l = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 2: repeat 3: (rl, Xl−1) ← draw a random data point from {(rt, Xt−1)} (A cross-sectional minibatch);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 4: g(l) ← ∇θµ � ℓµ(rl, Xl−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' G, θ(l) µ , λ∗) � (Gradients of minibatch estimator);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 5: θ(l+1) µ ← update parameters using learning rate γ and gradients g(l) (Adam);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 6: l ← l + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 7: until convergence of parameters θ(l+1) µ ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Output: The value of θ(l+1) µ , which is taken as the PLS estimator �θµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The QCM Learning for Higher-order Conditional Moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Let � Qt(τ1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', � Qt(τK) be the vectors of estimated conditional quantiles at K different quantile levels τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', τK, where � Qt(τk) ≡ ( �Q1,t(τk), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', �QN,t(τk))′ for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', K are computed as in (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Below, we elaborate on how to estimate hi,t, si,t, and ki,t by the QCM method in Zhang and Zhu (2022) for the fixed values of i and t, based on �Qi,t(τ1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', �Qi,t(τK).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The QCM method is motivated by the Cornish-Fisher expansion (Cornish and Fisher, 1938), which shows the following fundamental linkage between conditional quantiles and conditional moments: Qi,t(τk) = µi,t + z(τk) � hi,t + � z(τk)2 − 1 � � hi,tsi,t 6 + � z(τk)3 − 3z(τk) � � hi,t(ki,t − 3) 24 + � hi,tωi,t(τk) (21) for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', K, where z(τk) is the τk-th quantile of standard normal distribution, and ωi,t(τk) is the remainder of this expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Define ε• i,t,k = ε∗ i,t,k + ε◦ i,t,k with ε∗ i,t,k = � hi,tωi,t(τk) and ε◦ i,t,k = �Qi,t(τk) − Qi,t(τk), Zk = � z(τk), z(τk)2 − 1, z(τk)3 − 3z(τk) �′, βi,t ≡ (βi,t,1, βi,t,2, βi,t,3)′ = �� hi,t, � hi,tsi,t 6 , � hi,t(ki,t − 3) 24 �′ , where �Qi,t(τk) is the estimator of Qi,t(τk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Then, we can rewrite (21) as follows: �Qi,t(τk) = µi,t + Z′ kβi,t + ε• i,t,k for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', K, (22) 14 where ε• i,t,k is the gross error containing the expansion error ε∗ i,t,k and the quantile estimation error ε◦ i,t,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Clearly, the equation (22) is a linear regression model with the response variable �Qi,t(τk), explanatory variables Zk, parameter vector (µi,t, β′ i,t)′, and error term ε• i,t,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Since ε• i,t,k may not have zero mean for model identification, we add an additional deterministic intercept term γi,t into the equation (22) to form the following linear regression model: �Qi,t(τk) = (µi,t + γi,t) + Z′ kβi,t + εi,t,k ≡ ¯Z′ kθi,t + εi,t,k for k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', K, (23) where εi,t,k = ε• i,t,k − γi,t, ¯Zk = (1, Z′ k)′, and θi,t = (βi,t,0, β′ i,t)′ with βi,t,0 = µi,t + γi,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Let Yi,t be a K × 1 vector with entries �Qi,t(τk), ¯Z be a K × 4 matrix with rows ¯Z′ k, and εi,t be a K × 1 vector with entries εi,t,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Then, the ordinary least squares (OLS) estimator of θi,t in (23) is �θi,t ≡ (�βi,t,0, �β′ i,t)′ = ( ¯Z′ ¯Z)−1 ¯Z′Yi,t, (24) where �βi,t = (�βi,t,1, �βi,t,2, �βi,t,3)′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Zhang and Zhu (2022) show that �θi,t −→ θi,t in probability as K → ∞ under the following two classical conditions in the regression literature: Condition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' ¯Z′ ¯Z is positive definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Condition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' ¯Z′εi,t/K −→ 000 in probability as K → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Consequently, by the continuous mapping theorem, we have (25) �hi,t ≡ �β 2 i,t,1 −→ hi,t, �si,t ≡ 6�βi,t,2 �βi,t,1 −→ si,t, and �ki,t ≡ 24�βi,t,3 �βi,t,1 + 3 −→ ki,t in probability as K → ∞, where �hi,t, �si,t, and �ki,t are the QCMs of hi,t, si,t, and ki,t, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' In order to make sure that �hi,t, �si,t, and �ki,t are moments under certain distribution of ri,t, two necessary constraints are required: �hi,t ≥ 0 and �ki,t ≥ �s2 i,t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Clearly, the first constraint holds automatically, and the second constraint can be checked directly based on the values of �ki,t and �si,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' If the second constraint does not hold, we can easily replace �θi,t in (24) by a constrained least squares estimator, so that the resulting �ki,t and �si,t satisfy this constraint;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 15 see more detailed discussions in Zhang and Zhu (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Moreover, it should be noted that we are unable to estimate µi,t by the QCM method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The reason is that µi,t can not be estimated by �βi,t,0 in (24) due to the presence of γi,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Therefore, we have to estimate µi,t separately by other methods (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', the graph-based method in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='2 above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' As we observed, the core idea of QCM method is to transform the estimation of conditional moments to that of conditional quantiles, giving us two remarkable advantages particularly in the realm of high- dimensional data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' First, the QCM method is easy-to-implement, since it only requires the estimated conditional quantiles as the input to compute the OLS estimator �θi,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' When N is large, a direct estimation for the higher-order conditional moments hi,t, si,t, and ki,t via high-dimensional GARCH-type models is computationally infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The reason is that the high-dimensional GARCH-type models are fitted by the QML estimation method, which relies on a certain distribution of rt to write down the log- likelihood function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' However, the log-likelihood function is too complex to be optimized for large N cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' For example, the often used Gaussian log-likelihood function depends on the inverse of many N × N-dimensional variance-covariance matrices, making its optimization infeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Transforming the estimation of higher-order moments to that of quantiles circumvents this annoying difficulty, since the estimation of quantiles is a classical supervised learning but that of higher-order moments is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' To be more specific, the supervised learning is a machine learning paradigm, and it aims to learn a function f0 that maps features (say, x) to labels (say, y) supervised by a certain loss function without assuming the distribution of x or y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' For example, the τ-th quantile of y can be learned by f0(x) using the quantile loss function ρτ(y − f0(x)) as the supervisor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' However, it is unclear how to design an appropriate loss function as the supervisor for learning the variance, skewness, or kurtosis of y by f0(x), unless certain distributional assumption is made for x or y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' This indicates that the direct estimation of hi,t, si,t, and ki,t has to rely on a certain distribution of ri,t, as done by the QML estimation in the high-dimensional GARCH-type models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Owing to the supervised learning feature of quantiles, our indirect estimation of hi,t, si,t, and ki,t from the QCM method does not need any distributional assumption of ri,t, so it bypasses the computational difficulty raised in the direct estimation method to deal with large N cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Second, the QCM method can largely reduce the risk of model mis-specification, since �hi,t, �si,t, and �ki,t are simultaneously computed without any prior estimation of µi,t, and their consistency holds even when the specification of Qi,t(τ) is mis-specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' This advantage is far beyond our expectations, since 16 normally we have to first estimate µi,t and then hi,t, si,t, and ki,t using some parametric models that are needed to be correctly specified to generate consistent estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The reason leading to this advantage is that the QCM method is regression-based, so that the impact of µi,t is eliminated by absorbing it into the intercept term and the consistency of �hi,t, �si,t, and �ki,t is ensured by Conditions 1–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Note that Condition 1 holds for the often choices of quantile sequence {τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', τK}, and Condition 2 allows each model error εi,t,k (including the quantile estimation error ε◦ i,t,k) to deviate from zero to some extent, as long as the averages of {εi,t,k}, {εi,t,kz(τk)}, {εi,t,k[z(τk)2 − 1]}, and {εi,t,k[z(τk)3 − 3z(τk)]} across k are close to zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' see Zhang and Zhu (2022) for more discussions on this aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Implementation Details of the GRACE Method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Due to the use of FTGCN, the GRACE method first needs to alleviate the risk of overfitting, a prevalent deficiency of the neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Following the standard approach to circumvent overfitting, we chronologically partition the full data into three disjoint parts: training sample, validation sample, and testing sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The training and validation samples are taken to do parameter estimation, and the testing sample is used to evaluate the truly out-of-sample performance of the GRACE method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' To be more specific, we compute θ(l) τ at l-th iteration in Algorithm 1 based on the training sample, and then calculate its corresponding validation sample error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Here, the validation sample error is the value of the objective function in (15) based on the validation sample and θτ = θ(l) τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' To regularize against overfitting, we utilize the early stopping method to terminate the iteration process early in Algorithm 1 when the validation sample error increases for several iterations, and select the estimator θ(l) τ having the smallest validation sample error as the quantile estimator �θτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Similarly, the PLS estimator �θµ is computed from the training and validation samples under Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Next, the GRACE method uses �θτk and �θµ to predict the values of �Qi,t(τk) and µi,t on the test- ing sample, respectively, where τk = k/(K + 1), k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', K, for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' In the large pool of �Qi,t(τk), some of �Qi,t(τk) are inevitable to be invalid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Intuitively, it is reasonable to exclude those invalid �Qi,t(τk) for the computation of QCMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' To achieve this goal, we make use of the uncondi- tional coverage test LRuc in Kupiec (1995) and conditional coverage test LRcc in Christoffersen (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Specifically, we compute �Qi,t(τk) on the training and validation samples, and apply LRuc and LRcc to detect whether the sequence of estimated conditional quantiles Qi(τk) ≡ { �Qi,t(τk) : t ∈ training and validation samples} is valid at the significance level α for each stock i and quantile level 17 τk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Then, we build a valid quantile level set for stock i: (26) Ωi = {τk : the validity of Qi(τk) is accepted by both LRuc and LRcc at the level α}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' That is, Ωi groups all of those quantile levels τk, for which the sequence Qi(τk) is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Clearly, Ωi depends on α and K jointly, where its size (denoted by |Ωi|) is decreasing with the value of α while increasing with the value of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' In particular, we know that |Ωi| = K when α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' After having Ωi, we use the predicted values { �Qi,t(τk) : τk ∈ Ωi} to predict the values of hi,t, si,t, and ki,t for stock i via the related QCMs on the testing sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' As K is essentially replaced by |Ωi| from the above manipulation, we need a large value of |Ωi| to ensure the consistency of the QCMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' This motivates us to discard those stocks having the value of |Ωi| less than a predetermined tolerance K0 (say, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', K0 = 30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Finally, the GRACE method employs different performance measures from the predicted values of µi,t, hi,t, si,t, and ki,t to construct portfolios, based on all of remaining stocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Empirical Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' We apply our GRACE method to construct portfolios based on the stocks in two major exchanges: NASDAQ and NYSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The stock data we consider are the same as those in Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' (2019), and they contain daily prices from January 2, 2013 to December 8, 2017 for 1026 and 1737 stocks in NASDAQ and NYSE, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Alone with the stock price data, we also take the S2S relation data in Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' (2019) to describe the multiple types of S2S relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Based on the domain knowledge, the S2S relations can be divided into two groups: Sector-industry relations Esi stock and Wiki company-based relations Ewiki stock (see the Appendix A of Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' (2019) for their detailed definitions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Specifically, two stocks (say, stock i and stock j) have a sector-industry relation if they belong to the same industry, where the industries are classified by the GICS standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' For example, all 1026 stocks in NASDAQ are divided into 13 different sectors, where each sector contains several industries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' see Fig 2(a) for the sector-industry hierarchy of all 1026 stocks in this market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' From this figure, we know that MSFT LLC and Google LLC have an S2S relation since they belong to the same industry “Computer Software: Programming”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Meanwhile, two stocks can also have a Wiki company-based relation if they have an either first-order or second-order relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The first-order and second-order relations have the format of “company i R −→ company j” and “company i R1 −→ entity k R2 ←− company j”, respectively, where the companies i and j 18 Fig 2: Examples of the sector-industry, first-order and second-order relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' bridged by an entity k are corresponding to the stocks i and j, respectively, and the relations R, R1, and R2 are defined in Wikidata (https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='wikidata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='org/wiki/Wikidata:List of properties/all).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' It turns out that there are 5 and 53 different types of first-order and second-order relations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Fig 2(b) and 2(c) give some illustrating examples on the first-order and second-order relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' We see from this figure that BlackRock Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' has a first-order relation with Netflix Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' since BlackRock Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' owns Netflix Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', and United Airlines Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' and Boeing Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' have a second-order relation since Boeing Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' produces Boeing 747 that is sold to United Airlines Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' In sum, Table 1 lists the number of S2S relation types and the ratio of S2S relations to all possible stock pairs in NASDAQ and NYSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Since the ratio of S2S relations is always less than 10%, it indicates that the S2S relations in both markets are sparse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Table 1 Summary of S2S relations Sector-industry relation Wiki company-based relation Market Relation types Relation ratio (pairwise) Relation types Relation ratio (pairwise) NASDAQ 112 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='00% 42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='21% NYSE 130 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='37% 32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='30% Technology Finance Computer Software: Data Processing Programming19 In addition, the F2S relation data are also needed to facilitate our GRACE method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Among a great variety of factors, we use the prevalent daily Fama-French five factors (Fama and French, 2015) to spec- ify the related F2S relations Eff5 factor, based on the asset-pricing knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' These five factors are excess market return, RMW, HML, SMB, and CMA, and their daily data can be downloaded from the home- page of Kenneth French (https://mba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='tuck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='dartmouth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='edu/pages/faculty/ken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='french/data library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='html).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' For stock i, we let ri,t be its 1-day return and r(k∗) i,t = 1 k∗ �k∗ s=1 ri,t+1−s be its k∗-day moving average of returns at day t, and use the OLS method to calculate its 5-dimensional vector of factor exposures (denoted by λi,t = (λ(1) i,t , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', λ(5) i,t )′) based on the sample of 1-day returns in a half-year rolling window up to day t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' For factor b, we similarly let fb,t be its value, f(k∗) b,t be its k∗-day moving average, and ¯λ(b∗) b,t be its exposure on factor b∗ at day t, where we assume that factor b has one exposure on itself and zero exposure on other factors (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', b∗ ̸= b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Now, based on ten stock features r(k∗) i,t and λ(b∗) i,t together with ten factor features f(k∗) b,t and ¯λ(b∗) b,t for k∗ = 1, 5, 10, 20, 30 and b∗ = 1, 2, 3, 4, 5, our feature tensor Xt−1 = [X1,t−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', XN,t−1, XN+1,t−1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' XN+B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t−1] in (5) is taken as (27) Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t−1 = � � � � � � � � � � � � � � � � � � � � r(1) i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t−S · · r(1) i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t−1 · · · · · · r(30) i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t−S · · r(30) i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t−1 λ(1) i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t−S · · λ(1) i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t−1 · · · · · · λ(5) i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t−S · · λ(5) i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t−1 � � � � � � � � � � � � � � � � � � � � and XN+b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t−1 = � � � � � � � � � � � � � � � � � � � � f(1) b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t−S · · f(1) b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t−1 · · · · · · f(30) b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t−S · · f(30) b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t−1 ¯λ(1) b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t−S · · ¯λ(1) b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t−1 · · · · · · ¯λ(5) b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t−S · · ¯λ(5) b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t−1 � � � � � � � � � � � � � � � � � � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' for i = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', N and b = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', B, where N = 1026 (or 1737) for the NASDAQ (or the NYSE) market, B = 5, Xi,t−1 ∈ RP×S, and XN+b,t−1 ∈ RP×S with P = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' It is worthy noting that each entry in Xi,t−1 and XN+b,t−1 is normalized by its range in the training sample to reduce its skewness and leptokurtosis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' see the similar implementation in Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' With the full data sample {rt, Xt−1} in hand, we divide it into three disjoint parts in the same way as Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' (2019): The training sample from January 2, 2013 to December 31, 2015 has 756 trading days, the validation sample follows and ends on December 30, 2016 with 252 trading days, and the testing sample covers the remaining 237 trading days from January 3, 2017 to December 8, 2017 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', the out-of-sample period).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Now, based on the values of hyperparameters listed in Table 2, 20 our GRACE method is implemented as the details specified in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='4 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Here, the selection of λ∗, γ, S, and d for the FTGCN-based models is suggested by the tuning results in Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' (2019), and that of α and K for the QCM method will be examined in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Table 2 List of hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Hyperparameter Description Value λ∗ tuning parameter in the computation of PLS estimator 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='1 γ learning rate in the Adam algorithm 10−3 S length of lagged features in the LSTM 16 d number of hidden units in the LSTM 64 α significance level of LRuc and LRcc tests 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='01 K number of predetermined quantile levels 199 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Comparison Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Besides our GRACE method, other graph-based methods can also be adopted to select portfolios using the same idea, except for different models to predict the conditional quantiles and mean of ri,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Below, we introduce two alternative graph-based methods for the purpose of comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The first competitor is labeled as GRACE1, which replaces the factor-augmented hypergraph in the GRACE method with the hypergraph in Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' (2019), and leaves other mechanisms (including the input features and the selection of tuning hyperparameters) unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The comparison between the GRACE and GRACE1 methods is to verify whether incorporating the asset pricing knowledge into the hypergraph is informative for portfolio selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The second competitor is the simple GRACE (denoted by GRACE2) method, which is motivated by the network autoregression model in Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' (2017) and the FDNQR model in Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Specifically, the GRACE2 method predicts the conditional quantiles of ri,t based on the following specification: Qi,t(τ) = ατ + γτ N � j=1 wi,jrj,t−1 + ζ′ τxi,t−1 + ς′ 1,τFt−1, (28) where xi,t−1 is the last column of Xi,t−1 in (27), Ft−1 is the 5-dimensional vector containing the values of Fama-French five factors at t − 1, and other notations are inherited from (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' By construction, 21 model (28) uses the term ζ′ τxi,t−1 (replacing the term βτzi + ζri,t−1 in (17)) to account for the lag- 1 stock features, and it takes the term ς′ 1,τFt−1 to include the lag-1 factor features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Similarly, the GRACE2 method predicts the conditional mean of ri,t by the following factor-augmented network autoregressive specification: ri,t = αµ + γµ N � j=1 wi,jrj,t−1 + ζ′ µxi,t−1 + ς′ 1,µFt−1 + ε∗ i,t, (29) where ε∗ i,t is the error term with zero mean, and αµ, γµ, ζµ, and ς1,µ are unknown regression coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Here, wi,j = ai,j/ni in models (28) and (29) is determined by Estock in the GRACE method, such that ai,j = 1 if there is any S2S relation between stock i and stock j, and ai,j = 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' We estimate models (28) and (29) respectively via the the quantile loss and L2 loss functions using the data from the combination of training and validation samples, and then proceed the portfolio selection on the testing sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' One may extend both models to contain the stock and factor features up to lag- S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' However, our unreported analysis shows that this extension makes model estimation less stable, leading to the worse performance in portfolio selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Clearly, the comparison between the GRACE and GRACE2 methods aims to exhibit how inadequate are those simple settings in models (28) and (29) for portfolio selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Economic Performance Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' This subsection evaluates the out-of-sample performance of the long-short portfolios selected by the GRACE method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' After sorting all stocks via a certain performance measure into 10 deciles, the long-short portfolio is re-balanced on every trading day via buying the 10% highest ranking stocks (decile 10) and selling the 10% lowest ranking stocks (decile 1) with equal weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The performance measures to sort all stocks include the M (mean),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' MV (Mean– variance),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' MVSK (Mean–variance with skewness and kurtosis),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' SR (Sharpe ratio),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' and SRSK (Sharpe ratio with skewness and kurtosis),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' where the definitions of the last four performance measures are given in (1)–(4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' and the values of all five performance measures are computed based on the predicted values of µi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' hi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' and ki,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='t from the GRACE method with Estock = Ewiki stock,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Efactor = Eff5 factor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' and hyperparameters taken as in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' For the performance measures MV, MVSK, and SRSK, the hyperparameters λ1, λ2, and λ3 are tuned by the grid search within the sets A1, A2, and A3, respectively, to maximize the value of SR of each long-short portfolio re-balanced on training and 22 validation samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Here, due to different ranges of hi,t and si,t (or ki,t), we take A1 = {a × 10−b : a = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', 9 and b = −1, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', 3}, A2 = A3 = {a × 10−b : a = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', 9 and b = 2, 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', 6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' As a comparison, the GRACE1 and GRACE2 methods are also used to select the long-short portfolios under the similar procedure as above, where the GRACE1 method with the performance measure M is the benchmark method proposed by Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Since the QCM method used by each method relies on the choices of α and K, the stock pool for portfolio selection under the performance measures MV, MVSK, SR, and SRSK varies with the choice of α or K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' For the sake of consistency, the portfolio selection under the performance measure M will use the same stock pool as for other performance measures in the sequel, although the implementation of performance measure M is independent of higher-order conditional moments and not affected by the choice of α or K technically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' From an economic viewpoint, we compare all of the selected portfolios in terms of their annualized SR, which is the ratio of annualized excess return to annualized risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' To compute the annualized excess return, we use the Treasury bill rate as a proxy for the risk-free return, and take the transaction cost of 3‰ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', 30 basic points) for buying and selling into account as done in Engle, Ferstenberg and Russell (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Note that when there is no ambiguity, we omit the wording “annualized” below for ease of presentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Impacts of Method and Performance Measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' We first assess how the graph-based methods and performance measures affect the out-of-sample performance of portfolios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Table 3 reports the values of (excess) return, risk, and SR of the out-of-sample portfolios selected by three different methods with five different performance measures in NASDAQ and NYSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' From this table, we can have the following findings: (i) For the GRACE method, the SRSK and M portfolios have the largest and smallest values of SR, respectively, in both markets, implying the necessity of using three higher-order conditional moments for portfolio selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Particularly, the advantage of using higher-order conditional moments is more evident in NASDAQ than NYSE by observing that the value of SR for the SRSK portfolio is 61% (or 12%) higher than that for the M portfolio in NASDAQ (or NYSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Moreover, the values of return and risk indicate that the SRSK portfolio has a larger value of SR mainly because it can generate a much larger (or smaller) value of return (or risk) than 23 Table 3 Out-of-sample performances of long-short portfolios across different methods and performance measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' NASDAQ NYSE Method Measure Return (%) Risk (%) SR Return (%) Risk (%) SR GRACE M 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='40 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='53 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='98 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='04 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='74 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='10 MV 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='81 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='43 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='72 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='36 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='62 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='17 MVSK 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='04 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='43 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='39 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='01 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='62 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='25 SR 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='48 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='03 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='15 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='90 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='43 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='47 SRSK 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='21 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='81 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='95 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='43 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='48 GRACE1 M 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='08 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='43 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='74 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='97 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='87 MV 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='55 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='83 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='73 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='82 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='97 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='88 MVSK 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='45 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='24 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='06 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='82 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='97 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='88 SR 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='57 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='48 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='10 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='91 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='32 SRSK 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='09 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='86 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='75 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='97 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='33 GRACE2 M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='06 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='01 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='09 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='94 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='46 MV 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='55 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='45 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='18 MVSK 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='66 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='78 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='26 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='03 SR 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='53 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='50 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='28 SRSK 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='90 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='72 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='56 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='45 the M portfolio in NASDAQ (or NYSE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Another difference between NASDAQ and NYSE is the influence of conditional skewness and kurtosis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Specifically, using the conditional skewness and kurtosis can well decrease the portfolio risk in NASDAQ while only marginally increase the portfolio return in NYSE, according to the comparison between MV and MVSK (or SR and SRSK) portfolios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The aforementioned distinction in two markets may attribute to the fact that the NYSE has a relatively more “normal” environment than the NASDAQ, so that the function of higher-order conditional moments (particularly the conditional skewness and kurtosis) is relatively weaker for portfolio selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' (ii) For the GRACE1 and GRACE2 methods, both of them perform worse than the GRACE method, regardless of the choice of performance measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The advantage of GRACE method over GRACE1 method is exceptionally significant for M and MV portfolios in NASDAQ, since 24 the M and MV portfolios selected by GRACE method have 108% and 115% higher value of SR than those selected by GRACE1 method, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' This finding shows that incorporat- ing asset price information is more important for portfolio selection in NASDAQ than NYSE, especially when the conditional skewness and kurtosis are not taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Moreover, the value of SR for the best portfolio selected by the GRACE method is 236% and 21% higher than that selected by the benchmark method (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', the GRACE1 method with the performance measure M) in NASDAQ and NYSE, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' In all cases, the GRACE2 method has a much worse performance than other two methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' This is not unexpected, because the simple model settings in the GRACE2 method can not capture the effects of features on stock returns adequately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Overall, the above findings clearly demonstrate the importance of higher-order conditional moments as well as the asset pricing knowledge in portfolio selection through the GRACE method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Impact of EEEstock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Since all considered three graph-based methods depend on the S2S rela- tion set Estock, a natural question is what kind of S2S relation set is more informative for portfolio selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' To answer this question, we alter all three methods by choosing Estock = Esi stock or Eall stock while keeping other settings as for Table 3 unchanged, where Eall stock is the union set of Ewiki stock and Esi stock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Table 4 reports the values of out-of-sample SR for portfolios selected from three different choices of Estock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' From Table 4, we find that the value of SR for the Esi stock-based portfolio is smaller than that for the corresponding Ewiki stock-based portfolio in all cases, except for the M portfolios se- lected by the GRACE1 method in NASDAQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Particularly, the advantage of Ewiki stock-based portfolio over Esi stock-based portfolio is much more substantial for the GRACE1 method with the performance measures MVSK, SR, and SRSK in NASDAQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' This finding indicates that the S2S relations in Esi stock could be less informative than those in Ewiki stock to learn higher-order conditional moments, especially when the asset pricing knowledge is absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Moreover, we find from Table 4 that using a richer S2S relation set Eall stock to replace the single S2S relation set Ewiki stock gives no change or little change to the values of SR in the GRACE and GRACE1 methods, and this replacement even makes the portfolios have smaller values of SR for many cases in the GRACE1 method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The reason is probably that the long-term correlations between stocks are largely driven by the factors through the F2S relations in Eff5 factor, and they could be wrongly captured by the S2S relations in Esi stock when the asset pricing knowledge is absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' For example, the stocks 25 belonging to the sector “Basic Industries” tend to have large market capitalization, while the SMB factor in Fama and French (2015) represents the outperformance of small-cap stocks over large-cap ones during a long-term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Hence, the comovement of stocks in the sector “Basic Industries” is more properly captured by the F2S relations with respect to the SMB factor in Eff5 factor rather than the sector-industry relations in Esi stock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The unsatisfactory performance from the use of Eall stock becomes more evident in the GRACE2 method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' This conveys the information that it is inappropriate to ignore the type of S2S relation as done by models (28)–(29), when the domain knowledge on multiple types of S2S relation is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' In sum, we could reach a general conclusion that Esi stock is less informative than Ewiki stock for portfolio selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Hence, if Ewiki stock is accessible, we recommend it for practical use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Table 4 Out-of-sample SRs of long-short portfolios across different choices of Estock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' NASDAQ NYSE Method Measure Esi stock Ewiki stock Eall stock Esi stock Ewiki stock Eall stock GRACE M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='74 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='98 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='98 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='04 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='12 MV 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='86 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='72 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='74 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='04 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='17 MVSK 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='98 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='39 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='40 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='04 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='25 SR 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='81 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='18 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='04 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='47 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='47 SRSK 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='52 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='81 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='81 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='48 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='48 GRACE1 M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='58 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='43 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='52 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='76 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='87 MV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='60 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='73 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='55 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='77 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='87 MVSK 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='94 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='06 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='02 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='77 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='88 SR 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='14 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='01 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='32 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='32 SRSK 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='19 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='75 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='65 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='32 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='32 GRACE2 M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='24 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='46 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='81 MV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='43 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='77 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='18 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='80 MVSK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='71 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='78 SR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='34 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='28 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='63 SRSK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='72 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='29 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='45 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='58 26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Impact of α and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Before implementing the QCM method, a valid quantile level set Ωi in (26) is defined for each stock i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' As discussed in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='4 above, we only use the predicted conditional quantiles at those quantile levels in Ωi to predict higher-order conditional moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Intuitively, when the value of K is fixed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', the quantile sequence {τ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', τK} is given), a large value of α enhances the reliability of the conditional quantile prediction, but at the same time, it reduces the learning efficiency of the higher-order conditional moments as the size of Ωi becomes small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Hence, there is a trade-off between reliability and learning efficiency, in terms of the choice of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' To address this issue, we examine the impact of α on the performance of portfolio by changing the value of α while keeping other settings in the same way as for Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Specifically, we select the long-short portfolios for each method when α ∈ {0%, 1%, 5%, 10%}, and report the related results of out-of-sample SR in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' From this table, we find that except for the M and SR portfolios from the GRACE1 method in NASDAQ, all of other portfolios achieve the maximum value of SR at α = 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' When the value of α decreases from 1% to 0% (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', |Ωi| = K), the performance of most portfolios becomes worse, particularly for those selected by the GRACE1 method in NYSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' This finding indicates that the tests LRuc and LRcc can effectively exclude those extremely ill-behaved predictions of conditional quantiles to improve the performance of QCM method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' When the value of α increases from 1% to 5% or 10%, the performance of most portfolios also becomes worse but with different tendencies in two markets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' For example, the reduction in SR under GRACE1 method is much larger than that under the GRACE method in NASDAQ, whereas the reduction in SR has similar patterns in NYSE based on both methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' This observation shows the advantage of the factor-augmented hypergraph, which stabilizes the performance of the GRACE method across the choice of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Besides α, K is also related to the construction of Ωi for all three methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' A large value of K is needed to ensure the consistency of the QCMs in theory, however, it is inevitable to increase the cost in computation as the conditional quantile model has to be trained K different times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Clearly, the choice of K reflects the trade-off between the computational cost and learning efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' To study the impact of K, we select the long-short portfolios as for Table 3 but with different values of K, and report the values of out-of-sample SR for those portfolios in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The findings from this table generally match our expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' First, the value of SR increases with the value of K for each portfolio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' This is because a larger value of K can grab more information on the conditional distribution of ri,t by learning its conditional quantiles at more refined quantile levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Second, the 27 performance of portfolios from the GRACE method is more stable across the choice of K than that from the GRACE1 method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' This finding is consistent to that in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Third, the values of SR under the GRACE2 method are always less than one or even below zero in many cases, implying the inadequacy of models (28)–(29) for learning conditional quantiles and mean of ri,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Overall, the portfolios from the GRACE method show a robust performance over the choice of α and K, and they outperform those from the GRACE1 and GRACE2 methods for all examined choices of α and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' In practice, we recommend to take α = 1% and K = 199 for the GRACE method, since this seems a desirable choice to balance the reliability (or computational cost) and learning efficiency, as demonstrated by the aforementioned results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Table 5 Out-of-sample SRs of long-short portfolios across different choices of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' NASDAQ NYSE Method Measure 0% 1% 5% 10% 0% 1% 5% 10% GRACE M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='35 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='98 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='83 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='81 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='88 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='06 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='05 MV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='58 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='72 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='67 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='61 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='01 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='09 MVSK 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='39 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='21 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='09 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='03 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='22 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='21 SR 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='99 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='94 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='87 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='47 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='45 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='42 SRSK 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='78 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='81 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='76 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='62 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='48 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='43 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='43 GRACE1 M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='42 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='43 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='34 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='46 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='46 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='81 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='83 MV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='48 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='73 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='62 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='59 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='53 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='88 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='81 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='83 MVSK 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='77 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='06 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='03 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='67 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='43 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='04 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='32 SRSK 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='47 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='76 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='87 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='04 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='32 GRACE2 M 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='46 MV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='18 MVSK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='78 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='03 SR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='28 SRSK 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='45 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Impact of Transaction Cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The transaction cost is a non-negligible important factor for portfolio selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' In NASDAQ and NYSE, the cost of a transaction mainly consists of three compo- nents: Commission, stamp tax, and slippage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' While the rates of commission and stamp tax are easily accessible, the slippage is hard to be measured quantitatively as the liquidity issue could make it difficult to execute transactions at a pre-specified price without affecting the market price, especially for small-cap stocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Normally, the transaction cost of 30 basis points is appropriate to capture the total effect of commission, stamp tax, and slippage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' However, the slippage could be larger in some extreme circumstances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Therefore, it is necessary to conservatively consider some higher transaction costs than 30 basis points to investigate their impact on portfolio selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' To achieve this goal, Table 7 reports the values of out-of-sample SR for long-short portfolios selected from each method, when the transaction costs are 30, 50, 75, and 100 basis points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' From this table, we 29 can obtain some interesting findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' First, as expected, the value of SR for each portfolio decreases with the transaction cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Second, regardless of transaction cost, the SRSK portfolio is the best one followed successively by the SR, MVSK, MV, and M portfolios in the GRACE and GRACE1 methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Moreover, the SRSK portfolio from the GRACE method has a more stable performance to the level of transaction cost than other portfolios in NASDAQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Third, the GRACE method outperforms the GRACE1 method in all considered cases, except that the SR and SRSK portfolios from the GRACE1 method perform better than those from the GRACE method when the transaction cost is 100 basis points in NYSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Fourth, the GRACE2 method performs much worse than the GRACE method, and its performance becomes more unsatisfactory when the transaction cost is higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Overall, we find that the advantage of using the higher-order conditional moments together with the asset pricing knowledge from the GRACE method for portfolio selection is unchanged with the setting of transaction cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Statistical Performance Evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' From an economic viewpoint, the good performance of our GRACE method in portfolio selection has been demonstrated above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' From a statistical viewpoint, it is worthwhile to check whether the GRACE method can well estimate and predict the conditional moments of ri,t, shedding light on its success in portfolio selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The Comovement of Conditional Moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Intuitively, the conditional moments of two linked stocks should have a tendency of comovement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Since the number of stocks is large, there has no easy way to exhibit whether most pairs of linked stocks have this tendency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' For ease of illustration, we only plot ri,t, �µi,t, �hi,t, �si,t, and �ki,t of MSFT LLC and Google LLC based on the GRACE method in Fig 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Here, �µi,t, �hi,t, �si,t, and �ki,t are estimated values during the in-sample period (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', the period of the training and validation samples), and they are predicted values during the out-of-sample period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' From Fig 3, we find that not only the stock returns but also all estimated and predicted conditional moments have similar trends during the entire period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The Validity of Conditional Moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Since the true values of conditional moments are unobserved, no explicit quantity can measure the precision of �hi,t, �si,t, and �ki,t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' To circumvent this deficiency, we propose some hypothesis tests to detect whether �µi,t, �hi,t, �si,t, and �ki,t are valid by using a similar idea as in Gu, Kelly and Xiu (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Specifically, we define αµ i = E(eµ i,t), αh i = E(eh i,t), αs i = E(es i,t), and αk i = E(ek i,t), where eµ i,t = ri,t −µi,t, eh i,t = (ri,t −µi,t)2 −hi,t, es i,t = � ri,t−µi,t √ hi,t �3 −si,t, 30 Table 7 Out-of-sample SRs of long-short portfolios across different transaction costs (in basis points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' NASDAQ NYSE Method Measure 30 50 75 100 30 50 75 100 GRACE M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='98 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='54 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='40 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='79 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='83 MV 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='72 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='88 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='71 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='83 MVSK 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='39 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='87 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='84 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='27 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='85 SR 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='46 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='69 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='95 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='47 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='90 SRSK 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='81 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='01 3.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='72 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='22 and ek i,t = � ri,t−µi,t √ hi,t �4 − ki,t for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Based on the estimates {�eµ i,t}T t=1 with �eµ i,t = ri,t − �µi,t, we adopt the classical Student’s t test Tµ i to detect the null hypothesis Hµ i : αµ i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' If Hµ i is not rejected by Tµ i at the significance level α∗, then we regard that {�µi,t}T t=1 is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Similarly, the Student’s t tests Th i , Ts i, and Tk i to detect the null hypotheses Hh i : αh i = 0, Hs i : αs i = 0, and Hk i : αk i = 0, respectively, can be used to examine the validity of {�µi,t}T t=1, {�hi,t}T t=1, {�si,t}T t=1, and {�ki,t}T t=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' We apply each Student’s t test to check the validity of conditional moments of stock i at the significance level α∗ ∈ {1%, 5%, 10%}, and then report the percentage of stocks having valid con- ditional moments in Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' From this table, we first find that the percentage of stocks having valid conditional moments during the out-of-sample period is higher than that during the in-sample period in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' This may indicate that none of the methods has the problem of overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 31 Date Return MSFT Google Date Conditional Mean Date Conditional Variance Date Conditional Skewness Date Conditional Kurtosis Fig 3: The plots of ri,t, �µi,t, �hi,t, �si,t and �ki,t for MSFT LLC and Google LLC from January 2, 2013 to December 8, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Next, except for the out-of-sample results of Ts i in NYSE, all of the testing results show that both GRACE and GRACE1 methods deliver much better estimated and predicted conditional moments than the GRACE2 method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' This statistically explains why both GRACE and GRACE1 methods can select better portfolios than the GRACE2 method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Moreover, we observe that the GRACE method performs better than GARCE1 method according to the results at the significance level of 5% and 10%, although both methods have the same results at the significance level of 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' This advantage of GRACE method over GARCE1 method shows the necessity of using factor-augmented hypergraph for learning conditional moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Concluding Remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' This paper proposes a new GRACE method for big portfolio selection under different performance measures that are defined by four conditional moments (with respect to mean, variance, skewness, and kurtosis) of stock returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The GRACE method builds on the FTGCN and the QCM method: The former embeds the factor-augmented hypergraph within a graph neural network to obtain the estimates of mean and quantiles, and the latter transforms the estimates of 32 Table 8 Percentages of stocks having valid conditional moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' NASDAQ NYSE GRACE GRACE1 GRACE2 GRACE GRACE1 GRACE2 Test 1% 5% 10% 1% 5% 10% 1% 5% 10% 1% 5% 10% 1% 5% 10% 1% 5% 10% Panel A: In-sample period Tµ i 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='0 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='8 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='6 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='0 71.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='6 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='9 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='6 Tk i 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='3 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='9 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='1 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='3 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='9 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='5 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='5 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='1 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='1 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='0 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='6 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='9 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='0 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='6 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='9 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='3 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='1 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='4 quantiles into those of higher-order moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The most attractive feature of the GRACE method is its capacity to estimate conditional variance, skewness, and kurtosis of high-dimensional stock returns, so the big portfolios under four performance measures MV, MVSK, SR, and SRSK can be constructed from thousands of stocks or even more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' There are two reasons leading to this feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' First, the FTGCN takes the domain knowledge on S2S relations and the asset pricing knowledge on F2S relations to form the factor-augmented hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' The knowledge of S2S and F2S relations is sparse to make estimation tractable in high-dimension, and at the same time, it is informative to capture the interplay between two stocks as well as the driving force from common factors in all stocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Second, the QCM method only needs the estimates of quantiles to obtain those of higher-order moments, and the quantiles can be estimated by a super- vised learning without assuming any distribution of stock returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' If we intend to estimate variance, skewness, and kurtosis directly, a specific distribution of stock returns is inevitably needed, as done by the traditional method based on high-dimensional GARCH models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' However, due to the pres- ence of distribution of stock returns, the estimation of high-dimensional GARCH models becomes computationally infeasible when the dimension of stocks is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' 33 The importance of our GRACE method is further demonstrated by the empirical studies in NAS- DAQ and NYSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' From an economic viewpoint, we find that the SRSK portfolio selected by the GRACE method earns the out-of-sample SR of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='81 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='48 in NASDAQ and NYSE, respectively, and it beats the M portfolio (with the out-of-sample SR of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='43 in NASDAQ and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content='87 in NYSE) selected by the benchmark method in Feng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' (2019) by a wide margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Moreover, we find that regardless of the performance measure, the portfolios from the GRACE method have more stable and larger values of out-of-sample SR than those from the competing methods, across different settings of hyperparameter and transaction cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' From a statistical viewpoint, we find that the predicted condi- tional moments from the GRACE method are largely valid and capable of reflecting the comovement between linked stocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' In future, our GRACE method can be extended in several directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' First, it is interesting to design some new hypergraphs to incorporate subjective information from the experiences and beliefs of investors) or unstructured data information from financial news and social media contents (Ke, Kelly and Xiu, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Fan, Xue and Zhou, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Second, it is intriguing to examine whether other observed factors or data-driven factors (Giglio, Kelly and Xiu, 2022) are useful for specifying the F2S relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Third, it is desirable to utilize the information of S2S and F2S relations for proposing GARCH-type methods (Engle, Ledoit and Wolf, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Pakel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=', 2021), which are computationally feasible to learn higher-order moments when the dimension of the data is very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' References.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Ando, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' and Bai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/eNFKT4oBgHgl3EQfAC2F/content/2301.11697v1.pdf'} +page_content=' Quantile co-movement in financial markets: A panel 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the topological numbers for the singly-rotating Kerr-AdS black holes in all +dimensions and the four-dimensional Kerr-Newman-AdS black hole. We show that for uncharged AdS black +holes, the rotation parameter has a significant impact on the topological number, and for rotating AdS black +holes, the dimension of spacetimes has an important effect on the topological number. In addition, it indicates +that the electric charge parameter has no impact on the topological number of rotating AdS black holes because +of the four-dimensional Kerr-AdS and Kerr-Newman-AdS black holes have the same topological numbers. +Furthermore, we propose a conjecture that, given an AdS black hole, the difference between its topological +number and that of its corresponding non-AdS black hole is always unity. +I. +INTRODUCTION +In recent years, topology as an important mathematical tool +has been applied to various research fields in physics. As one +of the most fundamental and fascinating objects in nature, the +black hole has been the subject of extensive research, among +which the study of black hole topology has shed new light +on the nature of gravity through light rings [1–4], timelike +circular orbits [5], and thermodynamic properties [6–17]. +In particular, in Ref. [6], a new method to investigate the +thermodynamic topological properties of black holes is pro- +posed by considering black hole solutions as topological ther- +modynamic defects and constructing topological numbers in- +dependently of the intrinsic parameters of black holes, and +further, dividing all black hole solutions into three categories +according to their different topological numbers. +Because +these topological numbers are universal constants that are in- +dependent of the black hole solution parameters are very im- +portant for understanding the nature of black holes and grav- +ity, and further, may provide new clues to the establishment +of the quantum gravity theory. The topological approach pro- +posed in Ref. [6] quickly gained popularity due to its straight- +forwardness and adaptability, and it was subsequently effec- +tively used to explore the topological numbers of several well- +known black hole solutions [12–16], i.e., the Schwarzschild- +AdS black hole solution [12], the static black hole solutions +in Lovelock gravity [13], the static Gauss-Bonnet-AdS black +hole solutions [14], the static black hole solution in nonlinear +electrodynamics [15], and the static Born-Infeld AdS black +hole solution [16]. Very recently, we extended the topologi- +cal approach to rotating black hole cases and investigate the +topological numbers for the cases of rotating Kerr and Kerr- +Newman black hole solutions [17]. +Although there has been some recent progress in investi- +gating the topological number of black holes, the topological +number of the rotating AdS black holes remains virgin terri- +tory and thus needs to be explored deeply. On the other hand, +the study of rotating AdS black holes has shed light on the +nature of gravity through gauge-gravity dualities [18–20], so +∗ wdcwnu@163.com +it is very important and remarkable to investigate the topolog- +ical number of rotating AdS black holes. These two aspects +motivate us to conduct the present work. +In this paper, we investigate the topological number of the +d-dimensional singly-rotating Kerr-AdS black holes and the +four-dimensional Kerr-Newman-AdS black hole. We demon- +strate that for uncharged AdS black holes, the rotation param- +eter has a significant effect on the topological number, and +for rotating AdS black holes, the dimension of spacetimes +has a remarkable effect on the topological number. In ad- +dition, we also find that the electric charge does not play a +role in the topological classification, in the sense of having +no effect on the topological number of rotating AdS black +holes. The remaining part of this paper is organized as fol- +lows. In Sec. II, we first give a brief review of the topolog- +ical approach and investigate the topological number of the +four-dimensional Schwarzschild-AdS black hole as a warmup +exercise. In Sec. III, we focus on the topological number of +the four-dimensional rotating Kerr-AdS black hole. In Sec. +IV, we extend the discussions from Sec. III to the cases of +the d-dimensional singly-rotating Kerr-AdS black holes. In +Sec. +V, we turn to discuss the topological number of the +four-dimensional Kerr-Newman-AdS black hole. Finally, we +present our conclusions in Sec. VI. +II. +SCHWARZSCHILD-ADS4 BLACK HOLE +In this section, we first present a brief review of the topo- +logical approach proposed in Ref. [6], then investigate the +topological number of the four-dimensional Schwarzschild- +AdS black hole as a warmup exercise. +As shown in Ref. [6], one can introduce the generalized +off-shell free energy +F = M − S +τ , +(1) +for a black hole thermodynamical system with mass M and +entropy S, where τ is an extra variable that can be thought +of as the inverse temperature of the cavity inclosing the black +hole. The generalized free energy only converts to being on- +shell if τ = 1/T. +arXiv:2301.03002v1 [hep-th] 8 Jan 2023 + +2 +In Ref. [6], a core vector φ is defined as +φ = +�∂F +∂rh +, −cotΘcscΘ +� +, +(2) +where the two parameters obey 0 < rh < +∞, 0 ≤ Θ ≤ π, +respectively. The component φ Θ is divergent at Θ = 0,π, thus, +the direction of the vector points outward there. +A topological current can be described as follows using +Duan’s φ-mapping topological current theory [21–23]: +jµ = 1 +2π εµνρεab∂νna∂ρnb , +µ,ν,ρ = 0,1,2, +(3) +where ∂ν = (∂/∂xν) and xν = (τ, rh, Θ). +The unit vec- +tor n is obtained as n = (nr,nφ), where nr = φ rh/||φ|| and +nΘ = φ Θ/||φ||. It is simple to demonstrate that the topological +current (3) given above is conserved, allowing one to deduce +∂µ jµ = 0 with simplicity. It is then established that the topo- +logical current jµ is a δ-function of the field configuration +[3, 22, 23] +jµ = δ 2(φ)Jµ +�φ +x +� +, +(4) +where the 3-dimensional Jacobian Jµ (φ/x) is defined as: +εabJµ (φ/x) = εµνρ∂νφ a∂ρφ b. It is simple to indicate that +jµ equals zero unless when φ a(xi) = 0, hence the topological +number W can be determined as follows +W = +� +Σ j0d2x = +N +∑ +i=1 +βiηi = +N +∑ +i=1 +wi , +(5) +where βi is the positive Hopf index counting the number of +the loops of the vector φ a in the φ-space when xµ are around +the zero point zi, ηi = sign(J0(φ/x)zi) = ±1 is the Brouwer +degree, and wi is the winding number for the i-th zero point of +φ that is contained in Σ. It is important to keep in mind that +two loops Σ1 and Σ2 have the same winding number if they +both enclose the same zero point of φ. On the other hand, if +there is no zero point in the surrounded region, then one can +arrive at W = 0. +In the following, we investigate the the topological num- +ber of the four-dimensional Schwarzschild-AdS black hole +via the above topological approach. For the four-dimensional +Schwarzschild-AdS black hole, its metric has the form [24] +ds2 = −f(r)dt2 + dr2 +f(r) +r2(dθ 2 +sin2 θdφ 2), +(6) +where +f(r) = 1− 2m +r + r2 +l2 , +in which m is the mass parameter, and l is the cosmological +scale associated with the pressure P = 3/(8πl2) of the four- +dimensional AdS black holes [25–27]. The mass and entropy +associated with the above metric (6) can be computed via the +standard method and have the following exquisite forms: +M = m, +S = πr2 +h , +(7) +Schwarzschild-AdS +τ1 /r0 +τc /r0 +0 +10 +20 +30 +40 +50 +60 +0 +5 +10 +15 +20 +τ/r0 +rh/r0 +FIG. 1. Zero points of the vector φrh are shown on the rh − τ plane +for the Schwarzschild-AdS black hole with Pr2 +0 = 0.0022. The red +dot with τc represents the annihilation point for the Schwarzschild- +AdS black hole. There are two Schwarzschild-AdS black holes when +τ = τ1. Obviously, the topological number W = 1−1 = 0. +where rh are the locations of the event and Cauchy horizons +that satisfy the equation f(rh) = 0. +For the Schwarzschild-AdS4 black hole, one can obtain the +generalized free energy +F = rh +2 + 4π +3 Pr2 +h − πrh +τ . +(8) +The components of the vector φ can then be easily calculated +as +φ rh = 1 +2 +4πPr2 +h − 2πrh +τ +, +(9) +φ Θ = −cotΘcscΘ. +(10) +By solving the equation φ rh = 0, one can obtain a curve on +the rh−τ plane. For the four-dimensional Schwarzschild-AdS +black hole, one can obtain +τ = +4πrh +1+8πPr2 +h +. +(11) +Taking the pressure Pr2 +0 = 0.0022, where r0 is an arbitrary +length scale set by the size of a cavity enclosing the black +hole, we show zero points of φ rh in the rh − τ plane in Fig. +1. For small τ, such as τ = τ1, there are two intersection +points for the Schwarzschild-AdS black hole. The intersection +points exactly satisfy the condition τ = 1/T, and thus rep- +resent the on-shell Schwarzschild-AdS black hole solutions +with the characteristic temperature T = 1/τ. The two intersec- +tion points for the Schwarzschild-AdS black hole can coincide +each other when τ = τc, and then vanish when τ > τc, there- +fore τc is a annihilation point can be found at τc = 26.75r0 +which can be seen straightforward from Fig. 1. Furthermore, +the annihilation point τc divides the Schwarzschild-AdS black +hole into the upper and lower branches with the winding num- +ber w = 1 and w = −1, respectively. Based upon the local +property of the zero point, one can easily find that the topo- +logical number W = 0 for the Schwarzschild-AdS black hole, +which is consistent with the result given in Ref. [12]. + +3 +III. +KERR-ADS4 BLACK HOLE +In the section, we will focus on the topological number of +the four-dimensional Kerr-AdS black hole. For the Kerr-AdS4 +black hole, the metric has the form [24] +ds2 = −∆r +Σ +� +dt − a +Ξ sin2 θdφ +�2 ++ Σ +∆r +dr2 + Σ +∆θ +dθ 2 ++∆θ sin2 θ +Σ +� +adt − r2 +a2 +Ξ +dφ +�2 +(12) +in the Boyer-Lindquist coordinates, where +∆r = (r2 +a2) +� +1+ r2 +l2 +� +−2mr, +Ξ = 1− a2 +l2 , +∆θ = 1− a2 +l2 cos2 θ , +Σ = r2 +a2 cos2 θ , +in which a is the rotation parameter, m is the mass parameter, +and l is the AdS radius. +The mass M and entropy S associated with the above solu- +tion (12) are [28] +M = m +Ξ2 , +S = π(r2 +h +a2) +Ξ +. +(13) +According to the generalized off-shell free energy (1), and us- +ing l2 = 3/(8πP), thus one can easily yield +F = 3(r2 +h +a2) +� +2πrh(8πPa2 +4Prhτ −3)+3τ +� +2rhτ(8πPa2 −3)2 +(14) +for the Kerr-AdS4 black hole. The components of the vector +φ can be computed as +φ rh = 12π(8πPa2 −3)r3 +h +3a2(8πPr2 +h −3)τ +2r2 +hτ(8πPa2 −3)2 ++ 9(r2 +h +8πPr4 +h) +2r2 +h(8πPa2 −3)2 , +(15) +φ Θ = −cotΘcscΘ. +(16) +By solving the equation φ rh = 0, one can obtain +τ = +4πr3 +h(3−8πPa2) +a2(8πPr2 +h −3)+3(8πPr4 +h +r2 +h) +(17) +as the zero point of the vector field φ. +Taking the pressure Pr2 +0 = 0.0022 and the rotation param- +eter a = r0, we show zero points of φ rh in the rh − τ plane +in Fig. 2. Note that for these values of Pr2 +0 and a/r0, one +generation point and one annihilation point can be found at +τ/r0 = τa/r0 = 24.93 and τ/r0 = τb/r0 = 26.81, respectively. +We find that there are one large black hole branch for τ < τa, +three black hole branches for τa < τ < τb, and one small black +hole branch for τ > τb. Calculating the winding number w for +these three black hole branches, we indicate that the small and +large black hole branches both have w = 1, however the inter- +mediate black hole branch has w = −1. Thus, the Kerr-AdS4 +black hole always has the topological number W = 1, which +LBH +SBH +IBH +τa /r0 +τb /r0 +23 +24 +25 +26 +27 +28 +29 +0 +1 +2 +3 +4 +5 +6 +7 +τ/r0 +rh/r0 +FIG. 2. Zero points of φrh shown in the rh − τ plane for the Kerr- +AdS4 black hole with Pr2 +0 = 0.0022 and a = r0. The red solid, blue +dashed, and black solid lines are for the large black hole (LBH), inter- +mediate black hole (IBH), and small black hole (SBH), respectively. +The annihilation and generation points are represented by red and +black dots, respectively. +is same to the case of the RN-AdS4 black hole [6]. So the +Kerr-AdS4 black hole and RN-AdS4 black hole are the same +kind of black hole solutions. What is more, since the topolog- +ical number of the Schwarzschild-AdS black hole is 0, while +that of Kerr-AdS black hole is 1, it indicate that the rotation +parameter has a remarkable effect on the topological number +for the uncharged AdS black hole. +IV. +SINGLY-ROTATING KERR-ADS BLACK HOLES IN +ARBITRARY DIMENSIONS +In this section, we will extend the above discussions to the +cases of rotating black holes in higher dimensions by consid- +ering the singly-rotating Kerr-AdS black holes in arbitrary di- +mensions. For d-dimensional singly-rotating Kerr-AdS black +holes, the metric has the form [29] +ds2 = −∆r +Σ +� +dt − asin2 θ +Ξ +dφ +�2 ++ Σ +∆r +dr2 + Σ +∆θ +dθ 2 ++∆θ sin2 θ +Σ +� +adt − r2 +a2 +Ξ +dφ +�2 ++r2 cos2 θdΩ2 +d−4 , (18) +where dΩd denotes the line element of the d-dimensional unit +sphere, and +∆r = (r2 +a2) +� +1+ r2 +l2 +� +−2mr, +Ξ = 1− a2 +l2 , +∆θ = 1− a2 +l2 cos2 θ , +Σ = r2 +a2 cos2 θ , +are the same as their correponding ones in the four- +dimensional case in the last section. + +4 +The thermodynamic quantities are [30] +M = ωd−2m +4πΞ2 +�(d −4)Ξ +2 ++1 +� +, +J = ωd−2ma +4πΞ2 +, +Ω = a(r2 +h +l2) +l2(r2 +h +a2) , +S = A +4 = ωd−2 +4Ξ (r2 +h +a2)rd−4 +h +, +T = rh +2π +� +1+ r2 +h +l2 +�� +1 +r2 +h +a2 + d −3 +2r2 +h +� +− +1 +2πrh +, +V = rhA +d −1 +� +1+ a2(r2 +h +l2) +(d −2)Ξl2r2 +h +� +, P = (d −1)(d −2) +16πl2 +, +(19) +where ωd−2 = 2π(d−1)/2/Γ[(d − 1)/2], and rh is determined +by the horizon equation ∆r = 0. In the following, we will dis- +cuss the topological number of the singly-rotating Kerr-AdS +black holes from five to nine dimensions. +A. +d = 5 case +We first consider d = 5 case. From Eq. (19), one can obtain +the expression of the generalized free energy as +F = − +π(r2 +h +a2) +8τ(4πPa2 −3)2 +� +12πrh(3−4πPa2) ++τ(4πPr2 +h +3)(4πPa2 −9) +� +, +(20) +and the components of the vector φ can be computed as +φ rh = +π +4τ(4πPa2 −3)2 +� +6π(4πPa2 −3)(3r2 +h +a2) +−rhτ(4πPa2 −9) +� +4πP(2r2 +h +a2)+3 +�� +, +(21) +φ Θ = −cotΘcscΘ. +(22) +It is simple to obtain +τ = +6π(4πPa2 −3)(3r2 +h +a2) +rh(4πPa2 −9)[4πP(2r2 +h +a2)+3] +(23) +as the zero point of the vector field φ. +We plot the zero points of the component φ rh for the singly- +rotating Kerr-AdS5 black hole with Pr2 +0 = 0.02 and a/r0 = 1 +in Fig. 3. Note that for these values of Pr2 +0 and a/r0, one +generation point and one annihilation point can be found at +τ/r0 = τa/r0 = 5.96 and τ/r0 = τb/r0 = 7.36, respectively. +One can easily obtain the topological number W = 1 for the +singly-rotating Kerr-AdS5 black hole using the local property +of the zero points, which is the same as the case of the four- +dimensional Kerr-AdS black hole in the previous section. +LBH +SBH +IBH +τa /r0 +τb /r0 +5 +6 +7 +8 +9 +10 +0 +1 +2 +3 +4 +5 +τ/r0 +rh/r0 +FIG. 3. Zero points of φrh shown in the rh − τ plane for the singly- +rotating Kerr-AdS5 black hole with Pr2 +0 = 0.02 and a/r0 = 1. The red +solid, blue dashed, and black solid lines are for the large black hole +(LBH), intermediate black hole (IBH), and small black hole (SBH), +respectively. The annihilation and generation points are represented +by red and black dots, respectively. +B. +d = 6 case +Next, we consider d = 6 case. In this case, the generalized +free energy is +F = − 2πrh(r2 +h +a2) +3τ(4πPa2 −5)2 +� +5πrh(5−4πPa2) ++τ(4πPr2 +h +5)(2πPa2 −5) +� +. +(24) +Thus, the components of the vector φ are +φ rh = +2π +3τ(4πPa2 −5) +� +10πrh(4πPa2 −5)(2r2 +h +a2) +−τ(2πPa2 −5) +� +20πPr4 +h +3(4πPa2 +5)r2 +h ++5a2�� +, +(25) +φ Θ = −cotΘcscΘ. +(26) +So the zero point of the vector field φ is +τ = +10πrh(4πPa2 −5)(2r2 +h +a2) +(2πPa2 −5) +� +20πPr4 +h +3(4πPa2 +5)r2 +h +5a2� . +(27) +Taking Pr2 +0 = 0.1 and a/r0 = 1, we plot zero points of φ rh +in the rh −τ plane in Fig. 4. For small τ, such as τ = τ1, there +are two intersection points for the singly-rotating Kerr-AdS6 +black hole. The two intersection points for the black hole +can coincide each other when τ = τc, and then vanish when +τ > τc, therefore τc is an annihilation point can be found at +τc = 2.84r0. Based upon the local property of the zero point, +one can easily get to the topological number W = 0 for the +singly-rotating Kerr-AdS6 black hole, which is different from +the corresponding results W = 1 for the Kerr-AdS4 black hole +in Sec. III and the singly-rotating Kerr-AdS5 black hole in +Sec. IV A. Therefore, for rotating AdS black holes, the di- +mension of spacetimes has a remarkable effect on the topo- +logical number. On the other hand, the singly-rotating Kerr- +AdS6 black hole and Schwarzschild-AdS black hole are the + +5 +6d Kerr-AdS +τc /r0 +τ1 /r0 +0 +1 +2 +3 +4 +5 +0 +1 +2 +3 +4 +5 +τ/r0 +rh/r0 +FIG. 4. Zero points of φrh shown in the rh − τ plane for the singly- +rotating Kerr-AdS6 black hole with Pr2 +0 = 0.1 and a/r0 = 1. The +red dot with τc represents the annihilation point for the black hole. +There are two singly-rotating Kerr-AdS6 black holes when τ = τ1. It +is easy to obtain the topological number W = 1−1 = 0. +same kind of black hole solutions since both of their topologi- +cal numbers are equal to zero via the topological classification +method. +C. +d = 7 case +Then, we consider d = 7 case. The generalized free energy +has the form +F = − +3π2r2 +h(r2 +h +a2) +16τ(15−8πPa2)2 +� +20πrh(15−8πPa2) ++τ(8πPr2 +h +15)(8πPa2 −25) +� +. +(28) +Therefore, one can straightforwardly obtain +τ = +10πrh(8πPa2 −15)(5r2 +h +3a2) +(8πPa2 −25) +� +6r2 +h(4πPr2 +h +5)+a2(16πPr2 +h +15) +� +(29) +by solving the equation φ rh = 0. +We plot the zero points of φ rh in the rh − τ plane for the +singly-rotating Kerr-AdS7 black hole in Fig. 5. Note that +for the values of Pr2 +0 = 0.3 and a/r0 = 1, one annihilation +point can be found at τ/r0 = τc/r0 = 1.31. Based on the local +property of the zero points, we have the topological number +W = 0 for the singly-rotating Kerr-AdS7 black hole, which +is the same as the case of he singly-rotating Kerr-AdS6 black +hole in the subsection IV B. +D. +d = 8 case +Let us continue dealing with the d = 8 case. Similar to the +procedure done in the previous three subsections, one can give +7d Kerr-AdS +τc /r0 +τ1 /r0 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +0 +1 +2 +3 +4 +5 +τ/r0 +rh/r0 +FIG. 5. Zero points of φrh shown in the rh − τ plane for the singly- +rotating Kerr-AdS7 black hole with Pr2 +0 = 0.3 and a/r0 = 1. The red +dot with τc denotes the annihilation point for the black hole. There +are two singly-rotating Kerr-AdS7 black holes for τ = τ1. Obviously, +the topological number W = 0. +8d Kerr-AdS +τc /r0 +τ1 /r0 +0.0 +0.5 +1.0 +1.5 +2.0 +0 +1 +2 +3 +4 +5 +τ/r0 +rh/r0 +FIG. 6. Zero points of φrh shown in the rh − τ plane for the singly- +rotating eight-dimensional Kerr-AdS black hole with Pr2 +0 = 0.5 and +a/r0 = 1. The red dot with τc denotes the annihilation point for the +black hole. There are two singly-rotating Kerr-AdS8 black holes for +τ = τ1. It is easy to see that the topological number W = 0. +the generalized free energy as follows: +F = − +2π2r3 +h(r2 +h +a2) +15τ(8πPa2 −21)2 +� +42πrh(21−8πPa2) ++τ(8πPr2 +h +21)(16πPa2 −63) +� +. +(30) +As the result, by figuring out the equation φ rh = 0, one can +easily arrive at +τ = +84πrh(8πPa2 −21)(3r2 +h +2a2) +(16πPa2 −63) +� +56πPr4 +h +5(8πPa2 +21)r2 +h +63a2� +(31) +as the zero point of the vector field. +In Fig. 6, we plot the zero points of φ rh in the rh −τ plane +for the singly-rotating Kerr-AdS8 black hole. Note that for the +values of Pr2 +0 = 0.5 and a/r0 = 1, one annihilation point can +be found at τ/r0 = τc/r0 = 0.92. Based on the local prop- +erty of the zero points, which demonstrate that the topological +number W = 0 for the singly-rotating Kerr-AdS8 black hole. + +6 +9d Kerr-AdS +τc /r0 +τ1 /r0 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0 +1 +2 +3 +4 +5 +τ/r0 +rh/r0 +FIG. 7. Zero points of φrh shown in the rh − τ plane for the singly- +rotating Kerr-AdS9 black hole with Pr2 +0 = 1 and a/r0 = 1. The red +dot with τc represents the annihilation point for the black hole. There +are two singly-rotating Kerr-AdS9 black holes when τ = τ1. Further- +more, the topological number W = 0. +E. +d = 9 case +In the final of this section, we investigate the topologi- +cal number for the nine-dimensional singly-rotating Kerr-AdS +black hole. In this case, the generalized free energy is +F = − +π3r4 +h(r2 +h +a2) +48τ(2πPa2 −7)2 +� +28πrh(7−2πPa2) ++τ(10πPa2 −49)(2πPr2 +h +7) +� +. +(32) +Thus, the zero point of the vector constructed in the topologi- +cal approach can be written as +τ = +14πrh(2πPa2 −7)(7r2 +h +5a2) +(10πPa2 −49)[8πPr4 +h +3(2πPa2 +7)r2 +h +14a2] . (33) +In Fig. 7, we show the zero points of φ rh in the rh −τ plane +for the nine-dimensional singly-rotating Kerr-AdS black hole. +Note that for the values of Pr2 +0 = 1 and a/r0 = 1, one annihi- +lation point can be found at τ/r0 = τc/r0 = 0.279. Based on +the local property of the zero points, which indicate that the +topological number W = 0 in this case. +Taking into account the findings of the Secs. III and IV, for +d ≥ 6 singly-rotating Kerr-AdS black holes, they both have the +topological number W = 0, and for d ≤ 5 singly-rotating Kerr- +AdS black holes, their topological number is always unity. +V. +KERR-NEWMAN-ADS4 BLACK HOLE +Finally, we want to study the impact of the electric charge +parameter on the topological number of the four-dimensional +rotating AdS black holes. So in this section, we turn to inves- +tigate the topological number of the four-dimensional Kerr- +Newman-AdS black hole [24], whose metric and Abelian +gauge potential are [31] +ds2 = −∆r +Σ +� +dt − a +Ξ sin2 θdφ +�2 ++ Σ +∆r +dr2 + Σ +∆θ +dθ 2 ++∆θ sin2 θ +Σ +� +adt − r2 +a2 +Ξ +dφ +�2 +, +(34) +A = qr +Σ +�dt +Ξ − asin2 θdφ +Ξ +� +, +(35) +where +∆r = (r2 +a2) +� +1+ r2 +l2 +� +−2mr +q2 , +Ξ = 1− a2 +l2 , +∆θ = 1− a2 +l2 cos2 θ , +Σ = r2 +a2 cos2 θ , +in which a is the rotation parameter, m and q are the mass +and electric charge parameters, respectively, and l is the AdS +radius. The horizon radius rh are determined by equation ∆r = +0. +The mass and entropy associated with the above metric (34) +can be calculated via the standard method and the results are +M = m +Ξ2 , +S = π(r2 +h +a2) +Ξ +. +(36) +Then, one can straightforwardly obtain the generalized free +energy of the four-dimensional Kerr-Newman-AdS black hole +as +F = 24πP(r4 +h +r2 +ha2)+a2[16πPQ2(4πPa2 −3)+9] +2rh(8πPa2 −3)2 ++ +9(r2 +h +Q2) +2rh(8πPa2 −3)2 + 6πrh(r2 +h +a2)(8πPa2 −3) +2τrh(8πPa2 −3)2 +. (37) +Therefore, the zero point of the vector can be easily given as +τ = 12πr3 +h(8πPa2 −3) +X +9(Q2 −r2 +h) +, +(38) +where +X = a2[16πPQ2(4πPa2 −3)+9]−24πP(3r4 +h +r2 +ha2). +For the four-dimensional Kerr-Newman-AdS black hole, +we plot the zero points of the component φ rh in Fig. 8. Ob- +viously, there is only one stable Kerr-Newman-AdS4 black +hole for any value of τ. +Based on the local property of +the zero points, which implies that the topological number +W = 1 for the four-dimensional Kerr-Newman-AdS black +hole. Thus, it is the same as the result that was found for +the four-dimensional Kerr-AdS black hole in Sec. III, which +also seems to indicate that the electric charge parameter has no +effect on the topological number of rotating AdS black holes. +VI. +CONCLUSIONS +In this paper, we have extended our previous work [17] to +more general rotating AdS black hole cases and investigated + +7 +4d Kerr-Newman-AdS +0 +10 +20 +30 +40 +0 +1 +2 +3 +4 +5 +6 +7 +τ/r0 +rh/r0 +FIG. 8. Zero point of the vector φrh are shown on the rh − τ plane +for the Kerr-Newman-AdS4 black hole with Pr2 +0 = 0.02, a/r0 = 1, +and Q/r0 = 1. There is only one stable Kerr-Newman-AdS4 black +hole for any value of τ. For the Kerr-Newman-AdS4 black hole, the +topological number W = 1. +TABLE I. The topological number W, numbers of generation and +annihilation points for various AdS black holes. +BH solution +W Generation point Annihilation point +Schwarzschild-AdS BH [12] +0 +0 +1 +d ≥ 6 singly-rotating Kerr-AdS BH 0 +0 +1 +d = 5 singly-rotating Kerr-AdS BH 1 +1 or 0 +1 or 0 +RN-AdS BH [6] +1 +1 or 0 +1 or 0 +Kerr-AdS BH +1 +1 or 0 +1 or 0 +Kerr-Newman-AdS BH +1 +0 +0 +the topological numbers of the singly-rotating Kerr-AdS black +holes in arbitrary dimensions and the four-dimensional Kerr- +Newman-AdS black hole. Table I summarizes some inter- +esting results found in our present work. The d ≥ 6 singly- +rotating Kerr-AdS black holes and the Schwarzschild-AdS +black hole belong to the same kind of topological classes since +their topological number W = 0; the RN-AdS black hole, +the Kerr-Newman-AdS black hole, and the d = 4,5 singly- +rotating Kerr-AdS black holes belong to the another same kind +of topological classes due to their topological number W = 1; +So far, no AdS black hole solution with a topological num- +ber equal to −1 has been found in the pure Einstein-Maxwell +gravity theory. +In the present work, we have reached three really interest- +ing conclusions: i) the topological number of the uncharged +AdS black hole is significantly influenced by the existence of +the rotation parameter; ii) the dimension of spacetimes has a +remarkable effect on the topological numbers of the rotating +AdS black holes; iii) given that the four-dimensional Kerr- +AdS and Kerr-Newman-AdS black holes have the same topo- +logical numbers, it indicates that the electric charge parame- +ter has no impact on the topological number of rotating AdS +black holes. Furthermore, combining our results with those in +Refs. [6, 12, 17], we present Table II. From the results in Table +II, we can propose a conjecture that, given an AdS black hole, +the difference between its topological number and that of its +corresponding non-AdS black hole is always unity. Of course, +our conjecture needs to be further examined by investigating +the topological numbers of many other black holes and their +AdS extensions. +It is worth mentioning that, in the present paper, all we have +studied are rotating AdS black holes, whose rotation angular +velocity is less than the speed of light (i.e., a < l). Therefore, +the most related issue is to examine the correctness of our con- +jecture by investigating the topological numbers of the ultra- +spinning AdS black holes (i.e., a = l) [32–37] and the most +rapidly rotating Kerr-AdS black holes (i.e., a > l) [38, 39]. +We hope to report related progress soon. +TABLE II. The topological number W, numbers of generation and +annihilation points for various black holes and their AdS extensions. +BH solution +W Generation point Annihilation point +Schwarzschild BH [6] +-1 +0 +0 +Schwarzschild-AdS BH [12] +0 +0 +1 +RN BH [6] +0 +1 +0 +RN-AdS BH [6] +1 +1 or 0 +1 or 0 +Kerr BH [17] +0 +1 +0 +Kerr-AdS BH +1 +1 or 0 +1 or 0 +Kerr-Newman BH [17] +0 +1 +0 +Kerr-Newman-AdS BH +1 +0 +0 +d = 5 singly-rotating Kerr BH [17] 0 +1 +0 +d = 5 singly-rotating Kerr-AdS BH 1 +1 or 0 +1 or 0 +d ≥ 6 singly-rotating Kerr BH [17] -1 +0 +0 +d ≥ 6 singly-rotating Kerr-AdS BH 0 +0 +1 +ACKNOWLEDGMENTS +We thank Prof. +Yen Chin Ong for helpful discussions. +This work is supported by the National Natural Science Foun- +dation of China (NSFC) under Grant No. +12205243, No. +11675130, by the Natural Science Foundation of Sichuan +Province of China under Grant No. 2023NSFSC1347, by the +Sichuan Youth Science and Technology Innovation Research +Team (21CXTD0038), and by the Doctoral Research Initia- +tion Project of China West Normal University under Grant No. +21E028. +[1] P.V.P. 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D 103, 024053 (2021). + diff --git a/kNE1T4oBgHgl3EQfNgMl/content/tmp_files/load_file.txt b/kNE1T4oBgHgl3EQfNgMl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bb21f103169e8ad0607ffd5b70173078991835b0 --- /dev/null +++ b/kNE1T4oBgHgl3EQfNgMl/content/tmp_files/load_file.txt @@ -0,0 +1,597 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf,len=596 +page_content='Topological classes of rotating AdS black holes Di Wu∗ School of Physics and Astronomy, China West Normal University, Nanchong, Sichuan 637002, People’s Republic of China (Dated: January 10, 2023) In this paper, we investigate the topological numbers for the singly-rotating Kerr-AdS black holes in all dimensions and the four-dimensional Kerr-Newman-AdS black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' We show that for uncharged AdS black holes, the rotation parameter has a significant impact on the topological number, and for rotating AdS black holes, the dimension of spacetimes has an important effect on the topological number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' In addition, it indicates that the electric charge parameter has no impact on the topological number of rotating AdS black holes because of the four-dimensional Kerr-AdS and Kerr-Newman-AdS black holes have the same topological numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Furthermore, we propose a conjecture that, given an AdS black hole, the difference between its topological number and that of its corresponding non-AdS black hole is always unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' INTRODUCTION In recent years, topology as an important mathematical tool has been applied to various research fields in physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' As one of the most fundamental and fascinating objects in nature, the black hole has been the subject of extensive research, among which the study of black hole topology has shed new light on the nature of gravity through light rings [1–4], timelike circular orbits [5], and thermodynamic properties [6–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' In particular, in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' [6], a new method to investigate the thermodynamic topological properties of black holes is pro- posed by considering black hole solutions as topological ther- modynamic defects and constructing topological numbers in- dependently of the intrinsic parameters of black holes, and further, dividing all black hole solutions into three categories according to their different topological numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Because these topological numbers are universal constants that are in- dependent of the black hole solution parameters are very im- portant for understanding the nature of black holes and grav- ity, and further, may provide new clues to the establishment of the quantum gravity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' The topological approach pro- posed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' [6] quickly gained popularity due to its straight- forwardness and adaptability, and it was subsequently effec- tively used to explore the topological numbers of several well- known black hole solutions [12–16], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=', the Schwarzschild- AdS black hole solution [12], the static black hole solutions in Lovelock gravity [13], the static Gauss-Bonnet-AdS black hole solutions [14], the static black hole solution in nonlinear electrodynamics [15], and the static Born-Infeld AdS black hole solution [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Very recently, we extended the topologi- cal approach to rotating black hole cases and investigate the topological numbers for the cases of rotating Kerr and Kerr- Newman black hole solutions [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Although there has been some recent progress in investi- gating the topological number of black holes, the topological number of the rotating AdS black holes remains virgin terri- tory and thus needs to be explored deeply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' On the other hand, the study of rotating AdS black holes has shed light on the nature of gravity through gauge-gravity dualities [18–20], so ∗ wdcwnu@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='com it is very important and remarkable to investigate the topolog- ical number of rotating AdS black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' These two aspects motivate us to conduct the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' In this paper, we investigate the topological number of the d-dimensional singly-rotating Kerr-AdS black holes and the four-dimensional Kerr-Newman-AdS black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' We demon- strate that for uncharged AdS black holes, the rotation param- eter has a significant effect on the topological number, and for rotating AdS black holes, the dimension of spacetimes has a remarkable effect on the topological number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' In ad- dition, we also find that the electric charge does not play a role in the topological classification, in the sense of having no effect on the topological number of rotating AdS black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' The remaining part of this paper is organized as fol- lows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' II, we first give a brief review of the topolog- ical approach and investigate the topological number of the four-dimensional Schwarzschild-AdS black hole as a warmup exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' III, we focus on the topological number of the four-dimensional rotating Kerr-AdS black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' IV, we extend the discussions from Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' III to the cases of the d-dimensional singly-rotating Kerr-AdS black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' V, we turn to discuss the topological number of the four-dimensional Kerr-Newman-AdS black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Finally, we present our conclusions in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' SCHWARZSCHILD-ADS4 BLACK HOLE In this section, we first present a brief review of the topo- logical approach proposed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' [6], then investigate the topological number of the four-dimensional Schwarzschild- AdS black hole as a warmup exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' As shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' [6], one can introduce the generalized off-shell free energy F = M − S τ , (1) for a black hole thermodynamical system with mass M and entropy S, where τ is an extra variable that can be thought of as the inverse temperature of the cavity inclosing the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' The generalized free energy only converts to being on- shell if τ = 1/T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='03002v1 [hep-th] 8 Jan 2023 2 In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' [6], a core vector φ is defined as φ = �∂F ∂rh , −cotΘcscΘ � , (2) where the two parameters obey 0 < rh < +∞, 0 ≤ Θ ≤ π, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' The component φ Θ is divergent at Θ = 0,π, thus, the direction of the vector points outward there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' A topological current can be described as follows using Duan’s φ-mapping topological current theory [21–23]: jµ = 1 2π εµνρεab∂νna∂ρnb , µ,ν,ρ = 0,1,2, (3) where ∂ν = (∂/∂xν) and xν = (τ, rh, Θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' The unit vec- tor n is obtained as n = (nr,nφ), where nr = φ rh/||φ|| and nΘ = φ Θ/||φ||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' It is simple to demonstrate that the topological current (3) given above is conserved, allowing one to deduce ∂µ jµ = 0 with simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' It is then established that the topo- logical current jµ is a δ-function of the field configuration [3, 22, 23] jµ = δ 2(φ)Jµ �φ x � , (4) where the 3-dimensional Jacobian Jµ (φ/x) is defined as: εabJµ (φ/x) = εµνρ∂νφ a∂ρφ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' It is simple to indicate that jµ equals zero unless when φ a(xi) = 0, hence the topological number W can be determined as follows W = � Σ j0d2x = N ∑ i=1 βiηi = N ∑ i=1 wi , (5) where βi is the positive Hopf index counting the number of the loops of the vector φ a in the φ-space when xµ are around the zero point zi, ηi = sign(J0(φ/x)zi) = ±1 is the Brouwer degree, and wi is the winding number for the i-th zero point of φ that is contained in Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' It is important to keep in mind that two loops Σ1 and Σ2 have the same winding number if they both enclose the same zero point of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' On the other hand, if there is no zero point in the surrounded region, then one can arrive at W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' In the following, we investigate the the topological num- ber of the four-dimensional Schwarzschild-AdS black hole via the above topological approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' For the four-dimensional Schwarzschild-AdS black hole, its metric has the form [24] ds2 = −f(r)dt2 + dr2 f(r) +r2(dθ 2 +sin2 θdφ 2), (6) where f(r) = 1− 2m r + r2 l2 , in which m is the mass parameter, and l is the cosmological scale associated with the pressure P = 3/(8πl2) of the four- dimensional AdS black holes [25–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' The mass and entropy associated with the above metric (6) can be computed via the standard method and have the following exquisite forms: M = m, S = πr2 h , (7) Schwarzschild-AdS τ1 /r0 τc /r0 0 10 20 30 40 50 60 0 5 10 15 20 τ/r0 rh/r0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Zero points of the vector φrh are shown on the rh − τ plane for the Schwarzschild-AdS black hole with Pr2 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='0022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' The red dot with τc represents the annihilation point for the Schwarzschild- AdS black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' There are two Schwarzschild-AdS black holes when τ = τ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Obviously, the topological number W = 1−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' where rh are the locations of the event and Cauchy horizons that satisfy the equation f(rh) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' For the Schwarzschild-AdS4 black hole, one can obtain the generalized free energy F = rh 2 + 4π 3 Pr2 h − πrh τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' (8) The components of the vector φ can then be easily calculated as φ rh = 1 2 +4πPr2 h − 2πrh τ , (9) φ Θ = −cotΘcscΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' (10) By solving the equation φ rh = 0, one can obtain a curve on the rh−τ plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' For the four-dimensional Schwarzschild-AdS black hole, one can obtain τ = 4πrh 1+8πPr2 h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' (11) Taking the pressure Pr2 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='0022, where r0 is an arbitrary length scale set by the size of a cavity enclosing the black hole, we show zero points of φ rh in the rh − τ plane in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' For small τ, such as τ = τ1, there are two intersection points for the Schwarzschild-AdS black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' The intersection points exactly satisfy the condition τ = 1/T, and thus rep- resent the on-shell Schwarzschild-AdS black hole solutions with the characteristic temperature T = 1/τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' The two intersec- tion points for the Schwarzschild-AdS black hole can coincide each other when τ = τc, and then vanish when τ > τc, there- fore τc is a annihilation point can be found at τc = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='75r0 which can be seen straightforward from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Furthermore, the annihilation point τc divides the Schwarzschild-AdS black hole into the upper and lower branches with the winding num- ber w = 1 and w = −1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Based upon the local property of the zero point, one can easily find that the topo- logical number W = 0 for the Schwarzschild-AdS black hole, which is consistent with the result given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' 3 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' KERR-ADS4 BLACK HOLE In the section, we will focus on the topological number of the four-dimensional Kerr-AdS black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' For the Kerr-AdS4 black hole, the metric has the form [24] ds2 = −∆r Σ � dt − a Ξ sin2 θdφ �2 + Σ ∆r dr2 + Σ ∆θ dθ 2 +∆θ sin2 θ Σ � adt − r2 +a2 Ξ dφ �2 (12) in the Boyer-Lindquist coordinates, where ∆r = (r2 +a2) � 1+ r2 l2 � −2mr, Ξ = 1− a2 l2 , ∆θ = 1− a2 l2 cos2 θ , Σ = r2 +a2 cos2 θ , in which a is the rotation parameter, m is the mass parameter, and l is the AdS radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' The mass M and entropy S associated with the above solu- tion (12) are [28] M = m Ξ2 , S = π(r2 h +a2) Ξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' (13) According to the generalized off-shell free energy (1), and us- ing l2 = 3/(8πP), thus one can easily yield F = 3(r2 h +a2) � 2πrh(8πPa2 +4Prhτ −3)+3τ � 2rhτ(8πPa2 −3)2 (14) for the Kerr-AdS4 black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' The components of the vector φ can be computed as φ rh = 12π(8πPa2 −3)r3 h +3a2(8πPr2 h −3)τ 2r2 hτ(8πPa2 −3)2 + 9(r2 h +8πPr4 h) 2r2 h(8πPa2 −3)2 , (15) φ Θ = −cotΘcscΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' (16) By solving the equation φ rh = 0, one can obtain τ = 4πr3 h(3−8πPa2) a2(8πPr2 h −3)+3(8πPr4 h +r2 h) (17) as the zero point of the vector field φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Taking the pressure Pr2 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='0022 and the rotation param- eter a = r0, we show zero points of φ rh in the rh − τ plane in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Note that for these values of Pr2 0 and a/r0, one generation point and one annihilation point can be found at τ/r0 = τa/r0 = 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='93 and τ/r0 = τb/r0 = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='81, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' We find that there are one large black hole branch for τ < τa, three black hole branches for τa < τ < τb, and one small black hole branch for τ > τb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Calculating the winding number w for these three black hole branches, we indicate that the small and large black hole branches both have w = 1, however the inter- mediate black hole branch has w = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Thus, the Kerr-AdS4 black hole always has the topological number W = 1, which LBH SBH IBH τa /r0 τb /r0 23 24 25 26 27 28 29 0 1 2 3 4 5 6 7 τ/r0 rh/r0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Zero points of φrh shown in the rh − τ plane for the Kerr- AdS4 black hole with Pr2 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='0022 and a = r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' The red solid, blue dashed, and black solid lines are for the large black hole (LBH), inter- mediate black hole (IBH), and small black hole (SBH), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' The annihilation and generation points are represented by red and black dots, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' is same to the case of the RN-AdS4 black hole [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' So the Kerr-AdS4 black hole and RN-AdS4 black hole are the same kind of black hole solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' What is more, since the topolog- ical number of the Schwarzschild-AdS black hole is 0, while that of Kerr-AdS black hole is 1, it indicate that the rotation parameter has a remarkable effect on the topological number for the uncharged AdS black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' SINGLY-ROTATING KERR-ADS BLACK HOLES IN ARBITRARY DIMENSIONS In this section, we will extend the above discussions to the cases of rotating black holes in higher dimensions by consid- ering the singly-rotating Kerr-AdS black holes in arbitrary di- mensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' For d-dimensional singly-rotating Kerr-AdS black holes, the metric has the form [29] ds2 = −∆r Σ � dt − asin2 θ Ξ dφ �2 + Σ ∆r dr2 + Σ ∆θ dθ 2 +∆θ sin2 θ Σ � adt − r2 +a2 Ξ dφ �2 +r2 cos2 θdΩ2 d−4 , (18) where dΩd denotes the line element of the d-dimensional unit sphere, and ∆r = (r2 +a2) � 1+ r2 l2 � −2mr, Ξ = 1− a2 l2 , ∆θ = 1− a2 l2 cos2 θ , Σ = r2 +a2 cos2 θ , are the same as their correponding ones in the four- dimensional case in the last section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' 4 The thermodynamic quantities are [30] M = ωd−2m 4πΞ2 �(d −4)Ξ 2 +1 � , J = ωd−2ma 4πΞ2 , Ω = a(r2 h +l2) l2(r2 h +a2) , S = A 4 = ωd−2 4Ξ (r2 h +a2)rd−4 h , T = rh 2π � 1+ r2 h l2 �� 1 r2 h +a2 + d −3 2r2 h � − 1 2πrh , V = rhA d −1 � 1+ a2(r2 h +l2) (d −2)Ξl2r2 h � , P = (d −1)(d −2) 16πl2 , (19) where ωd−2 = 2π(d−1)/2/Γ[(d − 1)/2], and rh is determined by the horizon equation ∆r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' In the following, we will dis- cuss the topological number of the singly-rotating Kerr-AdS black holes from five to nine dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' d = 5 case We first consider d = 5 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' (19), one can obtain the expression of the generalized free energy as F = − π(r2 h +a2) 8τ(4πPa2 −3)2 � 12πrh(3−4πPa2) +τ(4πPr2 h +3)(4πPa2 −9) � , (20) and the components of the vector φ can be computed as φ rh = π 4τ(4πPa2 −3)2 � 6π(4πPa2 −3)(3r2 h +a2) −rhτ(4πPa2 −9) � 4πP(2r2 h +a2)+3 �� , (21) φ Θ = −cotΘcscΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' (22) It is simple to obtain τ = 6π(4πPa2 −3)(3r2 h +a2) rh(4πPa2 −9)[4πP(2r2 h +a2)+3] (23) as the zero point of the vector field φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' We plot the zero points of the component φ rh for the singly- rotating Kerr-AdS5 black hole with Pr2 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='02 and a/r0 = 1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Note that for these values of Pr2 0 and a/r0, one generation point and one annihilation point can be found at τ/r0 = τa/r0 = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='96 and τ/r0 = τb/r0 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='36, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' One can easily obtain the topological number W = 1 for the singly-rotating Kerr-AdS5 black hole using the local property of the zero points, which is the same as the case of the four- dimensional Kerr-AdS black hole in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' LBH SBH IBH τa /r0 τb /r0 5 6 7 8 9 10 0 1 2 3 4 5 τ/r0 rh/r0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Zero points of φrh shown in the rh − τ plane for the singly- rotating Kerr-AdS5 black hole with Pr2 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='02 and a/r0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' The red solid, blue dashed, and black solid lines are for the large black hole (LBH), intermediate black hole (IBH), and small black hole (SBH), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' The annihilation and generation points are represented by red and black dots, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' d = 6 case Next, we consider d = 6 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' In this case, the generalized free energy is F = − 2πrh(r2 h +a2) 3τ(4πPa2 −5)2 � 5πrh(5−4πPa2) +τ(4πPr2 h +5)(2πPa2 −5) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' (24) Thus, the components of the vector φ are φ rh = 2π 3τ(4πPa2 −5) � 10πrh(4πPa2 −5)(2r2 h +a2) −τ(2πPa2 −5) � 20πPr4 h +3(4πPa2 +5)r2 h +5a2�� , (25) φ Θ = −cotΘcscΘ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' (26) So the zero point of the vector field φ is τ = 10πrh(4πPa2 −5)(2r2 h +a2) (2πPa2 −5) � 20πPr4 h +3(4πPa2 +5)r2 h +5a2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' (27) Taking Pr2 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='1 and a/r0 = 1, we plot zero points of φ rh in the rh −τ plane in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' For small τ, such as τ = τ1, there are two intersection points for the singly-rotating Kerr-AdS6 black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' The two intersection points for the black hole can coincide each other when τ = τc, and then vanish when τ > τc, therefore τc is an annihilation point can be found at τc = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='84r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Based upon the local property of the zero point, one can easily get to the topological number W = 0 for the singly-rotating Kerr-AdS6 black hole, which is different from the corresponding results W = 1 for the Kerr-AdS4 black hole in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' III and the singly-rotating Kerr-AdS5 black hole in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' IV A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Therefore, for rotating AdS black holes, the di- mension of spacetimes has a remarkable effect on the topo- logical number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' On the other hand, the singly-rotating Kerr- AdS6 black hole and Schwarzschild-AdS black hole are the 5 6d Kerr-AdS τc /r0 τ1 /r0 0 1 2 3 4 5 0 1 2 3 4 5 τ/r0 rh/r0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Zero points of φrh shown in the rh − τ plane for the singly- rotating Kerr-AdS6 black hole with Pr2 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='1 and a/r0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' The red dot with τc represents the annihilation point for the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' There are two singly-rotating Kerr-AdS6 black holes when τ = τ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' It is easy to obtain the topological number W = 1−1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' same kind of black hole solutions since both of their topologi- cal numbers are equal to zero via the topological classification method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' d = 7 case Then, we consider d = 7 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' The generalized free energy has the form F = − 3π2r2 h(r2 h +a2) 16τ(15−8πPa2)2 � 20πrh(15−8πPa2) +τ(8πPr2 h +15)(8πPa2 −25) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' (28) Therefore, one can straightforwardly obtain τ = 10πrh(8πPa2 −15)(5r2 h +3a2) (8πPa2 −25) � 6r2 h(4πPr2 h +5)+a2(16πPr2 h +15) � (29) by solving the equation φ rh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' We plot the zero points of φ rh in the rh − τ plane for the singly-rotating Kerr-AdS7 black hole in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Note that for the values of Pr2 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='3 and a/r0 = 1, one annihilation point can be found at τ/r0 = τc/r0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Based on the local property of the zero points, we have the topological number W = 0 for the singly-rotating Kerr-AdS7 black hole, which is the same as the case of he singly-rotating Kerr-AdS6 black hole in the subsection IV B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' d = 8 case Let us continue dealing with the d = 8 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Similar to the procedure done in the previous three subsections, one can give 7d Kerr-AdS τc /r0 τ1 /r0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='5 0 1 2 3 4 5 τ/r0 rh/r0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Zero points of φrh shown in the rh − τ plane for the singly- rotating Kerr-AdS7 black hole with Pr2 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='3 and a/r0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' The red dot with τc denotes the annihilation point for the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' There are two singly-rotating Kerr-AdS7 black holes for τ = τ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Obviously, the topological number W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' 8d Kerr-AdS τc /r0 τ1 /r0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='0 0 1 2 3 4 5 τ/r0 rh/r0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Zero points of φrh shown in the rh − τ plane for the singly- rotating eight-dimensional Kerr-AdS black hole with Pr2 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='5 and a/r0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' The red dot with τc denotes the annihilation point for the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' There are two singly-rotating Kerr-AdS8 black holes for τ = τ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' It is easy to see that the topological number W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' the generalized free energy as follows: F = − 2π2r3 h(r2 h +a2) 15τ(8πPa2 −21)2 � 42πrh(21−8πPa2) +τ(8πPr2 h +21)(16πPa2 −63) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' (30) As the result, by figuring out the equation φ rh = 0, one can easily arrive at τ = 84πrh(8πPa2 −21)(3r2 h +2a2) (16πPa2 −63) � 56πPr4 h +5(8πPa2 +21)r2 h +63a2� (31) as the zero point of the vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' 6, we plot the zero points of φ rh in the rh −τ plane for the singly-rotating Kerr-AdS8 black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Note that for the values of Pr2 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='5 and a/r0 = 1, one annihilation point can be found at τ/r0 = τc/r0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Based on the local prop- erty of the zero points, which demonstrate that the topological number W = 0 for the singly-rotating Kerr-AdS8 black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' 6 9d Kerr-AdS τc /r0 τ1 /r0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='5 0 1 2 3 4 5 τ/r0 rh/r0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Zero points of φrh shown in the rh − τ plane for the singly- rotating Kerr-AdS9 black hole with Pr2 0 = 1 and a/r0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' The red dot with τc represents the annihilation point for the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' There are two singly-rotating Kerr-AdS9 black holes when τ = τ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Further- more, the topological number W = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' d = 9 case In the final of this section, we investigate the topologi- cal number for the nine-dimensional singly-rotating Kerr-AdS black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' In this case, the generalized free energy is F = − π3r4 h(r2 h +a2) 48τ(2πPa2 −7)2 � 28πrh(7−2πPa2) +τ(10πPa2 −49)(2πPr2 h +7) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' (32) Thus, the zero point of the vector constructed in the topologi- cal approach can be written as τ = 14πrh(2πPa2 −7)(7r2 h +5a2) (10πPa2 −49)[8πPr4 h +3(2πPa2 +7)r2 h +14a2] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' (33) In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' 7, we show the zero points of φ rh in the rh −τ plane for the nine-dimensional singly-rotating Kerr-AdS black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Note that for the values of Pr2 0 = 1 and a/r0 = 1, one annihi- lation point can be found at τ/r0 = τc/r0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='279.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Based on the local property of the zero points, which indicate that the topological number W = 0 in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Taking into account the findings of the Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' III and IV, for d ≥ 6 singly-rotating Kerr-AdS black holes, they both have the topological number W = 0, and for d ≤ 5 singly-rotating Kerr- AdS black holes, their topological number is always unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' KERR-NEWMAN-ADS4 BLACK HOLE Finally, we want to study the impact of the electric charge parameter on the topological number of the four-dimensional rotating AdS black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' So in this section,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' we turn to inves- tigate the topological number of the four-dimensional Kerr- Newman-AdS black hole [24],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' whose metric and Abelian gauge potential are [31] ds2 = −∆r Σ � dt − a Ξ sin2 θdφ �2 + Σ ∆r dr2 + Σ ∆θ dθ 2 +∆θ sin2 θ Σ � adt − r2 +a2 Ξ dφ �2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' (34) A = qr Σ �dt Ξ − asin2 θdφ Ξ � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' (35) where ∆r = (r2 +a2) � 1+ r2 l2 � −2mr +q2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Ξ = 1− a2 l2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' ∆θ = 1− a2 l2 cos2 θ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Σ = r2 +a2 cos2 θ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' in which a is the rotation parameter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' m and q are the mass and electric charge parameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' and l is the AdS radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' The horizon radius rh are determined by equation ∆r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' The mass and entropy associated with the above metric (34) can be calculated via the standard method and the results are M = m Ξ2 , S = π(r2 h +a2) Ξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' (36) Then, one can straightforwardly obtain the generalized free energy of the four-dimensional Kerr-Newman-AdS black hole as F = 24πP(r4 h +r2 ha2)+a2[16πPQ2(4πPa2 −3)+9] 2rh(8πPa2 −3)2 + 9(r2 h +Q2) 2rh(8πPa2 −3)2 + 6πrh(r2 h +a2)(8πPa2 −3) 2τrh(8πPa2 −3)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' (37) Therefore, the zero point of the vector can be easily given as τ = 12πr3 h(8πPa2 −3) X +9(Q2 −r2 h) , (38) where X = a2[16πPQ2(4πPa2 −3)+9]−24πP(3r4 h +r2 ha2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' For the four-dimensional Kerr-Newman-AdS black hole, we plot the zero points of the component φ rh in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Ob- viously, there is only one stable Kerr-Newman-AdS4 black hole for any value of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Based on the local property of the zero points, which implies that the topological number W = 1 for the four-dimensional Kerr-Newman-AdS black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Thus, it is the same as the result that was found for the four-dimensional Kerr-AdS black hole in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' III, which also seems to indicate that the electric charge parameter has no effect on the topological number of rotating AdS black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' CONCLUSIONS In this paper, we have extended our previous work [17] to more general rotating AdS black hole cases and investigated 7 4d Kerr-Newman-AdS 0 10 20 30 40 0 1 2 3 4 5 6 7 τ/r0 rh/r0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Zero point of the vector φrh are shown on the rh − τ plane for the Kerr-Newman-AdS4 black hole with Pr2 0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='02, a/r0 = 1, and Q/r0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' There is only one stable Kerr-Newman-AdS4 black hole for any value of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' For the Kerr-Newman-AdS4 black hole, the topological number W = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' The topological number W, numbers of generation and annihilation points for various AdS black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' BH solution W Generation point Annihilation point Schwarzschild-AdS BH [12] 0 0 1 d ≥ 6 singly-rotating Kerr-AdS BH 0 0 1 d = 5 singly-rotating Kerr-AdS BH 1 1 or 0 1 or 0 RN-AdS BH [6] 1 1 or 0 1 or 0 Kerr-AdS BH 1 1 or 0 1 or 0 Kerr-Newman-AdS BH 1 0 0 the topological numbers of the singly-rotating Kerr-AdS black holes in arbitrary dimensions and the four-dimensional Kerr- Newman-AdS black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Table I summarizes some inter- esting results found in our present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' The d ≥ 6 singly- rotating Kerr-AdS black holes and the Schwarzschild-AdS black hole belong to the same kind of topological classes since their topological number W = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' the RN-AdS black hole, the Kerr-Newman-AdS black hole, and the d = 4,5 singly- rotating Kerr-AdS black holes belong to the another same kind of topological classes due to their topological number W = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' So far, no AdS black hole solution with a topological num- ber equal to −1 has been found in the pure Einstein-Maxwell gravity theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' In the present work, we have reached three really interest- ing conclusions: i) the topological number of the uncharged AdS black hole is significantly influenced by the existence of the rotation parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' ii) the dimension of spacetimes has a remarkable effect on the topological numbers of the rotating AdS black holes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' iii) given that the four-dimensional Kerr- AdS and Kerr-Newman-AdS black holes have the same topo- logical numbers, it indicates that the electric charge parame- ter has no impact on the topological number of rotating AdS black holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Furthermore, combining our results with those in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' [6, 12, 17], we present Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' From the results in Table II, we can propose a conjecture that, given an AdS black hole, the difference between its topological number and that of its corresponding non-AdS black hole is always unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Of course, our conjecture needs to be further examined by investigating the topological numbers of many other black holes and their AdS extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' It is worth mentioning that, in the present paper, all we have studied are rotating AdS black holes, whose rotation angular velocity is less than the speed of light (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=', a < l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Therefore, the most related issue is to examine the correctness of our con- jecture by investigating the topological numbers of the ultra- spinning AdS black holes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=', a = l) [32–37] and the most rapidly rotating Kerr-AdS black holes (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=', a > l) [38, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' We hope to report related progress soon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' The topological number W, numbers of generation and annihilation points for various black holes and their AdS extensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='BH solution ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='W Generation point Annihilation point ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='Schwarzschild BH [6] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='Schwarzschild-AdS BH [12] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='RN BH [6] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='RN-AdS BH [6] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='1 or 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='1 or 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='Kerr BH [17] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='Kerr-AdS BH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='1 or 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='1 or 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='Kerr-Newman BH [17] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='Kerr-Newman-AdS BH ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='d = 5 singly-rotating Kerr BH [17] 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='d = 5 singly-rotating Kerr-AdS BH 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='1 or 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='1 or 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='d ≥ 6 singly-rotating Kerr BH [17] -1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='d ≥ 6 singly-rotating Kerr-AdS BH 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='ACKNOWLEDGMENTS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='We thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Yen Chin Ong for helpful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' This work is supported by the National Natural Science Foun- dation of China (NSFC) under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' 12205243, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' 11675130, by the Natural Science Foundation of Sichuan Province of China under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' 2023NSFSC1347, by the Sichuan Youth Science and Technology Innovation Research Team (21CXTD0038), and by the Doctoral Research Initia- tion Project of China West Normal University under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' 21E028.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Cunha, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content=' Berti, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/kNE1T4oBgHgl3EQfNgMl/content/2301.03002v1.pdf'} +page_content='A.' metadata={'source': 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+++ b/ktFJT4oBgHgl3EQfYywB/content/tmp_files/2301.11527v1.pdf.txt @@ -0,0 +1,1077 @@ +Opinion-aware Influence Maximization in Online +Social Networks +Ying Wang♯, Yanhao Wang† +School of Data Science and Engineering, East China Normal University, Shanghai, China +♯yingwang007@stu.ecnu.edu.cn +†yhwang@dase.ecnu.edu.cn +Abstract—Influence maximization (IM) aims to find seed users +on an online social network to maximize the spread of informa- +tion about a target product through word-of-mouth propagation +among all users. Prior IM methods mostly focus on maximizing +the overall influence spread, which assumes that all users are +potential customers of the product and that more exposure leads +to higher benefits. However, in real-world scenarios, some users +who dislike the product may express and spread negative opin- +ions, damaging the product’s reputation and lowering its profit. +This paper investigates the opinion-aware influence maximization +(OIM) problem, which finds a set of seed users to maximize the +positive opinions toward the product while minimizing the nega- +tive opinions. We propose a novel algorithm for the OIM problem. +Specifically, after obtaining the users with positive and negative +opinions towards the product from historical data, we design a +reverse reachable set-based method for opinion-aware influence +estimation and a sandwich approximation algorithm for seed +set selection. Despite the NP-hardness and non-submodularity of +OIM, our algorithm achieves a data-dependent approximation +factor for OIM. Experimental results on three real-world datasets +demonstrate that our algorithm improves the spread of positive +opinions while reducing the spread of negative opinions compared +to existing methods. +I. INTRODUCTION +With the boom in social networking, the problem of influ- +ence maximization [1] (IM) has been extensively studied over +the last two decades for its wide applications in viral market- +ing [2] and recommendation [3]. The vanilla IM problem aims +to find a set of seed users from an online social network (OSN) +to maximize the spread of information about a target product. +For example, when a company wants to launch a promotion +campaign on an OSN, instead of promoting its product to all +users at once, a more cost-effective strategy is to select a small +set of influential users on the network as seeds to initiate the +spread of promotion information, hoping that a larger number +of users can be reached through word-of-mouth propagation +along social connections. However, the vanilla IM problem +only considers maximizing the total exposure of the product +to all users but ignores users’ opinions. In real-life scenarios, +some users may express and spread negative opinions about +the product, damaging its reputation and lowering its profit. As +an illustrative example, a singer may want to promote her/his +new song to a group of audiences who are more likely to +comment on it positively and spread it to their friends who +may also like it while avoiding dispersing it to the ones who +may express negative opinions about it. +Motivated by the above application, opinion-aware influence +maximization [4]–[8] (OIM) problems consider users’ opin- +Historical Rating Matrix +p1 +p2 +p3 +p4 +p5 +u1 +u2 +u3 +u4 +u5 +u6 +Social Graph +u3 +u4 +u1 +u2 +u5 +u6 +Users’ opinions on item p +Opinion +positive +Items +Users +neutral +negative +u3 +u4 +u1 +u2 +u5 +u6 +Graph with positive, neutral, and +negative users for item p +S = {u1, u4} +Seed Selection +User & Item Embeddings +Graph +Neural +Network +Positive +Neutral +Negative +u1 +u6 +u4 +u5 +u3 +u2 +p +p2 +p1 +p3 +p5 +p4 +Fig. 1: Illustration for opinion-aware IM in an OSN. +ions towards specific items (e.g., products and events), which +might be positive, neutral, or negative, and their goals are +mostly to maximize the positive opinions while minimizing +the negative opinions among all influenced users. Although +OIM has been extensively studied in the literature, prior +work still has several limitations. Most existing methods [4]– +[6] are item-unaware in the sense that they do not consider +incorporating historical information into OIM. They typically +assume that user opinions are arbitrarily generated or formed +merely from their neighbors’ opinions based on some given +opinion models. Although a few studies [7], [8] on OIM are +based on historical data to estimate users’ opinions towards the +target product, their methods for opinion estimation and seed +set selection are heuristics without performance guarantees. +In this work, we investigate a new opinion-aware influence +maximization (OIM) problem on OSNs as illustrated in Fig. 1. +Specifically, we adopt any graph neural network model for +recommendation [3] that captures both users’ historical rating +information and social connections to learn latent vector +representations of items (products) and users, from which +the sets of users with positive, neutral, and negative opinions +towards the target item are obtained. Then, we generalize the +independent cascade model to be opinion-aware to describe +the information diffusion process of the item on the social +graph. Based on the above concepts, we formalize our OIM +problem, aimed to find a set of k seed users from the graph to +maximize the difference between the influence spread on the +positive and negative sets of users for the given item under the +opinion-aware IC model. We show that OIM is NP-hard and +its objective function is non-monotone, neither submodular nor +supermodular. Consequently, the classic greedy algorithm for +arXiv:2301.11527v1 [cs.SI] 27 Jan 2023 + +submodular maximization [9], which has been widely used +in existing IM methods [1], cannot provide any theoretical +guarantee on OIM. To address this issue, we propose a novel +algorithm for OIM, which consists of an extended reverse +reachable (RR) set-based method [10], [11] for opinion-aware +influence estimation and a sandwich approximation strategy +for greedy seed selection based on the upper- and lower-bound +submodular functions of the objective function of OIM derived +from the difference of submodular (DS) decomposition [12]. +Our proposed algorithm runs in polynomial time and achieves +a data-dependent approximation for OIM. Finally, we conduct +extensive experiments on three real-world datasets to compare +our proposed algorithm with existing IM methods. The results +demonstrate that our algorithm significantly improves the +spread of positive opinions about a target item and reduces +the negative ones over existing IM methods while running in a +comparable time. Our main contributions are listed as follows: +• We formally define the opinion-aware influence maximiza- +tion problem and show its theoretical hardness. (Section III) +• We propose a data-dependent approximation algorithm con- +sisting of RR set-based influence estimation and sandwich +approximation-based seed selection for OIM. (Section IV) +• We perform extensive experiments to verify the effectiveness +and efficiency of our proposed algorithm. (Section V) +II. RELATED WORK +In this section, we briefly discuss the literature related to +this work, namely influence maximization and opinion-aware +influence maximization. +Influence Maximization: There has been a vast amount of +literature on influence maximization in OSNs (see [13] for +an extensive survey). The IM problem was first formulated +by Kempe et al. [1], where two propagation models, i.e., the +independent cascade (IC) model and the linear threshold (LT) +model, were used to simulate the information diffusion process +on social graphs. They then showed the NP-hardness and +submodularity of IM under IC and LT models. Moreover, Chen +et al. [14] proved that computing the influence spread under IC +and LT models is #P-hard. Despite the theoretical hardness, a +bunch of heuristic and approximation algorithms [10], [11], +[14]–[19] were proposed to scale IM on massive graphs +with millions of nodes. However, all the above methods for +vanilla IM problems do not consider using customized seeding +schemes for different items. +Opinion-aware Influence Maximization: Unlike vanilla IM, +opinion-aware IM (OIM) problems consider maximizing total +influence spread while reducing negative opinions. Chen et +al. [4] proposed an IC-N model by considering that negative +opinions emerge randomly during the IC diffusion process as +well as efficient IM methods under the IC-N model. The OIM +problems in [5]–[8] were the most similar to ours since their +goals were to maximize the total opinion in the form of the +difference between positive and negative opinions under the IC +or LT model. Nonetheless, the OIM problems in [5], [6] still +do not consider how to obtain user opinions from historical +data. In addition, most of them [5], [7], [8] are specific for the +LT model but cannot work with the more popular IC model. +Finally, their proposed OIM algorithms were all heuristics +without performance guarantees. Therefore, none of the above +methods can work directly for the problem in this paper. +III. PROBLEM FORMULATION +Let [n] denote a set of integers {1, . . . , n}. An online +social network (OSN) is represented as a directed graph +G = (V, E, w), where V = {u1, . . . , un} is a set of n users +(nodes), E ⊆ V ×V is a set of social links (edges) among users +and e = (u, v) ∈ E is the edge from u to v, and w : E → R+ +assigns a nonnegative weight w(e) to each edge e ∈ E. We +treat an undirected graph as a special case of a directed graph +with symmetric directed edges. We have a set of m items +(products) P = {p1, . . . , pm} associated with graph G. Let +R = {rpu ∈ R | p ∈ P, u ∈ V } be the set of observed ratings, +where rpu is the rating of user u ∈ V on item p ∈ P. All rating +values are normalized to be nonnegative, and higher rating +values indicate more positive opinions. We assume that a graph +neural network (GNN) based model for recommendation [3] is +used to learn latent vector representations hu and hp of each +user u ∈ V and item p ∈ P from R and G to jointly capture +the effects of both a user’s interest and her/his neighborhoods’ +preferences on her/his rating on an item. Then, the preference +of user u on item p is estimated from their latent vector +representations, i.e., ˆrpu = s(hp, hu) ∈ R+, where the scoring +function s(·, ·) can be the inner product, cosine, multi-layer +perception, etc. We define the opinion opu of user u on item +p to be “positive” (1), “neutral” (0), or “negative” (−1) based +on the difference between the predicted rating value ˆrpu and +the neutral rating value r0 as follows: +opu = +� +� +� +� +� +1, +ˆrpu − r0 ≥ τ +0, +|ˆrpu − r0| < τ +−1, +r0 − ˆrpu ≥ τ +(1) +where τ > 0 decides the width of neutral zones. In practice, we +set the value of r0 to the average or median of all (observed) +rating values. For any item p, the set V of all users can be +divided into three subsets based on their opinions on p, i.e., +positive set V + +p = {u ∈ V | opu = 1}, neutral set V 0 +p = {u ∈ +V | opu = 0}, and negative set V − +p = {u ∈ V | opu = −1}. +We use an opinion-aware variant of the independent cascade +(OIC) model to describe the information propagation process +on G, as the IC model and its variants have been widely used in +the existing literature on influence and opinion maximization +problems [1], [4]. The influence diffusion process under the +OIC model is described as follows. There are four states, +namely “positively active”, “neutrally active”, “negatively ac- +tive”, and “inactive”, for each user to represent whether the +user is affected by the promotion campaign (i.e., “active” +vs. “inactive”) and her/his opinion to the promoted item (i.e., +“positive”, “neutral”, or “negative”). Initially, all the nodes +in V are inactive. Then, a set S ⊆ V of seed nodes are +selected as initiators to promote item p to others. For each + +u2 +u4 +u3 +u1 +u5 +u6 +u7 +u10 +u9 +u8 +Fig. 2: Counterexamples for the monotonicity and submodu- +larity of OIM. +u ∈ S, the state of u will be set to either “positively active”, +or “neutrally active”, or “negatively active” if opu = 1, 0, +or −1, respectively. At step t + 1, each node u activated +at step t will try to activate each of its neighbors v that is +inactive at step t along edge e = (u, v) with probability w(e). +If the activation is successful, v will be (positively, neutral, +or negatively) activated based on the value of opv. Note that +each node has only one chance to activate its neighbors. The +nodes activated at previous steps always remain active. This +process will terminate until no more nodes can be activated. +The total numbers of positive and negative nodes activated +after the diffusion process on G are defined as the positive +influence spread E[I+ +G(S)] = � +u∈V + +p Pr[S ⇝ u] and negative +influence spread E[I− +G(S)] = � +u∈V − +p Pr[S ⇝ u] of item +p, respectively, where Pr[S ⇝ u] is the probability that S +activates u under the OIC model on G. We aim to maximize +the positive influence while minimizing the negative influence +for a given item. We thus define the objective function E[IG(·)] +as the difference between the positive and negative influence +spread functions, i.e., +E[IG(S)] = E[I+ +G(S)] − E[I− +G(S)]. +(2) +In this paper, we are concerned about the problem of selecting +a set of seed nodes to maximize E[IG(·)] in Eq. (2), which is +formally defined by the opinion-aware influence maximization +(OIM) problem as follows. +Definition 1. [Opinion-aware Influence Maximization (OIM)] +For a graph G, a set P of items, a set R of ratings, a target +item p, the OIC model on G, and the seed set size k ∈ Z+, +select a set S∗ of k seed nodes from V such that E[IG(S∗)] +is maximized, i.e., S∗ = arg maxS⊆V,|S|=k E[IG(S)]. +Example 1. We present a graph G with 10 users in Fig. 2, +where the nodes in black (u1, u2, u4, u5), gray (u3, u10), and +white (u6, u7, u8, u9) denote the sets of positive, neutral, and +negative users for a target item, respectively. We assume that +the probabilities of all edges are equal to 1. For the vanilla +IM problem with k = 2, S1 = {u1, u8} is the optimal seed +set because S1 activates 7 users on G, which is the maximum +among all size-2 subsets. But we have E[IG(S1)] = −1 since +the numbers of positive and negative users activated by S1 are +2 and 3, respectively, and thus S1 is not the optimal solution to +OIM. Alternatively, OIM returns S2 = {u1, u5} as its optimal +solution with E[IG(S2)] = 3 − 0 = 3. +Hardness of OIM: First, OIM is NP-hard because the vanilla +influence maximization [1] (IM) problem, which is known to +be NP-hard under the IC model, is a special case of OIM when +V + +p += V and V 0 +p , V − +p += ∅. Moreover, unlike vanilla IM [1], +the objective function E[IG(·)] of OIM is non-monotone, +neither submodular nor supermodular1 due to the existence of +negative opinions. E[IG(·)] is non-monotone because its value +can decrease whenever a user who activates more negative +users than positive ones in expectation is added. Next, we can +show that E[IG(·)] is neither submodular nor supermodular by +providing two counterexamples from Fig. 2: on the one hand, +since E[IG({u4, u7})] − E[IG({u4})] = −1 > E[IG({u7})] − +E[IG(∅)] = −2, E[IG(·)] is not submodular; on the other hand, +since E[IG({u1, u2})] − E[IG({u1})] = 0 < E[IG({u2})] − +E[IG(∅)] = 1, E[IG(·)] is also not supermodular. Therefore, +the greedy algorithm for submodular maximization problems +cannot provide any theoretical guarantee for OIM anymore, +which implies that OIM is more challenging than vanilla IM +problems. Thus, we propose a novel algorithmic framework +for OIM in the subsequent section. +IV. OUR ALGORITHM +In this section, we describe our proposed algorithm for OIM. +We first introduce the background on the reverse reachable +(RR) sets to estimate the influence spread and how they are +generalized for opinion-aware influence estimation in Sec- +tion IV-A. We then propose a seed selection algorithm for +OIM in Section IV-B. We note that the proofs of all lemmas +and theorems are omitted due to space limitations and included +in a technical report [xx]. +A. Opinion-aware Influence Estimation +In this subsection, we consider generalizing the reverse +reachable (RR) set-based methods for opinion-aware influence +estimation. Generally, since computing the influence spread +E[IG(·)] exactly under the IC model is #P-hard [14], we +typically use Monte-Carlo simulations to obtain an unbiased +estimator for the influence spread. There have been many +different simulation methods for influence estimation, among +which the reverse reachable (RR) set-based method first pro- +posed by Borgs et al. [10] is the most widely used one. +After the seminal work of [10], more efficient RR set-based +methods [11], [16]–[19] have been proposed for influence +estimation in different IM problems. +Formally, an RR set Ru of a node u ∈ V on G is generated +in two steps: i) acquire the transpose graph G⊤ = (V, E⊤) +of G, where (vi, vj) ∈ E⊤ iff (vj, vi) ∈ E; ii) simulate +the diffusion process starting from u on G⊤ to obtain the +set of nodes activated by u as the RR set Ru. Under the +IC model, an equivalent method for RR set generation in the +existing literature [16] is as follows: We generate a graph G′ +by removing each edge e ∈ E with probability 1 − w(e) +independently. For any node u, an RR set Ru is the set of +nodes in G′ that can reach u. Intuitively, if a seed set S has a +higher probability of activating u, i.e., the value of Pr[S ⇝ u] +1A set function f : 2V +→ R is monotone if f(S) ≤ f(T) for any +S ⊆ T ⊆ V ; Furthermore, a set function f : 2V → R is submodular (or +supermodular) if f(S ∪ {v}) − f(S) ≥ (or ≤) f(T ∪ {v}) − f(T) for any +S ⊆ T ⊆ V and v ∈ V \ T. + +is greater, there will be a higher chance that the intersection +of S and Ru is non-empty. According to the analysis in [10], +we have the following lemma to formalize the above intuition. +Lemma 1. For a seed set S ⊆ V , a node u ∈ V , and an RR +set Ru of node u, we have Pr[S ⇝ u] = Pr[Ru ∩ S ̸= ∅]. +Lemma 1 is a special case of [10, Observation 1], which +implies that an unbiased estimation σu(S) of Pr[S ⇝ u] can +be obtained by estimating the probability of the event Ru ∩ +S ̸= ∅. Next, we analyze how many RR sets are needed to +get an accurate estimation σu(S) within a small error with +high confidence. We consider l instances G′ +1, G′ +2, . . . , G′ +l are +generated independently, each is used to compute an RR set +Ru,i of node u to estimate Pr[S ⇝ u] for any S ⊆ V . We +define a Bernoulli random variable XS,i for S and Ru,i as: +XS,i = +� +1, +if Ru,i ∩ S ̸= ∅ +0, +otherwise +(3) +Lemma 1 implies that Pr[S ⇝ u] = E[XS,i]. Thus, we have +the following lemma to indicate the number of RR sets needed +to get an (ε, δ)-approximation σu(S) of Pr[S ⇝ u] for any +S ⊆ V of size at most k and u ∈ V . +Lemma 2. Let l = O( k log n +ε2 +log 1 +δ ), Ru,1, . . . , Ru,l be the +RR sets for any u ∈ V , and XS,1, . . . , XS,l be the ran- +dom variables defined by Eq. (3) for any S ⊆ V +with +|S| ≤ k. Then, with probability at least 1 − δ, it holds that +|σu(S) − Pr[S ⇝ u]| < ε, where σu(S) = 1 +l +�l +i=1 XS,i. +Proof. First of all, we have XS,i ∈ [0, 1] for each i ∈ [l]. By +applying Hoeffding’s inequality [20] on XS,1, . . . , XS,l, we +have +Pr +� +|X − E[X]| ≥ ε +� +< 2 exp(−2lε2), +where X = σu(S) = 1 +l +�l +i=1 XS,i, as �l +i=1(bi − ai)2 = l. +Furthermore, it holds that σu(S) = E[X] from Lemma 1. The +probability that |σu(S) − Pr[S ⇝ u]| < ε is thus at least 1 − +2 exp(−2lε2) for a specific u and S. Taking the union bound +over all �k +i=1 +�n +i +� += O(nk) seed sets of size at most k and all +n nodes u ∈ V , the probability that |σu(S) − Pr[S ⇝ u]| < ε +holds for any S and u is at least 1 − 2 exp(−2lε2) · O(nk+1). +And we need l = O( k log n +ε2 +log 1 +δ ) RR sets to obtain an (ε, δ)- +approximation σu(S) of Pr[S ⇝ u] with probability at least +1 − δ. +Based on Lemma 2, we further obtain the estimations for +the expected values E[I+ +G(S)], E[I− +G(S)] of the positive and +negative influence spread functions as well as the expected +value E[IG(S)] of the opinion-aware influence function for +any S ⊆ V in the following lemma. +Lemma 3. Let σ+(S) := � +u∈V + +p σu(S) and σ−(S) := +� +u∈V − +p σu(S). We have |σ+(S) − E[I+ +G(S)]| ≤ ε|V + +p | and +|σ−(S) − E[I− +G(S)]| ≤ ε|V − +p | with probabilities at least +1 − δ|V + +p | and 1 − δ|V − +p |. Moreover, |σ(S) − E[IG(S)]| ≤ εn +with probability at least 1−δn where σ(S) = σ+(S)−σ−(S). +Algorithm 1 SANDWICHGREEDY +Input: Graph G = (V, E, w), user vectors HV = {hu : u ∈ V }, target +item vector hp, seed set size k ∈ Z+, parameters ε, δ ∈ (0, 1) +Output: A set S′ ⊆ V such that |S′| ≤ k +1: Compute opu based on Eq. (1) from hu for each u ∈ V and hp +2: Let V + +p += {u ∈ V | opu = 1} and V − +p += {u ∈ V | opu = −1} +3: Generate a set of l = O( k log n +ε2 +log 1 +δ ) RR sets of G +4: Initialize S, S, S ← ∅ +5: for i = 1 to k do +6: +Define ∆σ(u | S) := σ(S ∪ {u}) − σ(S), ∆σ(u | S) := σ(S ∪ +{u}) − σ(S), ∆σ(u | S) := σ(S ∪ {u}) − σ(S) w.r.t. σ, σ, σ +7: +Compute ∆σ(u | S), ∆σ(u | S), ∆σ(u | S) for each u ∈ V based on +the RR sets +8: +Find the nodes u∗ +i ← arg maxu∈V ∆σ(u | S), u∗ +i ← arg maxu∈V +∆σ(u | S), u∗ +i ← arg maxu∈V ∆σ(u | S) +9: +Update the seed sets S ← S ∪{u∗ +i }, S ← S ∪{u∗ +i }, S ← S ∪{u∗ +i } +10: return S′ ← arg max{σ(S), σ(S), σ(S)} +Proof. The first two inequalities are obtained by summing up +the (ε, δ)-approx-imations for all users u ∈ V + +p or u ∈ V − +p +in +Lemma 2 and applying the union bound on all the estimations. +Then, we have the third inequality by taking the difference +between the first two inequalities. +B. Seed Selection +Given the influence estimation results in Section IV-A, we +introduce our algorithm for seed selection. First of all, for a +target item p with its vector representation hp, we can compute +the positive and negative sets of users V + +p +and V − +p +based +on HV = {hu : u ∈ V } by computing the value of opu +for each u ∈ V based on Eq. (1). Then, we generate a set +of RR sets on graph G under the OIC model, from which +the value of the objective function for any seed set of size +at most k can be estimated with bounded absolute errors as +indicated in Lemma 3. Accompanied by the opinion-aware +influence estimation based on RR sets, we should identify +the seeds with high influence in V + +p +and low influence in +V − +p +to maximize E[IG(S)] for OIM. The classic approach +to seed selection in IM is to use the greedy strategy, i.e., +picking a node that maximally increases E[IG(S)] progres- +sively in k iterations. However, since the objective function +E[IG(S)], as well as its estimated function σ(S), are non- +submodular, the greedy algorithm no longer has any theoretical +guarantee. Thus, we consider extending the greedy selection +procedure using the sandwich strategy to provide a data- +dependent approximation for OIM. In particular, to apply the +sandwich strategy for greedy non-submodular maximization, +we need to find the upper-bound and lower-bound submodular +functions for E[IG(S)] and σ(S). According to the analysis +in [1], the influence spread function under the IC model is +monotone and submodular, i.e., E[I+ +G(S)] and E[I− +G(S)] are +both monotone and submodular. In addition, their estimated +functions σ+(S) and σ−(S) based on RR sets can be seen as +coverage functions, which are also monotone and submodular. +In this way, E[IG(S)] and σ(S) can be naturally written in the +form of the difference of two submodular functions. Following +the difference of submodular (DS) decomposition in [12], we +derive the modular (Note that a function is called a modular + +function if it is both submodular and supermodular) upper and +lower bounds of any submodular function f as follows. Given +a submodular function f : 2V → R and any subset X ⊆ V , +the modular upper bound function f of f is defined as +f(S) := f(X) − +� +u∈X\S +f(u|S \ {u}) + +� +u∈S\X +f(u|∅) +(4) +where f(u|S) = f(S ∪ {u}) − f(S). Let π be a permutation +of V and define Sπ +i += {π(1), π(2), . . . , π(i)} as π’s chain +containing X, where Sπ +0 = ∅ and Sπ +|X| = X. Then, we define +the modular lower bound function f of f as +f(S) = +� +u∈S +gπ +X(u), gπ +X(π(i)) = f(Sπ +i ) − f(Sπ +i−1) +(5) +Based on the above definitions, we formulate the submodular +upper and lower bounds of the estimated objective function +σ(S) as σ(S) = σ+(S)−σ−(S) and σ(S) = σ+(S)−σ−(S) +due to σ+(S) ≤ σ+(S) ≤ σ+(S), σ−(S) ≤ σ−(S) ≤ σ−(S), +and the modularity of f and f for any submodular function f. +Having the upper and lower bounds, the sandwich strategy is +to run the greedy selection procedure with respect to σ, σ, σ +independently in k rounds to find three sets S, S, S of seeds, +respectively. Especially, starting from empty sets S, S, S, it +finds the nodes u∗, u∗, u∗ that maximize the marginal increases +in σ(S), σ(S), σ(S) and adds them to S, S, S accordingly at +each round. Finally, it returns the seed set with the largest +(estimated) objective value among the three as the final result +S′. The above steps for seed selection are described in +Algorithm 1. +Theoretical Analysis: We then analyze the data-dependent ap- +proximation ratio of the seed set S′ provided by Algorithm 1. +First of all, we show that Algorithm 1 finds an approximation +to the best greedy selection with high probability at each +iteration in the following lemma. +Lemma 4. The i-th node u∗ +i selected by Algorithm 1 satisfies +that ∆σ(u∗ +i | S) ≥ E[IG(S ∪ {u∗ +i }) − IG(S)] − 2εn with +probability at least 1 − δn. +Proof. According to Lemma 3, |σ(S) − E[IG(S)]| ≤ εn with +probability at least 1 − δn for any S ⊆ V of size at most k. +Thus, σ(S) − E[IG(S)] < −εn and σ(S) − E[IG(S)] > εn +both with probability at most δn +2 . Taking S and S∪{u} into the +inequalities, we have |∆σ(u | S)−E[IG(S ∪{u})−IG(S)]| ≤ +2εn with probability at least 1 − δn for any u ∈ V . Naturally, +the above inequality holds for node u∗ +i , and we conclude the +proof. +Based on the sandwich approximation strategy, we have +the following data-dependent approximation factor for Algo- +rithm 1 on OIM. +Theorem 1. Let S′ be the set of seeds returned by Algo- +rithm 1. We have E[IG(S)] ≥ max +� +σ(S) +σ(S), σ(S∗) +σ(S∗) +� +· +� +1 − 1 +e +� +· +E[IG(S∗)] − O(εkn) with probability at least 1 − δkn, where +S∗ is the optimal solution to maximize E[IG(·)] under cardi- +nality constraint k. +Proof. Since σ and σ are both monotone submodular func- +tions, we have σ(S) ≥ (1 − 1/e) · σ(S +∗) and σ(S) ≥ +(1 − 1/e) · σ(S∗), where S +∗ and S∗ are the optimal solutions +to maximize σ and σ subject to cardinality constraint k, +respectively. For the solution S to the upper-bound function +σ, we have +σ(S) ≥ σ(S) +σ(S) · +� +1 − 1 +e +� +· σ(S +∗) +≥ σ(S) +σ(S) · +� +1 − 1 +e +� +· σ(S∗) ≥ σ(S) +σ(S) · +� +1 − 1 +e +� +· σ(S∗). +For the solution S to the lower-bound function σ, we have +σ(S) ≥ σ(S) ≥ +� +1 − 1 +e +� +· σ(S∗) +≥ +� +1 − 1 +e +� +· σ(S∗) ≥ σ(S∗) +σ(S∗) · +� +1 − 1 +e +� +· σ(S∗). +Furthermore, by applying the result of Lemma 3 k times for +each node u∗ +i ∈ S, we have σ(S) ≥ E[IG(S)] − 2εkn with +probability at least 1−δkn. Note that the same result also holds +for S, S, and S∗. Thus, we conclude the proof by combining +all the above results. +In terms of time complexity, computing opu as well as V + +p +and V − +p in Lines 1–2 takes O(nd) time where d is the dimen- +sionality of hu and hp. Then, generating the RR sets in Line 3 +requires O( mk log n +ε2 +log 1 +δ ) time. Next, it runs in k iterations for +seed selection and takes O( nk log n +ε2 +log 1 +δ ) time per iteration to +find and add the nodes with the maximum marginal gains. To +guarantee that the approximation factor in Theorem 1 satisfies +with a constant probability, i.e., δkn = O(1), we should +set 1/δ = O(nk). Therefore, the overall time complexity of +Algorithm 1 is O +� +ε−2(mk + nk2) log n(log n + log k) +� +. +V. EXPERIMENTS +To evaluate the performance of our approach, we conduct +extensive experiments on real-world data sets. Next, we intro- +duce our experimental setup in Section V-A. Then, we present +our experimental results in Section V-B. +A. Setup +Datasets: We use three publicly available real-world datasets, +i.e., Ciao, Epinion2, and MovieLens3. Ciao and Epinion are +two product review data sets containing users’ ratings on +products and their trust relationships. MovieLens is a rec- +ommendation datasets consisting of user ratings on movies. +As the social relationships between users are unavailable on +the MovieLens dataset, we generate an influence graph for it +by adding an edge between each pair of users whose rating +histories have a Jaccard similarity of greater than a threshold +(e.g., 0.2). As a pre-processing step, we normalize the ratings +of all datasets to the same scale [0, 5], where higher ratings +denote more positive opinions. We summarize the statistics of +the three datasets in Table I. + +TABLE I: Statistics of real-world datasets in the experiments. +Dataset +Type +# users +# edges +# items +# ratings +Ciao +directed, product +2,378 +57,544 +16,861 +36,065 +Epinion +directed, product +49,289 +355,813 +139,738 +664,824 +MovieLens +directed, movie +6,040 +136,122 +3,900 +1,000,209 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Opinion +Time(s) +0.0 0.1 0.2 0.3 0.4 0.5 +ε +172 +176 +180 +184 +Opinion +100 +101 +102 +103 +Time(s) +(a) Ciao +0.0 0.1 0.2 0.3 0.4 0.5 +ε +280 +300 +320 +340 +Opinion +101 +102 +103 +Time(s) +(b) Epinion +0.0 0.1 0.2 0.3 0.4 0.5 +ε +144 +148 +152 +156 +Opinion +100 +101 +102 +103 +Time(s) +(c) MovieLens +Fig. 3: Total opinions and running time of OIM by varying +the error parameter ε. +Algorithms: We compare our proposed OIM algorithm with +the following baselines under the same OIC model used in this +paper: (1) RAND that randomly picks k seed users; (2) degree +discount (DEGDIS) heuristic [15]; and (3) an RRS-based +algorithm IMM [11]. We do not compare with the algorithms +in [5], [7], [8] because they are specific for LT models and +cannot work under the OIC model. Other IM algorithms +are ignored since they are opinion and item-unaware, whose +results are mostly close to or worse than IMM’s. +Implementation: In our implementation, we extend DEGDIS +to be opinion-aware by choosing the node with the largest +difference between positive and negative degrees, instead of +the one with the largest overall degree, in each iteration. We +use the original IMM algorithm as its influence estimation +process cannot be generalized to the OIC model. On each +dataset, we utilize an existing GNN-based model [3] to embed +all users and items into the same vector space to estimate the +(unobserved) ratings of users on items. Then, the weight w(e) +of each edge e = (u, v) is computed based on the cosine +similarity between their vector representations hu and hv. +Moreover, we use the average of all observed ratings in each +dataset as the neutral rating value r0 and set the parameter τ to +0.5. Finally, for the seed set S computed by each algorithm, +we run 1,000 Monte Carlo simulations to estimate its total +opinion, i.e., E[IG(S)]. +All algorithms were implemented in Python 3, and PyTorch +was used for GNN training and inference. The experiments +were run on a server with an Intel Xeon E5-2650v4 2.2GHz +processor, 96GB main memory, and an NVIDIA Tesla V100 +GPU with 16GB HBM2 memory, running Ubuntu 18.04 LTS. +B. Experimental Results +Effect of Error Parameter ε: We present the total opinion +(i.e., difference between the numbers of positive and negative +users activated by the seed set) and running time of our OIM +algorithm on each dataset by varying the parameter ε from +0.05 to 0.5 in Fig. 3. Here, we follow existing RR set-based +2https://www.cse.msu.edu/∼tangjili/datasetcode/truststudy.htm +3https://grouplens.org/datasets +methods [11], [17] to set δ = 100kn. The size k of the +seed set is fixed to 10. Generally, we observe that the total +opinion and running time of OIM decrease with an increasing +ε across all datasets. Based on our analysis in Section IV, +the number of RR sets for OIM is linear with respect to +ε−2. Accordingly, on the one hand, a larger ε leads to a less +stable influence estimation, a greater error in seed selection, +and thus slightly lower seed quality (cf. Lemmas 2–4); on the +other hand, a quadratically fewer number of RR sets w.r.t. ε +also significantly improves the time efficiency of OIM. In the +remaining experiments, we use ε = 0.15 and δ = 100kn +to determine the number of RR sets to sample in the OIM +algorithm to strike a good balance between seed quality and +time efficiency. +Effect of Solution Size k: We show the total opinion and +running time of each algorithm by varying the size k of the +seed set from 1 to 50 in Fig. 4. We randomly sample 50 +items from each dataset, select a size-k seed set for each item +using different algorithms, and report the average opinion and +running time of each algorithm across all items. Our OIM +algorithm always provides seed sets with the most positive +opinions among all algorithms in all cases. Nevertheless, +its advantage over other algorithms depends on the item’s +opinion distribution. On the Ciao dataset, since almost all +items receive more positive opinions than negative ones, OIM +is only marginally better than DEGDES and IMM. On all other +three datasets, as positive and negative opinions are generally +balanced, the total opinions of RAND and IMM are close to or +lower than 0, as they are opinion-unaware. However, OIM still +provides seed sets with more positive opinions, thus achieving +significant improvements upon all baselines in terms of seed +quality. In terms of time efficiency, the running time of OIM is +generally close to that of IMM, which both much slower than +that of DEGDES and RAND for larger k’s. This is because +the numbers of RR sets in OIM and IMM are close to each +other when the same value of ε is used, and the time for seed +selection is nearly negligible compared to the time for RR set +generation for both OIM and IMM. Accordingly, DEGDES +and RAND run much faster than OIM and IMM because they +do not require the time-consuming RR set generation process, +yet come at the expense of lower seed quality. +VI. CONCLUSION +In this paper, we defined the opinion-aware influence max- +imization (OIM) problem to find a set of k seeds from an +online social network to maximize the difference between the +positive and negative opinions for a target item. Despite its +NP-hardness and non-submodularity, we proposed an efficient +data-dependent approximation algorithm for OIM based on +the RR sets for opinion-aware influence estimation and the +sandwich approximation for seed selection. Experimental re- +sults on three real-world datasets showed the effectiveness and +efficiency of our proposed algorithm. In future work, we would +like to extend the OIM problem to more general settings where +the diffusion model and user opinions can be unknown in +advance and should be learned from observations. + +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Rand +DegDis +IMM +OIM +1 +10 +20 +30 +40 +50 +k +0 +50 +100 +150 +200 +250 +Opinion +1 +10 +20 +30 +40 +50 +k +10−1 +100 +101 +102 +Time(s) +(a) Ciao +1 +10 +20 +30 +40 +50 +k +−100 +0 +100 +200 +300 +400 +Opinion +1 +10 +20 +30 +40 +50 +k +100 +101 +102 +103 +Time(s) +(b) Epinion +1 +10 +20 +30 +40 +50 +k +−40 +0 +40 +80 +120 +160 +200 +Opinion +1 +10 +20 +30 +40 +50 +k +100 +101 +102 +103 +Time(s) +(c) MovieLens +Fig. 4: Total opinions and running time of different algorithms by varying the seed set size k. +REFERENCES +[1] D. Kempe, J. M. Kleinberg, and ´E. Tardos, “Maximizing the spread of +influence through a social network,” in KDD, 2003, pp. 137–146. +[2] P. M. Domingos and M. Richardson, “Mining the network value of +customers,” in KDD, 2001, pp. 57–66. +[3] W. Fan, Y. Ma, Q. Li, Y. He, Y. E. Zhao, J. Tang, and D. Yin, “Graph +neural networks for social recommendation,” in WWW, 2019, pp. 417– +426. +[4] W. Chen, A. Collins, R. Cummings, T. Ke, Z. Liu, D. Rinc´on, X. Sun, +Y. Wang, W. Wei, and Y. Yuan, “Influence maximization in social +networks when negative opinions may emerge and propagate,” in SDM, +2011, pp. 379–390. +[5] H. Zhang, T. N. Dinh, and M. T. Thai, “Maximizing the spread of +positive influence in online social networks,” in ICDCS, 2013, pp. 317– +326. +[6] S. Galhotra, A. Arora, and S. Roy, “Holistic influence maximization: +Combining scalability and efficiency with opinion-aware models,” in +SIGMOD, 2016, pp. 743–758. +[7] X. Liu, X. Kong, and P. S. Yu, “Active opinion maximization in social +networks,” in KDD, 2018, pp. 1840–1849. +[8] J. Luo, X. Liu, and X. Kong, “Competitive opinion maximization in +social networks,” in ASONAM, 2019, pp. 250–257. +[9] G. L. Nemhauser, L. A. Wolsey, and M. L. Fisher, “An analysis of +approximations for maximizing submodular set functions—I,” Math. +Program., vol. 14, no. 1, pp. 265–294, 1978. +[10] C. Borgs, M. Brautbar, J. T. Chayes, and B. Lucier, “Maximizing social +influence in nearly optimal time,” in SODA, 2014, pp. 946–957. +[11] Y. Tang, Y. Shi, and X. Xiao, “Influence maximization in near-linear +time: A martingale approach,” in SIGMOD, 2015, pp. 1539–1554. +[12] R. K. Iyer and J. A. Bilmes, “Algorithms for approximate minimization +of the difference between submodular functions, with applications,” in +UAI, 2012, pp. 407–417. +[13] Y. Li, J. Fan, Y. Wang, and K. Tan, “Influence maximization on social +graphs: A survey,” IEEE Trans. Knowl. Data Eng., vol. 30, no. 10, pp. +1852–1872, 2018. +[14] W. Chen, C. Wang, and Y. Wang, “Scalable influence maximization for +prevalent viral marketing in large-scale social networks,” in KDD, 2010, +pp. 1029–1038. +[15] W. Chen, Y. Wang, and S. Yang, “Efficient influence maximization in +social networks,” in KDD, 2009, pp. 199–208. +[16] N. Ohsaka, T. Akiba, Y. Yoshida, and K. Kawarabayashi, “Fast and +accurate influence maximization on large networks with pruned Monte- +Carlo simulations,” in AAAI, 2014, pp. 138–144. +[17] Y. Tang, X. Xiao, and Y. Shi, “Influence maximization: near-optimal +time complexity meets practical efficiency,” in SIGMOD, 2014, pp. 75– +86. +[18] H. T. Nguyen, M. T. Thai, and T. N. Dinh, “Stop-and-stare: Optimal +sampling algorithms for viral marketing in billion-scale networks,” in +SIGMOD, 2016, pp. 695–710. +[19] Q. Guo, S. Wang, Z. Wei, and M. Chen, “Influence maximization revis- +ited: Efficient reverse reachable set generation with bound tightened,” +in SIGMOD, 2020, pp. 2167–2181. +[20] W. Hoeffding, “Probability inequalities for sums of bounded random +variables,” J. Am. Stat. Assoc., vol. 58, no. 301, pp. 13–30, 1963. + diff --git a/l9E1T4oBgHgl3EQfOANh/content/tmp_files/2301.03009v1.pdf.txt b/l9E1T4oBgHgl3EQfOANh/content/tmp_files/2301.03009v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3963f0fd1de67554e9979f296a6ba3746dece8da --- /dev/null +++ b/l9E1T4oBgHgl3EQfOANh/content/tmp_files/2301.03009v1.pdf.txt @@ -0,0 +1,2425 @@ +Comparing Three Generations of D-Wave Quantum Annealers for +Minor Embedded Combinatorial Optimization Problems +Elijah Pelofske ∗1 +1Los Alamos National Laboratory, CCS-3 +Abstract +Quantum annealing is a novel type of analog computation that aims to use quantum mechanical fluctuations +to search for optimal solutions of Ising problems. Quantum annealing in the Transverse Ising model, implemented +on D-Wave QPUs, are available as cloud computing resources. In this article we report concise benchmarks +across three generations of D-Wave quantum annealers, consisting of four different devices, for the NP-Hard +combinatorial optimization problems unweighted maximum clique and unweighted maximum cut on random +graphs. The Ising, or equivalently QUBO, formulation of these problems do not require auxiliary variables for +order reduction, and their overall structure and weights are not highly complex, which makes these problems +simple test cases to understand the sampling capability of current D-Wave quantum annealers. All-to-all minor +embeddings of size 52, with relatively uniform chain lengths, are used for a direct comparison across the Chimera, +Pegasus, and Zephyr device topologies. A grid search over annealing times and the minor embedding chain +strengths is performed in order to determine the level of reasonable performance for each device and problem +type. Experiment metrics that are reported are approximation ratios for non-broken chain samples and chain +break proportions. How fairly the quantum annealers sample optimal maximum cliques, for instances which +contain multiple maximum cliques, is also quantified using entropy of the measured ground state distributions. +The newest generation of quantum annealing hardware, which has a Zephyr hardware connectivity, performed +the best overall with respect to approximation ratios and chain break frequencies. +1 +Introduction +Quantum annealing (QA) is a quantum computational technology that can be viewed as a heuristic version of +Adiabatic Quantum Computing (AQC). QA uses quantum fluctuations in order to search for a minimum variable +assignment of a combinatorial optimization problem [1–7]. Quantum annealing has been physically implemented +using superconducting flux qubits in commercial devices by companies such as D-Wave [8]. +D-Wave quantum +annealers are available to be used in the cloud. The topic of benchmarking quantum annealers capabilities in the +NISQ era [9, 10] is a topic of considerable interest [11–19]. For reviews on quantum annealing, see refs. [10, 20]. +Quantum annealing in the transverse field Ising model works be initializing the system with the Hamiltonian: +Hi = +� +n∈V +σx +i +(1) +The system then applies a user specified problem Hamiltonian while reducing the magnitude of the transverse +field Hamiltonian: +H(t) = A(t)Hi + B(t)Hising +(2) +Combined, A(t) and B(t) define the anneal schedules. The total annealing time can be chosen by the user. The +problem Hamiltonian is defined as: +Hising = +n +� +i +hiσz +i + +n +� +i 0; it has support in the interval [0, 1], with density +Γ(a + b) +Γ(a)Γ(b)xa−1(1 − x)b−1, +(16) +where Γ(·) is the Euler Gamma function. Let YMP be a random variable with Marchenko- +Pastur distribution (6). Then, YMP is equal in distribution to the product of a uniform +random variable on the interval [0, 4] and an independent arcsine variable in the interval +[0, 1]. In formulae: +YMP +d= U(0, 4)B(1/2, 1/2). +(17) + +Random matrices associated to Young diagrams +8 +This is equivalent to the above mentioned factorisation of semicircular variables [18]. +Indeed, if U, U′ are independent and uniformly distributed on [0, 1], we can write +EY k +MP = E(UA)k, +(18) +where A +d= (2 cos πU′)2. The random variable A is the rescaled squared projection of +a uniform point on the unit semicircle, hence an arcsine random variable. See [8] for a +‘semiclassical’ interpretation for Gaussian random matrices. +Proposition 2. Let r ≥ 1. Set L(r) = (r + 1)r+1 +rr +> 0. +(i) The Stieltjes-Cauchy transform of F⟨r⟩ +G⟨r⟩(z) := +� +R +1 +z − xdF⟨r⟩(x), +(19) +has the hypergeometric representation +G⟨r⟩(z) = +1 +r + 1 +� +1 − rFr−1 +� +− +1 +r+1, − +2 +r+1, . . . , − +r +r+1 +− 1 +r, − 2 +r, . . . , − r−1 +r +; L(r) +z +�� +(20) +(ii) If Y⟨r⟩ is a real random variable with distribution F⟨r⟩, then the following identity +in distribution holds +Y⟨r⟩ +d= U (0, L(r)) +r� +j=1 +B +� +j +r + 1, j +r − +j +r + 1 +� +, +(21) +where the variables on the right are jointly independent. +(iii) Let U, U′ be independent and uniformly distributed on [0, 1]. Then, +EY k +⟨r⟩ = E +� +UA⟨r⟩ +�k . +(22) +where A⟨r⟩ +d= e(r−1)πiU′ (2 cos πU′)r+1. +Remark 2. For the first values of r ≥ 1 we get +Y⟨1⟩ +d= U +� +0, 22 +11 +� +B +�1 +2, 1 +2 +� +, +(23) +Y⟨2⟩ +d= U +� +0, 33 +22 +� +B +�1 +3, +1 +2 · 3 +� +B +�2 +3, +2 +2 · 3 +� +, +(24) +Y⟨3⟩ +d= U +� +0, 44 +33 +� +B +�1 +4, +1 +3 · 4 +� +B +�2 +4, +2 +3 · 4 +� +B +�3 +4, +3 +3 · 4 +� +, +(25) +etc. For r = 1, Eq. (23) coincides with the factorisation (17). For r = 2, Eq. (24) is +equivalent to a formula proved by M�lotkowski and Penson [24, Proposition 3.1]. + +Random matrices associated to Young diagrams +9 +0 +2 +4 +6 +8 +10 +12 +x +0.1 +0.2 +0.3 +0.4 +0.5 +F'(x) +r=1 +r=2 +r=3 +r=4 +F 0 +hri(x) +x +r = 1 +r = 2 +r = 3 +r = 4 +Figure 1. Plot of the densities F ′ +⟨r⟩(x) for several values of r. +We can use the decomposition (21) to write ‘explicit’ expressions for the densities +F ′ +r. For small values of r we have +F ′ +⟨1⟩(x) = 1 +2π +� +L(1) − x +x +χ[0,L(1)](x) +(26) +F ′ +⟨2⟩(x) = +1 +23+1/331/2πw2/3 +�� +31/2 + 2 +� +L(2) − w +� � +33/2 − 2 +� +L(2) − w +�1/3 +− +� +31/2 − 2 +� +L(2) − w +� � +33/2 + 2 +� +L(2) − w +�1/3� +χ[0,L(2)](w). +(27) +The first is the Marchenko-Pastur distribution F ′ +MP(x). The second is equivalent to +formula in [24, Theorem 3.1] proved by Mellin inversion. For generic values of r, a +‘direct’ way to numerically compute F ′ +⟨r⟩(x) is by applying the Stieltjes inversion formula +F ′ +⟨r⟩(x) = − 1 +π lim +ϵ↘0 G⟨r⟩(x + iϵ), +(28) +to Equation (20). This is how we made the numerical plots shown in Fig. 1. +In fact, one could, as in [30], write F ′ +⟨r⟩(x) as the inverse Mellin transform of +the moments. Since the moments are products of ratios of Pochhammer symbols, the +resulting density would be a Mejier-G function (see the note by Dunkl [10]). In fact, +from the previous Proposition we can get some precise informations on the support of +F ′ +⟨r⟩(x) and on its behavior at the edges. +Corollary 1. The measure F⟨r⟩ has a density, F ′ +⟨r⟩ with support +suppF ′ +⟨r⟩ := +� +x: F ′ +⟨r⟩(x) > 0 +� += [0, L(r)] . +(29) + +Random matrices associated to Young diagrams +10 +Moreover at the edges of the support the density behaves as +F ′ +⟨r⟩(x) ∼ +� +� +� +� +� +� +� +c x− +r +r+1, +as +x ↘ 0, +c′ � +L(r) − x, +as +x ↗ L(r), +(30) +for some constants c, c′ > 0. +The density vanishes as a square root at the ‘soft edge’ x = L(r). At the ‘hard edge’ +x = 0 the density has an integrable singularity x− +r +r+1. For r = 1 (Wishart matrices) +this is the classical x− 1 +2 divergence, but for r > 1 the singularity is stronger. +Remark 3. The generalised Catalan numbers +C⟨r⟩ +k += +r +k + 1 +�rk + k +k +� +(31) +are related to the most known Fuss-Catalan sequences +FCr,k = +1 +rk + 1 +�rk + k +k +� +. +(32) +The Fuss-Catalan sequences also appear in random matrix theory and free probability as +limiting moments of products of independent Wishart matrices [2, 19, 23]. As shown by +Penson and ˙Zyczkowski [30], the Fuss-Catalan numbers FCr,k are the moment sequence +of a distribution FFCr(x) whose density is a Meijer G-function, with support on the +interval [0, L(r)]. +5. Proofs +Theorem 1 was discovered by a combination of experimenting and guessing, and the +proof is based on matrix moments calculations combined with the exact solution for the +enumeration of r-plane trees (Proposition 1). +Proof of Theorem 1. Let +mk,N := +� +R +xkdFN(x), +and +mk = +1 +k + 1 +�(r + 1)k +k +� +. +(33) +Claim 1. Each FN has moments of all order. +Claim 2. For all k, the moment sequence mk,N converges to mk, almost surely. +Claim 3. The sequence (mk)k≥0 uniquely determine the distribution F⟨r⟩. +By the moment method, the three claims imply Theorem 1. +Note that FN +corresponds to a random measure with finite support. Hence, Claim 1 is immediate. +Claim 3 can be showed by checking Riesz’s condition [4, Lemma B.2], +lim inf +k +1 +km +1 +2k +2k < ∞. +(34) + +Random matrices associated to Young diagrams +11 +Indeed, Stirling’s approximation formula implies +lim +k→∞ +1 +k +� +1 +2k + 1 +�2(r + 1)k +2k +�� 1 +2k += 0. +(35) +It remains to prove Claim 2: +lim +N→∞ mk,N = mk, +almost surely. +(36) +Note that +mk,N = +� +R +xkdFN(x) = +1 +rN Tr W k +N. +(37) +We may first assume that the entries Xij are uniformly bounded. Under this assumption +we can prove the following two Lemmas. +Lemma 1. Let λ(N) = N +r. Then, for all k ∈ N, +lim +N→∞ Emk,N = mk, +(38) +Lemma 2. Let λ(N) = N +r. Then, for all k ∈ N, +E (mk,N − Emk,N)2 = O +� 1 +N 2 +� +. +(39) +Lemma 1 states the convergence in expectation of the moments. Lemma 2 implies, +through Borel-Cantelli, the almost sure convergence (36). +We now show that the Theorem holds true under the sole hypothesis of finite +second moment E|Xij|2 = 1. This is done by the standard procedure of truncation. +Fix a constant C and consider the symmetric matrix ˆXN whose elements satisfy, for +1 ≤, i, j ≤ rN +ˆXN(i, j) = +Xij1|Xij| 0: F(x − ϵ) − ϵ ≤ G ≤ F(x + ϵ) + ϵ, for all x ∈ R}. +(41) +For the reader’s convenience, we state explicitly a bound that can be extracted +from the book of Bai and Silverstein [4, Theorem 3.7, p. 47-48]. For all C > 0, +lim sup +N +� +dL´evy( ˆFN, FN) +�4 +≤ 4E +� +|Xij|2 1|Xij|>C +� ++ 2 +� +1 − +� +E +��Xij1|Xij| L(r).) +When computing the ratio of consecutive terms of the series we identify the claimed +hypergeometric representation (20). +In order to prove ii), recall the moments of Beta and uniform random variables, +EB(a, b)k = Γ(a + b)Γ(a + k) +Γ(a)Γ(a + b + k) = +k−1 +� +j=0 +a + j +a + b + j , +EU(0, ℓ)k = +ℓk +k + 1. +(50) +A little calculation shows that +mk = +[(r + 1)k]! +(rk)!(k + 1)! = +(r+1)k−1 +� +j=0 +((r + 1)k − j) +rk−1 +� +j=0 +(rk − j) +k +� +j=0 +(k + 1 − j) += (L(r))k +(r+1)k−1 +� +j=0 +� +k − +j +r + 1 +� +rk−1 +� +j=0 +� +k − j +r +� +k +� +j=0 +(k + 1 − j) += (L(r))k +k + 1 +r� +i=1 +k−1 +� +j=0 +� +i +r+1 + j +� +� i +r + j +� . +(51) +Compare now with (50) to conclude the proof. +Finally, for point iii) we notice that +mk = +1 +k + 1 +�(r + 1)k +k +� += +1 +k + 1 × coefficient of zk in (1 + z)(r+1)k += +1 +k + 1 · 1 +2πi +� +|z|=1 +(1 + z)(r+1)k +zk+1 +dz. +(52) + +Random matrices associated to Young diagrams +15 +Now change coordinates z �→ exp(2πiu) to get +mk = +� 1 +0 +ukdu · +� 1 +0 +� +exp(−2πiu′) (1 + exp(2πiu′))r+1�k +du′. +(53) +Acknowledgments +The authors acknowledge the partial support by the Italian National Group of +Mathematical Physics INdAM-GNFM. 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Stanley R P, Catalan Numbers, Cambridge University Press, 2015. + diff --git a/mdFRT4oBgHgl3EQfaDeS/content/tmp_files/load_file.txt b/mdFRT4oBgHgl3EQfaDeS/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7d262c9c6b0278e7eb54d572cd837d7ceaa6ba39 --- /dev/null +++ b/mdFRT4oBgHgl3EQfaDeS/content/tmp_files/load_file.txt @@ -0,0 +1,524 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf,len=523 +page_content='Random matrices associated to Young diagrams Fabio Deelan Cunden, Marilena Ligab`o, Tommaso Monni Dipartimento di Matematica, Universit`a degli Studi di Bari, I-70125 Bari, Italy E-mail: fabio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='cunden@uniba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='it E-mail: marilena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='ligabo@uniba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='it E-mail: tommaso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='monni@uniba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='it 1 February 2023 Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' We consider the singular values of certain Young diagram shaped random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' For block-shaped random matrices, the empirical distribution of the squares of the singular eigenvalues converges almost surely to a distribution whose moments are a generalisation of the Catalan numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' The limiting distribution is the density of a product of rescaled independent Beta random variables and its Stieltjes-Cauchy transform has a hypergeometric representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' In special cases we recover the Marchenko-Pastur and Dykema-Haagerup measures of square and triangular random matrices, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' We find a further factorisation of the moments in terms of two complex-valued random variables that generalises the factorisation of the Marcenko- Pastur law as product of independent uniform and arcsine random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Introduction Let X be a matrix whose entries are i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' complex random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' The nonnegative definite matrix XX∗ is known as random covariance matrix, and it is arguably one of the most studied models in random matrix theory with varied applications in physics, statistics and other areas [14, 20, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' In this paper we consider a class of random matrices, that we dub λ-shaped random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' They can be thought of as a generalisation of random covariance matrices where X has the ‘shape’ of a Young diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Instances of these matrix models in the special case of Gaussian entries were studied by Dykema and Haagerup [11] as a tool to construct certain non commutative random variables called DT-elements, and in the work of F´eray and Sniady [12] in relation to Stanley’s character formula [31] of the symmetric group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Under various names and in disguised form, Gaussian λ-shaped matrices recently resurfaced in connection to biorthogonal ensembles, last passage percolation and free probability [1, 6, 7, 15, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='13555v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='PR] 31 Jan 2023 Random matrices associated to Young diagrams 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Partitions and Young diagrams We begin by reviewing some of the basic terminology of integer partitons and Young diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' A standard reference is [21, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' I].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' A partition is any (finite or infinite) sequence λ = (λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' , λm, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=') of weakly decreasing nonnegative integers, λ1 ≥ λ2 ≥ · · · ≥ λm ≥ · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' It is convenient not to distinguish between two such sequences which differ only by a string of zeros at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Thus, for example, we regard (5, 4, 4, 1), (5, 4, 4, 1, 0), (5, 4, 4, 1, 0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=') as the same partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' We denote the set of partitions by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' The number of nonzero elements λi of a partition λ is called number of parts or length of λ, denoted by ℓ(λ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' the sum of the parts is the weight of λ, denoted by |λ| = λ1 + λ2 + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' If |λ| = n, we say that λ is a partition of n and we write λ ⊢ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' The Young diagram of a partition λ is the set of points {(i, j) ∈ N2 : 1 ≤ j ≤ λi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' A Young diagram is drawn usually as a set of boxes, not of points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' In drawing such diagrams we shall adopt the English convention, as with matrices, that the first coordinate i (the row index) increases as one goes downwards, and the second coordinate j (the column index) increases from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' With this convention the diagram can be visualised as a diagram of left-justified rows of boxes where the i-th row contains λi boxes (hence each row is not longer than the row on top of it).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' For instance, the diagram of λ = (5, 4, 4, 1) ⊢ 14 is: It is customary to identify a partition λ ⊢ n and its Young diagram with n boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' For λ, µ ∈ P, we shall write µ ⊂ λ to mean that the diagram of λ contains the diagram of µ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' that µi ≤ λi for all i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' We write (i, j) ∈ λ to mean that the box (i, j) is in the diagram of λ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' λi ≥ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' The conjugate of a partition λ is the partition λ′ whose diagram is the transpose of the diagram λ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' the diagram obtained by reflection in the main diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' It is possible to define an operation of left multiplication by positive integers on the set of Young diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' (In fact, it is possible to define a multiplication by positive real numbers using a natural identification between diagrams and their border path when drawing a partition in ‘Russian notation’ [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=') If N ∈ N and λ ∈ P, then Nλ is the partition with Nℓ(λ) parts given by (Nλ)i = Nλj iff jN ≤ i < (j + 1)N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' In other words Nλ = � �Nλ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' , Nλ1 � �� � N times , Nλ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' , Nλ2 � �� � N times , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Random matrices associated to Young diagrams 3 The Young diagram of Nλ is a dilation of the Young diagram of λ obtained by replacing each box in λ by a grid of N × N boxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Hence, if λ ⊢ n, then Nλ ⊢ N 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' For instance, if λ = (5, 4, 4, 1) ⊢ 14, then 3λ = (15, 15, 15, 12, 12, 12, 12, 12, 12, 3, 3, 3) ⊢ 32 · 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Its diagram is: 3 · = We use the following pictorial notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' The square partition with r parts all equal to r is denoted r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' For the staircase partition (r, r − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' , 2, 1) we use the symbol r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' λ-shaped random matrices Fix a field K, and a partition λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' We denote by M(K, λ) the set of all ℓ(λ) × ℓ(λ′) matrices A = (aij) over K with entries aij = 0 if (i, j) /∈ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' The set M(K, λ) is a K- vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' These sets appeared in applied mathematics and computer science mostly when K is a finite field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' See [5] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' We call matrices elements of M(K, λ), λ-shaped matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' In the present paper we will consider spectral properties of random matrices in M(C, λ) defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Let � λ(N)� N≥1 a nested sequence in P, λ(1) ⊂ λ(2) ⊂ · · · ⊂ λ(N) ⊂ · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' (1) Let {Xij : i, j ∈ N} be i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' complex random variables with zero mean EXij = 0 and second moment E|Xij|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' For N ∈ N let XN be the λ(N)-shaped matrix whose (i, j)-th entry is Xij if (i, j) ∈ λ(N) and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Set ℓN := ℓ(λ(N)) and consider the ℓN × ℓN complex Hermitian matrix WN := 1 N XNX∗ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' (2) Denote the eigenvalues of WN by x(N) 1 ≤ x(N) 2 ≤ · · · ≤ x(N) ℓN , and FN(x) := 1 ℓN #{j ≤ ℓN : x(N) j ≤ x} (3) Random matrices associated to Young diagrams 4 the empirical distribution of the eigenvalues x(N) i of WN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' It is natural to ask whether the sequence (FN)N≥1 converges in distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' The limit, when it exists, will be referred to as the limiting spectral distribution of WN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' We begin by recalling two known cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Full matrices Classical Wishart matrices fit naturally in the setting of λ-shaped random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Let λ(N) = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' We have 1 ⊂ 2 ⊂ · · · ⊂ N ⊂ · · · Then, the N-shaped random matrix XN is simply a N × N matrix with i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' entries Xij, so that WN is a Wishart matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Recall that the Catalan numbers are Ck = 1 k + 1 �2k k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' (4) Catalan numbers count hundreds of different combinatorial objects [32], such as rooted plane trees of k+1 vertices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' It is a classical result that (FN)N≥1 converges in distribution to a deterministic distribution function FMP whose moments are the Catalan numbers � R xkdFMP(x) = Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' (5) The limiting spectral distribution FMP is called Marchenko-Pastur distribution [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' It is supported on the interval [0, 4] with density F ′ MP(x) = 1 2π � 4 − x x χ[0,4](x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' (6) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Triangular matrices Let λ(N) = N be the staircase partition of length N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Again N ⊂ N+1 for all N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Then, XN is a N × N triangular random matrix (with entry (i, j) = Xij if i + j ≤ N and 0 otherwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' These matrices where first considered by Dykema and Haagerup [11] who proved the existence of the limiting spectral distribution FDH whose moments are � R xkdFDH(x) = 1 k + 1 kk k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' (7) The Dykema-Haagerup distribution FDH comes from a density supported on the interval [0, e] and defined by (see [11, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='9]) F ′ DH �sin v v exp(v cot v) � = 1 π sin v exp(−v cot v), 0 < v < π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' (8) This density can be also written in terms of Lambert function, see [6, Corollary 1] (arXiv version) and [15, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Random matrices associated to Young diagrams 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Balanced shapes We would like to study spectral properties of λ-shaped random matrices for more general increasing sequence � λ(N)� N≥1 of Young diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' This amounts to understand the large-N limit of moments lim N→∞ 1 ℓN E Tr W k N, k = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' , (9) where WN is defined is (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' We expect to find a nontrivial limiting spectral distribution if the sequence λ(N) ‘converges’ to a limit shape (the ‘macroscopic shape’ of λ(N)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' The first moment (k = 1) calculation can makes this a bit more precise, 1 ℓN E Tr WN = 1 ℓN 1 N � (i,j)∈λ(N) E|Xij|2 = ��λ(N)�� Nℓ(λ(N)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' (10) Therefore, in order to have a nontrivial limit distribution one needs (at least) the length of the partitions to scale like the square root of the weight NℓN ∼ |λ(N)|, as N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' (11) Young diagrams satisfying such a growth condition are called balanced Young diagrams [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' If we view a Young diagram as a geometric object, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' (11) suggests to consider sequences that, after rescaling as 1 NℓN λ(N), tend to a limit shape λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' The easiest example of balanced Young diagrams is the sequence of dilations of a fixed partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Let λ ∈ P and consider the sequence λ(N) = Nλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' For such a sequence the ratio |λ(N)| NℓN = |λ| ℓ(λ) is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Block-shaped random matrices Fix a positive integer r, consider the staircase partition r of length r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Define the sequence λ(N) = N r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' In this case ℓN = Nr, and λ(N) ⊂ λ(N+1) for all N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' The matrix XN is a block-shaped random matrix with �r+1 2 � nonzero blocks of size N × N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Random matrices associated to Young diagrams 6 Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' For r = 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' we have X1 = � � X11 X12 X13 X21 X22 0 X31 0 0 � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' X2 = � � � � � � � � � X11 X12 X13 X14 X15 X16 X21 X22 X23 X24 X25 X26 X31 X32 X33 X34 0 0 X41 X42 X43 X44 0 0 X51 X52 0 0 0 0 X61 X62 0 0 0 0 � � � � � � � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' X3 = � � � � � � � � � � � � � � � X11 X12 X13 X14 X15 X16 X17 X18 X19 X21 X22 X23 X24 X25 X26 X27 X28 X29 X31 X32 X33 X34 X35 X36 X37 X38 X39 X41 X42 X43 X44 X45 X46 0 0 0 X51 X52 X53 X54 X55 X56 0 0 0 X61 X62 X63 X64 X65 X66 0 0 0 X71 X72 X73 0 0 0 0 0 0 X81 X82 X83 0 0 0 0 0 0 X91 X92 X93 0 0 0 0 0 0 � � � � � � � � � � � � � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Let FN be the empirical distribution (3) of the eigenvalues of WN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Let λ(N) = N r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Then, the sequence (FN)N≥1 converges, with probability 1, to the deterministic distribution F⟨r⟩ with moments � R xkdF⟨r⟩(x) = 1 k + 1 �(r + 1)k k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' (12) The limiting moments mk (multiplied by r) have a combinatorial interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' They enumerate plane trees whose vertices are given labels from the set {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' , r} in such a way that the sum of the labels along any edge is at most r+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' These combinatorial object were invented by Gu, Prodinger and Wagner [17], extending a previous definition by Gu and Prodinger [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' A r-plane tree is a pair (T, c), where T = (V, E) is a plane tree, and c: V → {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' , r} is a colouring such that c(u) + c(v) ≤ r + 1 whenever {u, v} ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Proposition 1 (Gu, Prodinger and Wagner [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' The number of r-plane trees on k +1 vertices is C⟨r⟩ k = r k + 1 �(r + 1)k k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' (13) For recent refined formulae see [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' The integers � C⟨r⟩ k � k≥0 are generalised Catalan Random matrices associated to Young diagrams 7 numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Here are a few values of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='C⟨1⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='C⟨2⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='C⟨3⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='C⟨4⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='C⟨5⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='C⟨6⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='3 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='3735373880 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='34270012530 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='216384285936 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='The sequences ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='C⟨1⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='k≥0 for r = 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' 4 are the entries A000108,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' A007226,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' A007228,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' and A124724 in The On-Line Encyclopedia of Integer Sequences [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' For r = 1, the moments coincide with the Catalan sequence C⟨1⟩ k = 1 k + 1 �2k k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' (14) For r = 2, XN is a ‘three-blocks’ matrix model studied by Flynn-Connolly [13] who proved that the moments are 1 k+1 �3k k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' The sequence entry A005132 in The On-Line Encyclopedia of Integer Sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' For large r, we recover the moments (7) of the Dykema-Haagerup measure, lim r→∞ 1 rk+1C⟨r⟩ k = 1 k + 1 kk k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' (15) We now present a few results on the limiting measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' They are motivated by the observation (by Ledoux [18]) that a semicircular variable is equal in distribution to the product of the square root of a uniform random variable and an independent arcsine random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' We denote by U(0, ℓ) a random variable uniformly distributed in the interval [0, ℓ], and simply by U a random variable uniformly distribute in the unit interval [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' With B(a, b) we generally denote a beta random variable with parameters a, b > 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' it has support in the interval [0, 1], with density Γ(a + b) Γ(a)Γ(b)xa−1(1 − x)b−1, (16) where Γ(·) is the Euler Gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Let YMP be a random variable with Marchenko- Pastur distribution (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Then, YMP is equal in distribution to the product of a uniform random variable on the interval [0, 4] and an independent arcsine variable in the interval [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' In formulae: YMP d= U(0, 4)B(1/2, 1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' (17) Random matrices associated to Young diagrams 8 This is equivalent to the above mentioned factorisation of semicircular variables [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Indeed, if U, U′ are independent and uniformly distributed on [0, 1], we can write EY k MP = E(UA)k, (18) where A d= (2 cos πU′)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' The random variable A is the rescaled squared projection of a uniform point on the unit semicircle, hence an arcsine random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' See [8] for a ‘semiclassical’ interpretation for Gaussian random matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Let r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Set L(r) = (r + 1)r+1 rr > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' (i) The Stieltjes-Cauchy transform of F⟨r⟩ G⟨r⟩(z) := � R 1 z − xdF⟨r⟩(x), (19) has the hypergeometric representation G⟨r⟩(z) = 1 r + 1 � 1 − rFr−1 � − 1 r+1, − 2 r+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' , − r r+1 − 1 r, − 2 r, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' , − r−1 r ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' L(r) z �� (20) (ii) If Y⟨r⟩ is a real random variable with distribution F⟨r⟩, then the following identity in distribution holds Y⟨r⟩ d= U (0, L(r)) r� j=1 B � j r + 1, j r − j r + 1 � , (21) where the variables on the right are jointly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' (iii) Let U, U′ be independent and uniformly distributed on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Then, EY k ⟨r⟩ = E � UA⟨r⟩ �k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' (22) where A⟨r⟩ d= e(r−1)πiU′ (2 cos πU′)r+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' For the first values of r ≥ 1 we get Y⟨1⟩ d= U � 0, 22 11 � B �1 2, 1 2 � , (23) Y⟨2⟩ d= U � 0, 33 22 � B �1 3, 1 2 · 3 � B �2 3, 2 2 · 3 � , (24) Y⟨3⟩ d= U � 0, 44 33 � B �1 4, 1 3 · 4 � B �2 4, 2 3 · 4 � B �3 4, 3 3 · 4 � , (25) etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' For r = 1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' (23) coincides with the factorisation (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' For r = 2, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' (24) is equivalent to a formula proved by M�lotkowski and Penson [24, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Random matrices associated to Young diagrams 9 0 2 4 6 8 10 12 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content="5 F'(x) r=1 r=2 r=3 r=4 F 0 hri(x) x r = 1 r = 2 r = 3 r = 4 Figure 1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Plot of the densities F ′ ⟨r⟩(x) for several values of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' We can use the decomposition (21) to write ‘explicit’ expressions for the densities F ′ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' For small values of r we have F ′ ⟨1⟩(x) = 1 2π � L(1) − x x χ[0,L(1)](x) (26) F ′ ⟨2⟩(x) = 1 23+1/331/2πw2/3 �� 31/2 + 2 � L(2) − w � � 33/2 − 2 � L(2) − w �1/3 − � 31/2 − 2 � L(2) − w � � 33/2 + 2 � L(2) − w �1/3� χ[0,L(2)](w).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' (27) The first is the Marchenko-Pastur distribution F ′ MP(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' The second is equivalent to formula in [24, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='1] proved by Mellin inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' For generic values of r, a ‘direct’ way to numerically compute F ′ ⟨r⟩(x) is by applying the Stieltjes inversion formula F ′ ⟨r⟩(x) = − 1 π lim ϵ↘0 G⟨r⟩(x + iϵ), (28) to Equation (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' This is how we made the numerical plots shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' In fact, one could, as in [30], write F ′ ⟨r⟩(x) as the inverse Mellin transform of the moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Since the moments are products of ratios of Pochhammer symbols, the resulting density would be a Mejier-G function (see the note by Dunkl [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' In fact, from the previous Proposition we can get some precise informations on the support of F ′ ⟨r⟩(x) and on its behavior at the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' The measure F⟨r⟩ has a density, F ′ ⟨r⟩ with support suppF ′ ⟨r⟩ := � x: F ′ ⟨r⟩(x) > 0 � = [0, L(r)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' (29) Random matrices associated to Young diagrams 10 Moreover at the edges of the support the density behaves as F ′ ⟨r⟩(x) ∼ � � � � � � � c x− r r+1, as x ↘ 0, c′ � L(r) − x, as x ↗ L(r), (30) for some constants c, c′ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' The density vanishes as a square root at the ‘soft edge’ x = L(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' At the ‘hard edge’ x = 0 the density has an integrable singularity x− r r+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' For r = 1 (Wishart matrices) this is the classical x− 1 2 divergence, but for r > 1 the singularity is stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' The generalised Catalan numbers C⟨r⟩ k = r k + 1 �rk + k k � (31) are related to the most known Fuss-Catalan sequences FCr,k = 1 rk + 1 �rk + k k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' (32) The Fuss-Catalan sequences also appear in random matrix theory and free probability as limiting moments of products of independent Wishart matrices [2, 19, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' As shown by Penson and ˙Zyczkowski [30], the Fuss-Catalan numbers FCr,k are the moment sequence of a distribution FFCr(x) whose density is a Meijer G-function, with support on the interval [0, L(r)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Proofs Theorem 1 was discovered by a combination of experimenting and guessing, and the proof is based on matrix moments calculations combined with the exact solution for the enumeration of r-plane trees (Proposition 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Let mk,N := � R xkdFN(x), and mk = 1 k + 1 �(r + 1)k k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' (33) Claim 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Each FN has moments of all order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' For all k, the moment sequence mk,N converges to mk, almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Claim 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' The sequence (mk)k≥0 uniquely determine the distribution F⟨r⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' By the moment method, the three claims imply Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Note that FN corresponds to a random measure with finite support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Hence, Claim 1 is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Claim 3 can be showed by checking Riesz’s condition [4, Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content='2], lim inf k 1 km 1 2k 2k < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' (34) Random matrices associated to Young diagrams 11 Indeed, Stirling’s approximation formula implies lim k→∞ 1 k � 1 2k + 1 �2(r + 1)k 2k �� 1 2k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' (35) It remains to prove Claim 2: lim N→∞ mk,N = mk, almost surely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' (36) Note that mk,N = � R xkdFN(x) = 1 rN Tr W k N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' (37) We may first assume that the entries Xij are uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Under this assumption we can prove the following two Lemmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Let λ(N) = N r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Then, for all k ∈ N, lim N→∞ Emk,N = mk, (38) Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Let λ(N) = N r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Then, for all k ∈ N, E (mk,N − Emk,N)2 = O � 1 N 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' (39) Lemma 1 states the convergence in expectation of the moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Lemma 2 implies, through Borel-Cantelli, the almost sure convergence (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' We now show that the Theorem holds true under the sole hypothesis of finite second moment E|Xij|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' This is done by the standard procedure of truncation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mdFRT4oBgHgl3EQfaDeS/content/2301.13555v1.pdf'} +page_content=' Fix a constant C and consider the symmetric matrix ˆXN whose elements satisfy, for 1 ≤, i, j ≤ rN ˆXN(i, j) = Xij1|Xij| S[ρ]. +We construct a geometric interpretation to the previously mentioned thoughts such +that the “variation of the relevant entropy represents a transfer of information between the +relevant and the irrelevant variables.” and “dissipation appears as a leakage of the relevant +information” [19]. With the help of the algebraic structure endowed in the observable space, +the geometric construction in the language of manifolds6 would help us in explaining Eq. +(7-8) in a purely quantum information way. The observable ˆO has eigenvectors |αk⟩ in the +space states, and therefore, it can be written as +⟨ ˆO⟩ = +� +k +⟨αk|ρ|αk⟩Tr +� +|αk⟩⟨αk|O +� +. +(16) +To define the vector space of the observable ˆO, we set the components and the bases to +be +oµ := ⟨αk|ρ|αk⟩, +(17a) +f µ := |αk⟩⟨αk|O, +(17b) +respectively as the components and the bases of the Liouville vector ⃗O = oµf µ. +6In that sense, the density matrix ρ should be seen as a pre-probability [52]. +9 + +As the set of all observables ˆO is characterized by the ρ, and ρ itself is an operator +that can be written in terms of any set of orthonormal bases {|i⟩}. Then, we can rewrite +Eq. (15–17) such that we define the density operator as a vector ⃗ρ with components +ρµ := ⟨f µ⟩ = Tr [f µρ] . +(18) +Then, when O becomes ρ, Eq. (15) becomes +−→ +⟨ˆρ⟩ := ⃗ρ = eµρµ, +(19) +where ρµ act as the components of the averaged observable ⟨ˆρ⟩ in its Liouville vector +representation −→ +⟨ˆρ⟩, and the basis components for such vector are defined as eµ := |i⟩⟨i| +that are dual to f µ, i.e. eµ · f ν = δ ν +µ . In such representation, Eq. (15) can be seen as a +bilinear relation +|| ⃗O|| = ⟨ ⃗O, ⃗ρ⟩ := oµρµ. +(20) +ΣP +ΣQ +t0 +ri +t0 + δt +λ(t) +f∗ : ΣQ → ΣP +P(ρµ +0) +Q(ρµ) +P ′(ρµ) +ρ1 +ρ2 +ρ3 +1 +Figure 1: Density manifold and the evolution of entropy surfaces away from reduced state ρ0. +10 + +To make the above construction more elaborate, let’s say that ˆO commutes with the +spatial observable ˆx, i.e. the ⃗O is a function in the components of ˆx. Then, the density ρµ +would correspond to the averaged components ⟨xµ⟩. As the density space is made of all +the ⟨xµ⟩ including the irrelevant ones, this means it is valid to embed the function ⃗O as a +hypersurface in the space spanned by ρµ after being expressed in terms of the dual basis of +⃗ρ. This surface is called the the surface of reduced states, and it is extremized at ρ0. At the +same time, it is much easier to describe the density in terms of the microscopic variables +ξµ and the stochastic variables Xµ, both as functions parameterized by the spacetime +observables xν or their averages, and we will do that later in subsection (3.5). But for +now, we just finished setting the stage up for the debut of the density space. Next, we +show this space could be promoted to a density manifold. +3.2 +Density manifold +In the density space, ρ is guaranteed to be a function in time such that Eq. (3) is satisfied. +Geometrically this means that ρ(t), as a point in such space, evolves from the point P(ρ0) +on the surface ΣP, where S[ρ] = S[ρ0], to another point on the same ΣP with S[ρ] along +some trajectory λ [53]. This is defined as an exponential function just like how we relate, +in Riemannian geometry, a vector to the points belonging to the trajectory that the vector +is tangent to. Thus +expP : t · ρµ → λρµ(t). +(21) +As t ∈ [0, 1], then exp(1.ρµ) = λρµ(1) is the final point P(ρ)(t) at the trajectory. Meanwhile +exp(0.ρµ) = λρµ(0) is the initial point P(ρ0). The ρµ +0 can take any direction. Therefore, +its ith component ρi along the tangent of the trajectory is given by +ρi +0 ≡ dλ(t) +dt +���� +t=0 +. +(22) +Moreover, the ith component of ρi +0 is allowed to be in the direction of any irrelevant basis +ρµ as long as it is tangent to the trajectory λ. This motivates considering the exponential +function (21) as a diffeomorphism between the neighborhood of point P, which belongs to +the state space, and the vector ρµ +0, which belongs to the tangent space TPΣ at point P. If +we normalize ρi +0∀i, then the orthonormal basis {ρµ +0} of TPΣ provides the isomorphism +E : Rn +∼−→ TPΣ, +(23) +such that E(ξ1, · · · , ξn)|P = ciρi +0. +The previous manifold-related definitions are valid regardless whether the surface is ex- +tremized on not. Therefore, there exists charts ψ on manifold M containing all λ’s and +TQΣ, for all Q ∈ M, such that +ψ := (exp ·E)−1 : M → Rn. +(24) +11 + +Also, the exponential map of the density manifold acts similarly to how the known +the exponential map exp(tV ) = λV (t) works between any Lie group and its Lie algebra, +where V ∈ the algebra, λV (t) ∈ the group, and t ∈ R. Then, there exists an analytic +diffeomorphism in a neighborhood U of V = 0 such that, for the coordinates ξµ ∈ Rn +defined by the isomorphism E in Eq. (23), we find that exp(ξiV i) ∈ exp(U). This would +define the necessary canonical chart. +Moreover, the entropy S[ρ] never loses information as long as the initial point P(ρ0) +evolves in a Hamiltonian trajectory. However, when we disregard the irrelevant informa- +tion, we do something similar to the shift and lapse such that the entropy S[ρ] = S[ρ0] +evolves in time through the extemized and non-extremized surfaces, i.e. we can practice +push-forward, f∗ : ΣQ → ΣP, (or pull-back) transformations between ΣP and ΣQ, together +with applying the rules of Lie derivatives in order to relate the surfaces to each others. +Now we are ready to introduce a metric on this information manifold. +3.3 +Manifold metric +We notice that the exponent function in Eq. (21) transfers ρµ to a tangent space with +same properties but at different point. Since the logarithmic function is the opposite to +the exponent, then, in light of Eq. (20), we can safely say that ln ρ does the same to ρ +except that the domain here becomes the dual tangent space. This implies that the entropy +S[ρ] acts as a bilinear map between the density ρ and the information content (− ln ρ). +As both ρ and (− ln ρ) are unique geometric vectors, the entropy provides us with a linear +isomorphic relationship between the tangents and the dual tangent spaces, i.e. the duality +ρ ≡ � +(ln ρ). +(25) +is legitimate. Therefore, there exists a map G on the ρ space such that +G : (ln ρ)µ �→ Gµνρν, +(26a) +G : (ρ)µ �→ Gµν(ln ρ)ν . +(26b) +The map G is shown to be symmetric, real, and has positive eigenvalues in the Liouville +vector representation [19]. It is the best function to play the role of metic in the ρ space. +Then, we can use Eq. (26) to demonstrate the entropy as the bilinear function +S⟨ρ, ln ρ⟩ := −Gµν(ρ(ξ))ρµρν. +(27) +The ρ space is dense enough–satisfying the topological features of manifolds–such that we +can introduce the infinitesimal change dρ. Therefore, Eq. (27) can be redefined infinitesi- +mally such that the second differential in the entropy becomes the metric itself. And the +ρ space becomes eligible for a promotion to be a Riemannian manifold endowed with the +invariant distance +−ds2 := d2S⟨ρ, ln ρ⟩ = −Gµν(ρ(ξ))dρµdρν, +(28) +12 + +where the map (26) is explicitly defined as +G(eµ, eν) := Gµν(ρ(ξ)) = − d2S +dρµdρν . +(29) +We now can introduce the Hamiltonian in Eq. (7-8) as a superoperator 7 H such that +H := H µ +ν eµ ⊗ f ν, +(30a) +H ν +µ = ⟨Heµ, f ν⟩. +(30b) +If we set O = H, then Eq. (3), or the commutator in Eq. (15), becomes the Liouville +superoperator +dρ +dt = L ρ = −i[H, ρ]. +(31) +Eq. (30) helps defining the components of L as +L ν +µ += −iTrf ν[H, eµ] = −iTr[f ν, H]eµ. +(32) +Reintroducing the Liouville operator as a superoperator excavates its “super power” such +that it manifestly plays the role of the Lie derivatives on the Riemannian ρ manifold. The +Jacobian of transformations between ΣP and ΣQ is given by +J := det +�∂ρ(t + δt) +∂ρ(t) +� += 1 − iL δt +(33) +or +iL µ +ν ρν := lim +δt→0 +ρµ(Q)(t + δt) − ρµ +0(P)(t) +δt +. +(34) +It is obvious that, with the above geometric interpretation, we also can introduce the +evolution superoperator +U := exp[−iL (δt)], +(35) +which plays a role similar to that of the Lie algebra, or the Killing fields, over the usual +Riemannian manifolds. We find Eq. (29) leads us to define the dual vector (− ln ρ) as +−(ln ρ) := ϱµf µ, +(36a) +ϱµ := (− ln ρ)µ ≡ ∂S +∂ρµ. +(36b) +Then, we introduce the Legendre transformation +S(ϱµ) += +S − ⟨ln(ρ)⟩ += +S + ϱµρµ. +(37) +7The prefix super has no Grassmann rings, i.e. has nothing to do with Supersymmetry or graded +algebra in general. +13 + +This transform reintroduces the ρ components to be +ρµ := ∂S +∂ϱµ +. +(38) +Therefore, the metric in Eq.(29) can contravariantized as +Gµν(ρ(ξ)) +:= +∂2S +∂ϱµϱν += +∂ρκ +∂ϱµ +∂ρλ +∂ϱν +∂2S +∂ρκ∂ρλ. +(39) +Consequently, Eq. (28) is transformed into +d2S = d2S − +� +d2ϱµρµ + 2dϱµdρµ + ϱµd2ρµ +� +. +(40) +The metricity ∇G = 0, or the parallel transport along geodesics ∇ρρ = 0, implies that +there exists a set of connection coefficients {} on the ρ manifold similar to the Levi-Civita +connections on the Einstein manifold. This means that both metric and connections are +related through +G(∇eλeµ, eν) := +� κ +λµ +� +Gκν(ρ(ξ)), +(41a) +� κ +λµ +� += 1 +2Gκι(ρ(ξ)) +� +∂µGιλ(ρ(ξ))+∂λGιµ(ρ(ξ)) − ∂ιGµλ(ρ(ξ)) +� +, +(41b) +which is enough to introduce geodesic equations, Riemann curvature tensor and related +other tensors. Additionally, Eq. (29) and Eq. (39) reveal a Hessian structure on the density +manifold such that Eq. +(41a) can be rearranged to get the corresponding connection +coefficients of the first kind +� +λµν +� += −1 +2 +∂3S +∂ρλ∂ρµ∂ρν += 1 +2 +∂ Gλµ(ρ(ξ)) +∂ρν +. +(42) +In light of Balian et al. metric, Eq. (31) and Eq. (36a) yield +L ν +µ ϱν = GµλL λ +ν ρν. +(43) +In light of Eq. (26), the symmetric property of Balian et al metric, and the covariant form +of the Liouville superoperator Lµν = Gµλ(ρ(ξ))L λ +ν , we differentiate Eq. (43) such that +0 = 1 +2 +∂ Gµλ(ρ(ξ)) +∂ρκ +L λ +ν ρν + 1 +2 (Lµκ + Lκµ) . +(44) +Then, we substitute Eq. (42) and Eq. (31) in Eq. (44) such that +−idGµν(ρ(ξ)) +dt += Lµν + Lνµ , +(45) +which is the reason why we said before that U , as defined in Eq. (35), plays a role similar +to that of Lie algebra or the Killing fields over the usual Riemannian manifolds. +14 + +3.4 +Euclidean structure of the space of observables +The density manifold reveals the Euclidean structure in the example we mentioned in sub- +section vector space construction. It means for an observable ⃗W there exists a component8 +wi such that the orthogonal projection P defines the components wj, j ̸= i, as +⟨⃗W − P ⃗W, ⃗W⟩ = +� � +⃗W − ow +�j +, wi +� += δ j +i +, +(46) +where the bilinear form is defined according to the map (26), i.e. we can infer that P is +a superoperator, with Greek indices, such that +P ⃗W = P µ +ν wµf ν. +(47) +Therefore, we can connect this Euclidean space to the Riemannian density manifold by +introducing a vielbein structure9 e µ +i +≡ (w) µ +i +such that the Euclidean flat metric corre- +sponding to this structure is defined as +gij(ρ(ξ)) := Gµν(ρ(ξ))e µ +i e ν +j , +(48a) +gjk(ρ(ξ))gki(ρ(ξ)) = δ j +i . +(48b) +In an information manifold this metric plays the same role the spatial spacetime metric γij +does in define the expansion parameter in Eq. (11). It is worth noting that the vielbein +acts on the density vector as a projector operator to yield the components of the density +vector in the Liouville space, which is another way to define the axes in Fig. (1), i.e. we +could start from the projector operator and the vielbein structure backward until we reach +the Liouville vector representation of the density operator; both approaches therefore are +equivalent. And if we get back to Fig. (1) and choose a point P ′(t0) ∈ ΣQ along the curve +t0, then +Pρ(t0) = ρ0(t0), +(49a) +ρi := ⟨wi⟩ = ⟨wi, ρ⟩ = ⟨wi, ρ0⟩, +∀i. +(49b) +Now it is safe to infer that the distance on the surfaces Σ in Fig. (1) are given by +ds2 +Σ += +gij(ρ(ξ))dρidρj += +gij(ρ(ξ))dρidρj , +(50) +where gij(ρ(ξ)) could be not equal to gij(ρ(ξ)) in general, i.e. dρi ̸= gij(ρ(ξ))dρj necessarily +as dρi is more like the component oi of ⃗O as defined previously. Last important manifold +structure can be obtained by combining Eq. (22) and Eq.(49) such that +⟨wi,r⟩ = 0 , +(51) +where the vector r = (ρ − ρi +0) is the tangent along the curve t0, see Fig. (1). Thus, we +have a vector bundle structure, where the base is the surface ΣP and the fibres are the +curves t0 + nδt, n ∈ N. See Ref. [54] for more about the relation with the blurred space. +8The components are those of the measured w of the observable ⃗W. +9Vielbein structure in information manifold is similar to those of the pseudo-Riemannian manifolds. +15 + +3.5 +Fisher metric and Kullback-Leibler divergence +The previously constructed density manifold has Euclidean signature. So, we target con- +structing an information manifold with a Lorentzian signature diag(−1, +1, +1, · · · ), and +we achieve this goal in subsection (3.7). For now, we focus on relating Balian et al metric +to Fisher metric. As the vacuum spacetime is in a “continuous” experience of quantum +fluctuations, it can be optimized stochastically such that the expectation values of the +operators over spacetime, i.e. the stochastic variables 10, are invariant under coordinate +transformations between different frames of references [56]. This means we can define the +density vectors ρµ, which is a function in the classical variables xµ characterizing the space- +time itself, as function in the stochastic variables Xµ ≡ Xµ(⟨xµ⟩, σxµ) that are functions in +a 2D space of averages Xµ(⟨xµ⟩) and standard deviations σxµ 11. Then, we may guess that +Balian et al. metric Gµν (ρ(ξµ)) in Eq. (28) could be expressed as an explicit function of +ξµ and Xµ variables, i.e. Gµν ≡ Gµν(ξµ; Xµ). In order to check the validity of this guess, +we need to check that the probability density stays the same when coordinate transforma- +tions are considered. The best candidate to test this requirement is the Kullback-Leibler +divergence +DKL = +� +ξi;Xi +ρ(ξµ; Xµ) ln +ρ(ξµ; Xµ) +ρ(ξµ + dξµ; Xµ)∆Xi. +(52) +As the spacetime variables are infinitesimally changing and ρ → dρ, then Eq. (52) becomes +DKL = − +� +M +ρ(ξ; X) +� +1 +ρ(ξ; X) +∂ρ(ξ; X) +dξµ +dξµ − 1 +2 +1 +ρ2(ξ; X) +∂ρ(ξ; X) +dξµ +∂ρ(ξ; X) +dξν +dξµdξν� +dX, +(53) +where ρ(ξ; X) ≡ ρ(ξµ; Xµ). The second term in Eq. (53) is nothing but the Fisher metric +ds2 := +� +i +(dρi)2 +ρi +, +(54) +which is the same metric in Eq. (28) but as an explicit function in ξµ and Xµ [19]. Applying +Fisher metric, chain rule, and the completeness relation of the density to Eq. (52) yields +DKL = +� +M +1 +2Gµν(ξ; X)dξµdξνdX. +(55) +Eq. (52) can be seen as +DKL ∼ −d2S = d +� ∂S +∂ρµ +dρµ +� += +∂2S +∂ρµ∂ρν +∂ρµ +∂ξκ +∂ρν +∂ξλdξκdξλ. +(56) +10For more on the properties of the stochastic variables, see Ref. [55]. +11See Ref. [57] for more on the 2D manifold of averages and standard deviations in the exponential +families of normal distribution. +16 + +By comparing Eq. (39), Eq. (55), and Eq. (56), and as the Kullback-Leibler divergence +is nothing but a modified Shannon entropy, the covariant form of Eq. (39) says that +Gµν(ϱ(ξ; X)) = ∂ϱ +∂ξµ +∂ϱ +∂ξν . +(57) +As we notice, the density is no longer a vector, it is just a function, and the vectors of +the new manifold are ∂µ ≡ ∂/∂ξµ. If we strict expressing all densities as as functions in +stochastic variables Xµ rather than the canonical ones ξµ, i.e. ρ ≡ ρ(Xµ), then, with help +of Eq. (15–20), we suppress Fisher metric density ξµ-dependence, i.e. from now on the +Balian et al. metric12 Gµν is not the metric we use. And Eq. (57) should be improved as +⟨Gµν(ξ; X)⟩ = ⟨ ∂ϱ +∂ξµ +∂ϱ +∂ξν ⟩. +(58) +In order to understand the last result, we need to get back to the Fisher metric. Without +loss of generality, Eq. (53) describes the relation between the entropy at the reduced state +ρ0 and any other state ρ, see the Points P and Q in Fig. (1). Then, in information +manifold the probabilistic average of the stochastic Fisher metric gµν plays the same role +the spacetime metric does in the Riemannian manifolds, i.e. gµν ≃ gµν. Thus, +gµν := ⟨Gµν(ξ, X)⟩ . +(59) +This is crucial for defining Einstein tensor and the analogue of the gravitational constant +in the information manifold as we will see by the end of the next subsection (3.6). +3.6 +Hessian structure and Einstein tensor +For an information manifold endowed with exponential family of distribution, there exists +a potential function φ such that its Hessian defines the metric of that manifold [59]. Non- +exponential families do the same but in a less straightforward way13, for more on that see +Ref. [62, 63] and references there. For simplicity we discuss the Hessian structure of ex- +ponential families, but our discussion is applicable to non-exponential ones too. We share +with Ref. [64] constructing Einstein tensor from the Fisher metric in information mani- +fold. Also, we find Einstein tensor to be endowed with relevant information to construct +the energy-momentum tensor from the varying cumulant partition function defined as a +scalar field. This is guaranteed naturally since the entropy data of the system underpin +the field strength of the system. Thus, we can consider Einstein equations in information +manifold as the equations of coarse-grained states for the original microscopic system of +12We move from the information manifold of Balian et al. [19] to the information manifold of Amari [58]. +13This is mainly because the Bayesian hypothesis [60] related to the Hessian in non-exponential fam- +ilies needs variational Bayesian approximations, like Laplace approximation and parameter separation +parameterization, to obtain the posterior distribution [61]. +17 + +quantum field theories behind the classical ones. The major difference between this work +and the endeavor of Ref. [64] is that, without assuming any family of exponentials, we find +a positively definite cosmological-like background term in the coarse-grained Einstein equa- +tions, particularly in the terms containing the derivatives of the Christoffel connections. +This suggests redefining Einstein tensor in information manifold to become a Lovelock +tensor. Consequently, coarse-graining the states could reveal extra disguised higher or- +dered curvature terms in the theory. Additionally, in contrary to Ref. [64] choice of real +probability distributions that suit AdS/CFT, we impose the family of exponentials to be +defined as complex probability distributions such that we construct Einstein tensor in a +information manifold with Lorentzian signature. This does not change the result that the +gravitational constant is indeed dynamical. However, it changes the sign of the extremal +area and the entropy. This result is consistent with the fact that complex probabilities +are associated with non-Hermitian Hamiltonians and their non-unitary transformations. +Ref. [64] admits the problem of controlling the energy scale of the quantum field theory +in the information geometric approach. In order to resolve that, we suggest repeating the +process coarse-graining until fine-graining is achieved [33]. Consequently, the information +approach would have a renormalization process, and the von Neumann equation will be +fine-grained as suggested in Ref. [11]. Thus, our modifications may introduce quantum +information geometry approach to dS/CFT [65,66]. +As shown in Appendix 2.7 of Ref. [61], the variational Bayesian inferences limited to +exponential families, like the KL divergence, can be generalized through the technique of +parameter separation parameterization. The technique is applicable to both exponential +and non-exponential families of distributions by linearly relating ρ to a real valued potential +function14. The potential function φ would help defining the metric of the information +manifold as shown in the Appendix. Without loss of generality, the equality ∂µ∂νφ = gµν +in Eq. (A20) is very similar to the bilinear relation Hess(φ) = ∇∇φ, or the Hessian, in +Riemannian geometry [58]. We are allowed to do this comparison because of the bundle +structure in the density manifold we referred to at the end of subsection (3.4). Remember +that ϱ ∈ T∗ +PΣ, i.e. the Hessian acts on the potential function φ to get an element in the +sections of tangent bundles Hess(φ) ∈ Γ(T∗Σ ⊗ T∗Σ). Also, the density manifold and +the canonical variable manifold, which could be the conventional spacetime, are related +through transforming ϱ into ∂µϱ. Then, we can apply a derivative on Eq. (A21) to get +∂λgµν = ∂λ∂µ∂νφ = ⟨∂λ(∂µ∂νϱ)⟩ = ⟨∂λ(∂µϱ∂νϱ)⟩ = −⟨∂λϱ∂µϱ∂νϱ⟩, +(60) +where the tensor relation between the vectors ∂µϱ is suppressed, the last equality comes +from ⟨∂µ(∂νϱ)⟩ = ⟨∂µϱ∂νϱ⟩, and the negative sign in last equality comes from the metric +definition in Eq.(27). More obviously from the behavior of the derivatives, we have +∂λgµν = ⟨∂λ∂µϱ∂νϱ⟩ + ⟨∂µϱ∂λ∂νϱ⟩. +(61) +14See function h(y) defined in Eq. 2.14 in Ref. [61] +18 + +Using the symmetric property of gµν, the last two equations lead us to +∂λgµν = 1 +2 +� +⟨∂λ∂µϱ∂νϱ⟩ + ⟨∂µϱ∂λ∂νϱ⟩ − ⟨∂λϱ∂µϱ∂νϱ⟩ +� +. +(62) +In light of Eq. (41b), the last result is very enticing to define the Christoffel connection +corresponding to the canonical variable manifold as [57,58] +Γλ +µν = −1 +2gλκ∂κ∂µ∂νφ += gλκ� +⟨∂κϱ∂µ∂νϱ⟩ − 1 +2⟨∂κϱ∂µϱ∂νϱ⟩ +� +. +(63) +Next, we use Eq. (63) to calculate the Ricci tensor and the Ricci scalar as functions in the +dual density variables and their derivatives. The Ricci tensor is obtained as usual from +the contracted Riemann tensor +Rµν = gαβRαµβν = ∂αΓα +µν − ∂νΓα +µα + Γα +βαΓβ +µν − Γα +βνΓβ +µα. +(64) +Meanwhile the Ricci scalar also is obtained as usual for the contacted Ricci tensor +R = gµνRµν. +(65) +Then, the Einstein equation in information manifold becomes +Gµν = Rµν − 1 +2gµνR. +(66) +As shown in the Appendix, we deconstruct Eq. (66) into two main pieces. We analyze +these pieces using the definitions of the metric and Christoffel connections in information +manifolds such that the Hessian structure is realized. So, we can reintroduce Eq. (66) as +Gµν = Λgµν + (˜Rµν − 1 +2gµν ˜R) + 1 +2gµν ˜R, +(67a) +or ˜Gµν + Λgµν = Gµν − 1 +2gµν ˜R, +(67b) +where ˜Rµν is explained in Eq. 69, and ˜Gµν + Λgµν plays a rule similar to that of Lovelock +tensor Aµν = Gµν + Λgµν in Einstein spacetime manifold [67]. +The first new piece obtained from rearranging then deconstructing Eq. (66), which is +Eq. (A9-A9′), gives a cosmological-like term in the information manifold as +� +∂αΓα +µν − ∂νΓα +µα +� +− 1 +2gµνgκλ� +∂αΓα +κλ − ∂κΓα +λα +� += 1 +2D(D +2 − 1)gµν ≡ 1 +2Λgµν , +(68) +where Λ = D(D/2 − 1) is defined for D ⩾ 2 dimensional manifold as in Lovelock theory +of gravity [68]. This helps our information manifold pass the necessary condition, but not +19 + +sufficient on its own, to develop an additional Gauss-Bonnet term for the corresponding +Riemann tensor that is expressed in its information form. The appearance of D(D/2 − 1) +term in the theory is very tempting to study the effects of having higher-curvature terms +in the context of holography. We leave that to be discussed hopefully in a future study. +We notice that the cosmological constant we obtained does not depend on the manifold +parameters. It only depends on the number of dimensions, and it vanishes when we consider +D = 2. This may point to background symmetry behind the cosmological constant, and it +may have a relation with vanishing cosmological constant in the context of some M/string +theory [69]. +More importantly, this piece in the ˜Gµν tensor should exist for all kinds +of exponential or non-exponential probability distributions. It neither demands ρ to be +exponential, Gaussian or complex one, nor to be in dS or AdS spaces. And this is our +major difference between this work and the endeavors in Ref. [64]. +The second piece after deconstructing Eq. (66) appears written in Eq. (A14) in the +Appendix. This piece of non differentiated Christoffels, obtained from both Rµν and gµνR, +defines a new tensor +�Rµν = +� +Γα +βαΓβ +µν − Γα +βνΓβ +µα +� +− 1 +2gµνgκλ� +Γα +βαΓβ +κλ − Γα +βκΓβ +λα +� += 1 +4gακgβλ� +⟨∂αϱ∂βϱ∂κϱ⟩⟨∂µϱ∂νϱ∂λϱ⟩ − ⟨∂νϱ∂βϱ∂κϱ⟩⟨∂µϱ∂αϱ∂λϱ⟩ +� +. +(69) +Without loss of generality 15, we follow the exponential family example in Ref. [64]. We +use Eq. (A15) and after so Eq. (69) becomes +�Rµν = 1 +4gακgβλ +�� +GαβGκλ − gαβgκλ +� ++ Gκλ +� +˙Gαβ⟨X − ⟨x⟩⟩ + 1 +2 +¨Gαβ⟨(X − ⟨x⟩)2⟩ + · · · +� ++ Gαβ +� +˙Gκλ⟨X − ⟨x⟩⟩ + 1 +2 +¨Gκλ⟨(X − ⟨x⟩)2⟩ + · · · +�� +× ∂µφ∂νφ. +(70) +Or +�Rµν = 1 +2GD ∂µφ∂νφ. +(71) +where the dynamical entity GD is defined as +GD = 1 +2gακgβλ +�� +GαβGκλ − gαβgκλ +� ++ Gκλ +� +˙Gαβ⟨X − ⟨x⟩⟩ + 1 +2 +¨Gαβ⟨(X − ⟨x⟩)2⟩ + · · · +� ++ Gαβ +� +˙Gκλ⟨X − ⟨x⟩⟩ + 1 +2 +¨Gκλ⟨(X − ⟨x⟩)2⟩ + · · · +�� +. +(72) +15We do not lose generality because the crucial steps in Eq.(A20–A21) are guaranteed by the previously +mentioned technique of parameter separation parametrization to work even with non-exponential families +[61]. +20 + +Using the cumulant partition function φ as a classical scalar field in the information man- +ifold, we can define a Lagrangian in D = 4 for an effective field theory as +L = gµν∂µφ∂νφ, +(73) +and the corresponding energy-momentum tensor is +Tµν = gµλ +∂L +∂(∂λφ)∂νφ − gµνL = 1 +G4 +� +�Rµν − 1 +2gµν�R +� += 1 +G4 +�Gµν , +(74) +where obviously the last equation is Einstein equation with its reduced tensors �Gµν and +�Rµν as functions in the stochastic variables X in such information geometry. Also, we can +add the cosmological constant term we obtained before in Eq. (68). Comparing Eq. (74) +with Eq. (14), we get +G4 ≃ 2πℓ2 +P +ℏ +, +(75) +which again is nothing but the gravitational constant in the information form for such +4D geometry. This indicates quantum information geometry induces gravity phenomena, +and the gravitational constant is no longer constant besides its ℏ dependence. This may +support studies that implies inducing gravity from quantum mechanics [70–74]. Also, It +could be connected with realizing space and time as an approximate macroscopic concepts +stem fundamentally from quantum field theories [75]. Additionally, a varying gravitational +constant may give a clue to Dirac’s large numbers hypothesis [76] that also implies varying +gravitational constant is based on the simple analysis of dimensionless constants that are +provided by nature. +The results obtained in Eq. (63–68) are discussed in details in the Appendix of the +manuscript. There, the family of exponentials is not assumed to be either complex, which +is what we discuss in the next subsection, or real as in Ref. [64], since we keep everything +in the appendix in terms of an arbitrary ϱ. Additionally in Ref. [64], after the Gaussian +exponential was imposed, the “whole” Einstein tensor is equated to a “negative” cosmo- +logical constant. This is not what we obtain. Rather, we say that the Einstein tensor +Gµν should be split into ∂Γ terms that correspond to a “positively” cosmological constant +defined in terms of the dimension D as in Eq. (68), and ΓΓ terms that introduce a modi- +fied Ricci tensor ˜Rµν as the kinetic term ∂µφ∂νφ in Eq. (70). Notice that the Lagrangian +L ∼ gµν ˜Rµν ∼ �φ renders the reduced Einstein tensor ˜Gµν upon applying the variational +principle with respect to φ, while the Λgµν piece is produced from varying L with re- +spect to gµν = ∂µ∂νφ. This reminds us with the Y piece in the modified Lagrangian of +gravity in the framework of superstrings [77], both share the same Hessian structure. In +brief, the Einstein tensor Gµν in Ref. [64] should rather be treated as a Lovelock tensor +Aµν = Gµν + Λgµν. Such tensor should be split into a cosmological term Λgµν and a modi- +fied Ricci tensor ˜Rµν, and the later can be used to introduce a new modified Einstein tensor +˜Gµν that has no cosmological terms in it. All that can be obtained without assuming the +densities to be expressed as family of exponentials or any other family. +21 + +3.7 +Obtaining a pseudo-Riemannian information manifold +A question remains about how to construct an arbitrary (D + 1)-dimensional information +geometry with a Lorentzian signature diag(−1, +1, · · · ). And without loss of generality, we +follow the complex16 Gaussian ansatz in Ref. [78] to get a (1 + 1)-dimensional information +manifold with a Lorentzian signature diag(−1, +1) +ρ(t) = +1 +√ +2πσ exp +� +−(t − i⟨t⟩)2 +2σ +� += exp +� +− ln( +√ +2πσ) − t2 +2σ2 + it⟨t⟩ +σ2 + ⟨t⟩2 +2σ2 +� +, +(76) +that requires components of the corresponding Fisher metric to be +g00 = +� +d2ξ 1 +ρ(ξ) +�∂ρ(ξ) +∂ξ0 +�2 += −1, +(77a) +g11 = +� +d2ξ 1 +ρ(ξ) +�∂ρ(ξ) +∂ξ1 +�2 += 1, +(77b) +g01 = +� +d2ξ 1 +ρ(ξ) +�∂ρ(ξ) +∂ξ0 +� �∂ρ(ξ) +∂ξ1 +� += 0. +(77c) +As the Gaussian distribution requires defining a complex exponential family using the +following distribution +ρ(ξµ) = exp [ξµEµ(Xν) − φ(ξν)] , +(78) +where the function Eµ(Xν) plays the role of any physical property related to the corre- +sponding canonical or intensive variables ξν, and φ(ξν) is the cumulant partition function +of the Eµ(Xν) states [58], see Eq. (A15) and after in the Appendix. By comparing Eq. +(76) to Eq. (78), it is easy to notice that +Eµ := (E0, E1) = (t, t2), +(79a) +ξµ := (ξ0, ξ1) = +�i⟨t⟩ +σ2 , − 1 +2σ2 +� +, +(79b) +φ(ξν) := 1 +2 ln +� +− π +ξ1 +� +− (ξ0)2 +4ξ1 = ln +�√ +2πσ +� ++ ⟨t⟩2 +2σ2 , +(79c) +which defines the cumulant distribution as +ϱ := − ln(ρ(ξ)) ≡ φ(ξν) − ξµEµ(Xν) += ln( +√ +2πσ) + t2 +2σ2 − it⟨t⟩ +σ2 − ⟨t⟩2 +2σ2 . +(80) +Notice that Eµ(Xν) is a function in t, which is one of the components xµ of the classical +spacetime, i.e. the stochastic variables Xν can be parameterized generally by the classical +16We comment on the consequences of is ansatz in the discussion section. +22 + +xµ components of the spacetime as we emphasized before in the beginning of subsection +(3.5) when we mentioned the coarse-graining the properties of the manifold. +Also, as +ξµ variables are functions in the averages and the standard deviations of xν, i.e. ξµ are +functions in statistical variables, we can suppress the variable ξ in the metric such that +Gµν (ξ, X) ≡ Gµν (X). Moreover, defining the exponential family of probability distributions +to be either complex, like how we have just done, or a real, as in Ref. [64], will not change +the definition of the cumulant probability distribution17 ϱ = φ(ξν) − ξµEµ(Xν). Therefore, +the results in the previous subsection hold correct in both approaches, where our approach +is suitable for dS spaces while Ref. [64] fits for AdS. Constructing arbitrary D-dimensional +pseudo-Riemannian information manifold from a classical spacetime is left as an exercise +to the reader18. +4 +Entropy of the Information Manifold +As we have seen in section (2) for non dissipative systems, the time rate change of the en- +tropy is directly related to the content of information expressed in Liouville-von Neumann +equation. But in the previous section we found that there is a correspondence between the +spacetime and the statistical information manifold, this correspondence makes the space- +time metric corresponds to Fisher metric even when we compare the Lorentzian signature +of both [78,81,82]. As the spatial entropic area A is part of the spacetime, we can construct +an area A in the information manifold that corresponds to the A, and the spatial metric of +that information area exists in light of our discussion in subsection (3.4). More about the +area A of such blurred spatial space from quantum entanglement contours can be found in +Ref. [54], or after complexifying the exponential distribution families 19. And if we accept +the existence of such correspondence, then we can find a equation for the spatial-like ex- +pansion rate in the information manifold θ that corresponds to θ as mentioned before in +section (2), i.e. +θ = 1 +A +d +dt (A) , +(81) +where, in comparison to Eq. (11) and Eq. (48a), the spatial-like expansion rate in the +information manifold θ is defined using the determinant of the averaged spatial components +of the Fisher metric γ = det(γij) = det(⟨gij⟩), i.e. +θ = +1 +√γ +d +dt (√γ) . +(82) +Since the exponential family is assumed to be either real or complex, the Hamiltonian is +expected to be non-Hermitian with non-unitary transformation, and it is expected to be a +17Notice ρ = exp(−βH)/Z gives φ − ϱ = −βH where the energy H is related to the entropy S [15]. +18This can be obtained in a similar fashion of how AdS is obtained from complexifying dimensions of +CFT, see Eq. (9-12) in Ref. [79], or see the (u, v) thermal coordinates in Ref. [80] +19A could be developed for AdS as in Ref. [83]. +23 + +stochastic Hamiltonian in comparison with regular Hamiltonians in regular phase spaces. +Thus, the Hamiltonian should be modified to obey Lindblad master equation [84], which +is the most general form of Liouville equation, and we comment on this in the discussion. +Then for a stochastic Hamiltonian H as a function in X and its conjugate momentum, Eq. +(8) in the information manifold becomes +i 1 +4G4 +θA = Tr +� +iℏdρ +dt ln ρ + [H, ρ] +� +. +(83) +And for dissipative systems, Eq. (7) +−iℏdSBH +dt += −i 1 +4G4 +θA + Tr +� +iℏdρ +dt ln ρ + [H, ρ]Lb +� +, +(84) +which is the entropy of the black hole in the information manifold with no classical com- +ponents from the spacetime itself, just information geometry. Based on Lindblad master +equation, the [H, ρ]Lb term is the time evolution of the density under the influence of the +interaction Hamiltonian Hi in an open dissipative system [85] +dρ(t) +dt +≡ [H, ρ]Lb := 1 +iℏ[Hi(t), ρ(0)] − 1 +ℏ2 +� t +0 +dt′[Hi(t), [Hi(t′), ρ(t)]]. +(85) +The question now is: what is the meaning of the area A in the information manifold, +and how is it related to the spatial extremal area A in the spacetime? The answer lies +in the definition of extremal surface in dS space, which could be compared with that RT +surface in the AdS space [86, 87]. In order to get the extremal surfaces in dS space, we +must complexify time such that average speed on the timelike surfaces, defined by the +ratio of the shortest spatial angular length l of a dS space to the shortest time ϵ, becomes +the determining factor of the size of the extermal surface, i.e. +AdS = −πR2 +� l +cϵ − 1 +� +, +(86) +The last equation shows that for some cases, l/ϵ > 1, the area could be negative20, which +would lead to negative or even complex valued entropy! Before the last comment gets +“frown upon”, this could be necessary to avoid the disappearance of the spatial surfaces of +dS Rindler wedge [88]. This might not be well-appreciated as it says there could be non- +unitary states in CFT, i.e, the corresponding Hamiltonian H could be non-Hermitian. Very +recently, it is proved that the non-Hermiticity stems from the fact that the non-unitary +CFT, dual to dS, lives on a space-like surface and the time coordinate emerging from an +Euclidean CFT [89,90] related to the previously mentioned blurred space. When we shift +to the language of information manifolds, we surprisingly find some “untimely meditations” +about the necessity of complexfying the spacetime and probability distributions so that we +20The area could be imaginary for dS spaces with different dimensions, see Ref. [87]. +24 + +get a Fisher metric as an averaged metric over spacetime fluctuations with a Lorentzian +signature [78]. Consequently, claiming the necessity of non-Hermitian Hamiltonian corre- +sponding to the Einstein-Hilbert formulation of GR in information manifold suggests that +we can study the dynamics of a Wheeler-deWitt Hamiltonian [91, 92] described by the +spatial metric γ on information manifold. An example of the pseudo-Hermitian Wheeler- +deWitt Hamiltonian is discussed in details in [93], and detailed calculations of such Hamil- +tonian in information manifold could be followed from Ref. [94–96] but they are left for a +future study. +One last thing to be said about the RT formula. By comparing the approach21 followed +in [97], the RT formula in 4D dS becomes +SRT = AdS +4G4 += −πR2 +4G4 +ln( l +cϵ) ∼ −πR2 +4G4 +� l +cϵ − 1 +� +, +(87) +which is the formula of the entropy of black hole when the horizon coincides with the +RT extremal surface. And to get the expression of the entropy of the extremal surface in +information manifold, simply replace A → A and G4 → G4, where G4 is the gravitational +constant in 4D spacetime as given in Eq. (75). For more details on the leading divergent +term in the previous equation and its relation to the holographic entanglement entropy, +see Ref. [98]. +5 +Discussions and Conclusions +As we have seen, our study suggests reducing the geometrical properties, including space- +time itself, to an information geometry language in a way that could evolve the insight +on the deep connection between physics and information. In this work, we studied in +details the coarse grained entropy of the black hole that obeys the second law of ther- +modynamics. We analyzed the entropy-area law corrected by von Neumann entropy of +the quantum matter outside its event horizon in order to obey second law of thermo- +dynamics and to preserve information. We constructed the corresponding form of this +corrected entropy-area law in quantum information geometric language. Consequently, a +corresponding spacetime emerges from the quantum information. We discussed the link +between Wald-Jacobson approaches of thermodynamic/gravity correspondence and Fisher +pseudo-Riemannian metric of information manifold that guarantees extending the geomet- +ric interpretation to any quantum theory. We formulated Einstein’s field equations in infor- +mation geometry forms, and we obtained a modified Ricci tensor that helped constructing +a Lagrangian of such theory. Also, we used the modified Ricci tensor to introduce the +reduced Einstein tensor ˜Gµν, which is directly related to the energy-momentum tensor in +such manifold. The formulated Einstein’s field equations led into two interesting outcomes +stemming fundamentally from information geometry. The first result is finding a quantum +21See Ref. [87] for the analysis of RT formula in 2D information manifolds. +25 + +origin of a positive cosmological constant that is founded on Fisher metric. This cosmolog- +ical constant resembles those found in Lovelock’s theories in de Sitter background due to +complexifying time and the Gaussian exponential families of probability distributions. The +second result is a time varying gravitational constant that resembles the idea of Dirac’s +large number hypothesis and predicts varying of gravitational constant based on simple +analysis of nature constants. We extended our analysis into the information manifold and +write down a dynamical equation for the entropy in quantum information manifold using +Liouville-von Neumann equation for the quantum system Hamiltonian. According to our +results, the Hamiltonian in the information manifold could be non-Hermitian. The result- +ing dynamical equation provides a clue to a direction that could ameliorate the problem +of time. +It is worth noting that relating Jacobson endeavors to information geometry requires +considering non-equilibrium thermodynamics of spacetime and its associated dissipative +gravity [99,100]. This relation comes from the coarse-graining process and the suggested +quantum thermodynamical origin of spacetime. +To achieve fine-graining, the observa- +tional entropy will match von Neumann entropy after several consecutive coarse-graining +processes [33]. And for finite-dimensional systems like spacetime itself, xµ are obviously +finite, the observational entropy can be expressed as relative entropy. Moreover, a non- +local heat fluxes is proven to take place in non-equilibrium thermodynamical gravitational +systems, for both Einstein’s general relativity and the scalar-tensor modifications, due to +their dissipative characters [100]. When applied to Rindler spacetime, The thermal char- +acter of such heat flux extends the vacuum thermal state to include the whole Rindler +wedge not just the single observer. So, an accelerated observer would access information, +due to entanglement entropy, on spacelike slices. Consequently, the restricting the neigh- +borhood of the Rindler wedge origin, as spacelike slice around it, determines the expansion +coefficient. As the time rate change of the expansion coefficient is related to the dissipative +energy coupled to the bulk and shear viscosity, then it relates the entropy, both internal at +the irreversible level and exchanged at the reversible level, to the Equivalence Principle in +such dissipative systems. Moreover, the time rate change of the expansion coefficient states +a universal relation between viscosity and internal viscous part of the entropy density as +found in the AdS/CFT [26]. +We know that the collapse of a pure state results in a mixed state, which is a process +the unitary transformations cannot achieve in irreversible processes due to the problem of +preserving the norm of the wave functions. Moreover, there is no such realistic quantum +system that could be described by a pure state [101,102]. So, we are left with non-unitary +transformations. As the quantum time-reversible processes governed by Schrodinger evolu- +tion equation are always unitary, then there is an obvious contradiction in assuming those +processes to comprise the non-reversible macro physics such as the second law of ther- +modynamics. There is a long debate on the optimal epistemic and/or ontological way to +resolve this contradiction, it is usually discussed under the umbrella of wave function jump +and dissipative systems, see Ref. [103] for more details. One of the remarkable suggestions +is that the microscopic variables indeed evolve according to non-unitary processes [104]. +26 + +So, based on the non-unitary irreversible von Neumann’s measurement transitions that +render mixed states [105], Ref. [106] provides an account on assuming that wave func- +tion collapse takes place spontaneously and randomly in space and time. We know that +von Neumann entropy is a function in basis-independent density operators, meanwhile +the reversible Shannon entropy, related to Gibbs and Boltzmann ones, is a function in +the probability density matrices. +If von Neumann and Shannon entropies are related, +then there must be some stochastic variables that correspond to coarse-graining Liouville +equation. Classically this is associated with a loss in information, which is discussed in +Balian et al. endeavor [19]. At the quantum level, this is associated with a loss in phase +coherence in quantum states. This is suggested to happen in a non-unitary transforma- +tion from a pure state to mixed one [106]. This guarantees the validity of the second +law of thermodynamics despite having Liouville equation to be governed by phase-space +Hamiltonians. However, this does not demand making change to the Schrodinger equation; +rather it says that Liouville equation is physically broken at the quantum level. The price +could be replacing the regular Liouville equation with the more general Lindblad master +equation of open quantum systems [84], which allows the corresponding Hamiltonian to +have non-Hermitian parts [85]. These non-unitary quantum processes are found in many +high energy systems, such as CFTs with zero and negative central charges that exhibit +entanglement entropy [107,108], in condensed matter systems [109], and even in quantum +electronics [110]. +A final comment on complexifying time related to what is mentioned in Isham’s report +on the problem of time [111]. In the example, if we focus only on the properties of an +open quantum system, then the relevant state becomes that of the reduced density matrix +obtained by summing over–in another word by tracing out–the states of the surroundings. +If the states of the surroundings are approximately orthogonal–Balian et al. assume that +too—then the density of quantum system exhibits decoherence. This is guaranteed to +be true as the reduced density matrix is proved to be governed in different examples of +Lindblad master equation [112] such as spatial decoherence [113] and quantum Brownian +motion [114]. Isham emphasizes that the inability to find a satisfactory unitary Hamil- +tonian for Wheeler-de Witt equation should not be considered necessarily as a disaster. +Rather, this “might reflect something of genuine physical significance”. For example, in +Hartle and Hawking [115] and Vilenkin [116] endeavors, time becomes complex due to the +non-unitary evolution. +Aknowledgement +The author would like to thank A. F. Ali for for discussions and comments during the +preparation of this work. +27 + +6 +Appendix +This appendix is dedicated for the detailed calculations of the connection terms and their +derivatives in Einstein equation in information manifold. +If we substitute Eq. (63) in the derivative terms of Eq. (64) we get +∂αΓα +µν − ∂νΓα +µα = gαβ∂α +� +⟨∂µ∂νϱ∂βϱ⟩ − 1 +2⟨∂µϱ∂νϱ∂βϱ⟩ +� +− gαβ∂µ +� +⟨∂α∂νϱ∂βϱ⟩ − 1 +2⟨∂αϱ∂νϱ∂βϱ⟩ +� ++ ∂α(gαβ) +� +⟨∂µ∂νϱ∂βϱ⟩ − 1 +2⟨∂µϱ∂νϱ∂βϱ⟩ +� +− ∂µ(gαβ) +� +⟨∂α∂νϱ∂βϱ⟩ − 1 +2⟨∂αϱ∂νϱ∂βϱ⟩ +� +(A1) +Also, if we substitute Eq. (63) in the multiplicative terms of Eq. (64) we get +Γα +βαΓβ +µν − Γα +βνΓβ +µα = gακgβλ +� � +⟨∂α∂βϱ∂κϱ⟩ − 1 +2⟨∂αϱ∂βϱ∂κϱ⟩ +� +× +� +⟨∂µ∂νϱ∂λϱ⟩ − 1 +2⟨∂µϱ∂νϱ∂λϱ⟩ +� +− +� +⟨∂ν∂βϱ∂κϱ⟩ − 1 +2⟨∂νϱ∂βϱ∂κϱ⟩ +� +× +� +⟨∂µ∂αϱ∂λϱ⟩ − 1 +2⟨∂µϱ∂αϱ∂λϱ⟩ +� � +(A2) +We distribute the outer derivative applied on first line in Eq. (A1), together with exploiting +the properties in Eq. (60), to get +gαβ∂α +� +⟨∂µ∂νϱ∂βϱ⟩ − 1 +2⟨∂µϱ∂νϱ∂βϱ⟩ +� +− gαβ∂µ +� +⟨∂α∂νϱ∂βϱ⟩ − 1 +2⟨∂αϱ∂νϱ∂βϱ⟩ +� += +gαβ +� +⟨∂µ∂νϱ∂α∂βϱ⟩ − ⟨∂αϱ∂µ∂νϱ∂βϱ⟩ − 1 +2 +� +⟨∂µ∂αϱ∂νϱ∂βϱ⟩ + ⟨∂µϱ∂α∂νϱ∂βϱ⟩ + ⟨∂µϱ∂νϱ∂α∂βϱ⟩ +� +− ⟨∂α∂νϱ∂µ∂βϱ⟩ + ⟨∂µϱ∂α∂νϱ∂βϱ⟩ + 1 +2 +� +⟨∂α∂µϱ∂νϱ∂βϱ⟩ + ⟨∂αϱ∂µ∂νϱ∂βϱ⟩ + ⟨∂αϱ∂νϱ∂µ∂βϱ⟩ +�� += +gαβ� +⟨∂µ∂νϱ∂α∂βϱ⟩ + 1 +2⟨∂µϱ∂α∂νϱ∂βϱ⟩ + 1 +2⟨∂µϱ∂β∂νϱ∂αϱ⟩ +− ⟨∂µ∂βϱ∂ν∂αϱ⟩ − 1 +2⟨∂αϱ∂µ∂νϱ∂βϱ⟩ − 1 +2⟨∂µϱ∂α∂βϱ∂νϱ⟩ +� +(A3) +28 + +We contract Eq. (A3) using gµν to get +gαβ∂α +� +⟨∂κ∂κϱ∂βϱ⟩ − 1 +2⟨∂κϱ∂κϱ∂βϱ⟩ +� +− gαβ∂κ +� +⟨∂α∂κϱ∂βϱ⟩ − 1 +2⟨∂αϱ∂κϱ∂βϱ⟩ +� += +gαβ� +⟨∂κ∂κϱ∂α∂βϱ⟩ + 1 +2⟨∂κϱ∂α∂κϱ∂βϱ⟩ + 1 +2⟨∂κϱ∂β∂κϱ∂αϱ⟩ +− ⟨∂κ∂βϱ∂κ∂αϱ⟩ − 1 +2⟨∂αϱ∂κ∂κϱ∂βϱ⟩ − 1 +2⟨∂κϱ∂α∂βϱ∂κϱ⟩ +� +(A4) +Now, we multiply Eq. (A4) by − 1 +2gµν then pick terms from the result of such multiplication +that match with the terms in Eq. (A3) to show that Eq. (A1), i.e. the derivative parts +in Eq. (64), corresponds to the cosmological coupling constant term in Lovelock theory of +gravity [68]. The middle steps are obtained using the definition of the Gµν in Eq. (A20) and +the definition of gµν in Eq. (A21). The gµνgµν = GµνGµν = D, where D is the dimension +of the manifold. Collect the fourth term in the last two lines of Eq. (A3) with − 1 +2gµν×the +fourth term in the last two lines of Eq. (A4) to get +−gαβ⟨∂µ∂αϱ∂ν∂βϱ − 1 +2gµν∂κ∂αϱ∂κ∂βϱ⟩ = −(1 − D/2)gµν +(A5) +Collect the second and the third terms in the last two lines of Eq. (A3) with − 1 +2gµν×the +third term in the last two lines of Eq. (A4) to get +1 +2gαβ⟨∂α∂νϱ∂β∂µϱ + ∂α∂µϱ∂β∂νϱ − gµν∂κ∂αϱ∂κ∂βϱ⟩ = (1 − D/2)gµν +(A6) +It is obvious that the last two results, Eq. (A5) and Eq. (A6), cancel each other. Collect +the last term in the last two lines of Eq. (A3) with − 1 +2gµν×the last term in the last two +lines of Eq. (A4) to get +−1 +2gαβ⟨∂α∂βϱ(∂µ∂νϱ − 1 +2gµν∂κ∂κϱ)⟩ = −(D/2 − D2/4)gµν +(A7) +Collect the last term in the last two lines of Eq. (A3) with − 1 +2gµν×the last term in the +last two lines of Eq. (A4) to get +1 +2gαβ⟨∂α∂βϱ(∂µ∂νϱ − 1 +2gµν∂κ∂κϱ)⟩ = (D/2 − D2/4)gµν +(A8) +It is obvious that the last two results, Eq. (A7) and Eq. (A8), cancel each other. We are +left with the fifth term in the last two lines of Eq. (A3) and the − 1 +2gµν×fifth term in the +last two lines of Eq. (A4). We combine both to get +−1 +2gαβ⟨∂αϱ∂βϱ(∂µ∂νϱ − 1 +2gµν∂κ∂κϱ)⟩ = −(D/2 − D2/4)gµν +(A9) +29 + +Therefore, Eq. (A9) is the only part that contributes to Eq. (A1) +� +∂αΓα +µν − ∂νΓα +µα +� +− 1 +2gµνgκλ� +∂αΓα +κλ − ∂κΓα +λα +� += 1 +2D(D +2 − 1)gµν +(A9′) +where Λ = D(D/2 − 1) is defined for D ⩾ 2 dimensional manifold. +For the second line in Eq. (A1), we use +∂µ(gαβ) = −gακgβλ∂µgκλ = gακgβλ⟨∂µϱ∂κϱ∂λϱ⟩ +(A10) +so that the second line in Eq. (A1) is +gακgβλ +� +⟨∂αϱ∂κϱ∂λϱ⟩ +� +⟨∂µ∂νϱ∂βϱ⟩ − 1 +2⟨∂µϱ∂νϱ∂βϱ⟩ +� +− ⟨∂µϱ∂κϱ∂λϱ⟩ +� +⟨∂α∂νϱ∂βϱ⟩ − 1 +2⟨∂αϱ∂νϱ∂βϱ⟩ +� � +(A11) +Next, we expand Eq. (A2). The terms with no 1/4 resulting from such expansion are +gακgβλ� +⟨∂α∂βϱ∂κϱ⟩⟨∂µ∂νϱ∂λϱ⟩ − 1 +2⟨∂α∂βϱ∂κϱ⟩⟨∂µϱ∂νϱ∂λϱ⟩ − 1 +2⟨∂αϱ∂βϱ∂κϱ⟩⟨∂µ∂νϱ∂λϱ⟩ +− ⟨∂ν∂βϱ∂κϱ⟩⟨∂µ∂αϱ∂λϱ⟩ + 1 +2⟨∂ν∂βϱ∂κϱ⟩⟨∂µϱ∂αϱ∂λϱ⟩ + 1 +2⟨∂νϱ∂βϱ∂κϱ⟩⟨∂µ∂αϱ∂λϱ⟩ +� +(A12) +For the first line in Eq. (A12), exchange β and λ in the first term, change the sign in that +term according to Eq. (60). Then, collect that term with the third term in the same line. +And for the second line in Eq. (A12), exchange κ and α in the first term, change the sign +in that term according to Eq. (60). Then, collect that term with the third term in the +same line. Also for the second and fifth term in Eq.(A12), change the sign in those terms +according to Eq. (60). Thus, Eq. (A12) becomes +− gακgβλ +� +⟨∂αϱ∂κϱ∂λϱ⟩ +� +⟨∂µ∂νϱ∂βϱ⟩ − 1 +2⟨∂µϱ∂νϱ∂βϱ⟩ +� +− ⟨∂µϱ∂κϱ∂λϱ⟩ +� +⟨∂α∂νϱ∂βϱ⟩ − 1 +2⟨∂αϱ∂νϱ∂βϱ⟩ +� � ++ 1 +2gακgβλ +� +⟨∂αϱ∂βϱ∂κϱ⟩⟨∂µϱ∂νϱ∂λϱ⟩ − ⟨∂νϱ∂βϱ∂κϱ⟩⟨∂µϱ∂αϱ∂λϱ⟩ +� +(A13) +We see that the first line in Eq. (A13) cancels with Eq. (A11), which is obtained from the +second line in Eq.(A1). Moreover, we add the second line in Eq. (A13) to the terms with +1/4 in Eq. (A2). Then, we contract the µν indices of the result of such addition, multiply +it with − 1 +2gµν, and add it to the original terms before the µν contraction such that we +introduce a new tensor +�Rµν = +� +Γα +βαΓβ +µν − Γα +βνΓβ +µα +� +− 1 +2gµνgκλ� +Γα +βαΓβ +κλ − Γα +βκΓβ +λα +� += 1 +4gακgβλ� +⟨∂αϱ∂βϱ∂κϱ⟩⟨∂µϱ∂νϱ∂λϱ⟩ − ⟨∂νϱ∂βϱ∂κϱ⟩⟨∂µϱ∂αϱ∂λϱ⟩ +� +(A14) +30 + +Here we follow Ref. [64]. In order to relate the Fisher metric with the entropy defined +as in Eq. (29), or Eq. (39), we know the density means also the relative share of certain +energy state E(Xµ) from the total collection of all energy states in the partition function +Z [57], i.e. +ρ(ξµ; Xµ) = exp [−βE(Xµ) − ln Z(ξµ)] +(A15) +which reintroduces the probability distributions to the family of exponentials. Then, we +can define the density generally as +ρ(ξµ) = exp [ξµEµ(Xν) − φ(ξν)] +(A16) +which is Eq. (78). The corresponding dual density becomes +ϱ = − ln ρ = φ(ξν) − ξµEµ +(A17) +Applying the first and the second derivative with respect to ξµ on Eq. (A17) yields +∂µϱ = ∂µφ − Eµ(Xν) +(A18) +∂µ∂νϱ = ∂µ∂µφ +(A19) +In light of Eq. (29), Eq. (39) and Eq. (58), the last Eq. (A18–A19) can be rearranged to +get +⟨Gµν⟩ = ⟨∂µ∂νϱ⟩ = ∂µ∂νφ = gµν +(A20) +Since ⟨∂µϱ⟩ = 0, then we apply another differentiation and use Eq. (A20) to get +gµν = ⟨∂µ∂νϱ⟩ = ⟨∂µϱ∂νϱ⟩ +(A21) +despite that ∂µ∂νϱ ̸= ∂µϱ∂νϱ. Moreover, since +⟨Eµ⟩ = ∂µφ , +(A22) +then the last three equations give +gµν = ⟨EµEν⟩ − ⟨Eµ⟩⟨Eν⟩ +(A23) += ⟨EµEν⟩ − ∂µφ⟨Eν⟩ = −⟨∂µϱEν⟩ +(A24) +The last three relations will help us to construct the Christoffel symbol [57,58], from the +connections in Eq. (41b), and consequently the Riemann curvature tensor as functions in +the density vectors as we will see in a little bit. +Now, we substitute Eq. (57) and Eq. (A18) into Eq. (A14), then expand, we obtain +�Rµν = 1 +4 +� +D∂µφ∂νφ − gβκ∂µφ⟨EνGβκ⟩ − gαλ∂νφ⟨EµGαλ⟩ ++ gακgβλ� +⟨EµGαλ⟩⟨EνGβκ⟩ − ⟨∂αϱ∂βϱ∂κϱ⟩⟨EµEν∂λϱ⟩ +� ++ gακgβλ⟨∂αϱ∂βϱ∂κϱ⟩ +� +∂µφ⟨Eν∂λϱ⟩ + ∂νφ⟨Eµ∂λϱ⟩ +�� +(A25) +31 + +We focus on the last line of Eq. (A25). Eq. (A18) and Eq. (A20) yield ⟨∂αϱ∂βϱ∂κϱ⟩ = +⟨∂βϱGακ⟩ = ∂βφgακ −⟨EβGακ⟩. And the terms ⟨∂µϱEν⟩ = −gµν as we infer from Eq. (A24). +Then, we expand Eq. (A25) to get +�Rµν = 1 +4gακgβλ� +⟨EµGαβ⟩⟨EνGκλ⟩ − ⟨Eµ⟩⟨Eν⟩gαβgκλ − ⟨∂αϱ∂βϱ∂κϱ⟩⟨EµEν∂λϱ⟩ +� +(A26) +The last term in Eq. (A26) is negligable as ⟨EµEν∂λϱ⟩ ∼ −⟨∂λ(EµEν)⟩ = −2⟨∂λ[E(µEν)]⟩, +and Eq. (A19) says that ∂µEν = 0. Therefore, Eq. (A26) becomes +�Rµν = 1 +4gακgβλ� +⟨EµGαβ⟩⟨EνGκλ⟩ − ⟨Eµ⟩⟨Eν⟩gαβgκλ +� +(A27) +As we defined the stochastic variables Xµ ≡ Xµ(⟨xν⟩, σxν) in the beginning of subsection +(3.5), the same can be done for the stochastic metric Gµν(ξ, X) as we expand it around the +⟨xµ⟩ while we keep σxµ as it is. So, X ≡ Xµ(⟨x⟩), And the metric becomes +Gµν(X) = Gµν(⟨x⟩) + ˙Gµν(⟨x⟩) +� +X − ⟨x⟩ +� ++ 1 +2 +¨Gµν(⟨x⟩) +� +X − ⟨x⟩ +�2 ++ · · · +(A28) +where +˙Gµν(⟨x⟩) = lim +X→⟨x⟩ +∂ +∂XGµν(X) +(A29) +and ¨Gµν(⟨x⟩) is the usual second derivative of the above equation. +Defining Gµν as a +function in ⟨x⟩ allows us get ⟨Gµν(⟨x⟩)⟩ = Gµν(⟨x⟩) = Gµν as averaging the average is a +redundant process. Now we substitute Eq. (A28) in Eq. (A27), together with the help of +Eq. 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D 37 (1988) 888. +40 + diff --git a/ntFPT4oBgHgl3EQfKDRL/content/tmp_files/load_file.txt b/ntFPT4oBgHgl3EQfKDRL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..df2d7001e99914b3b2aabe49c9d8ca01a119fbac --- /dev/null +++ b/ntFPT4oBgHgl3EQfKDRL/content/tmp_files/load_file.txt @@ -0,0 +1,1553 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf,len=1552 +page_content='Einstein’s Equations and the Entropy of pseudo-Riemannian Information Manifolds Hassan Alshal � Department of Physics, Santa Clara University, 500 El Camino Real, Santa Clara, CA 95053, United States Abstract Motivated by the corrected form of the entropy-area law, and with the help of von Neumann entropy of quantum matter, we construct an emergent spacetime by the virtue of the geometric language of statistical information manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' We discuss the link between Wald–Jacobson approaches of thermodynamic/gravity correspondence and Fisher pseudo–Riemannian metric of information manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' We derive in detail Einstein’s field equations in statistical information geometric forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This results in finding a quantum origin of a positive cosmological constant that is founded on Fisher metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This cosmological constant resembles those found in Lovelock’s theories in a de Sitter background as a result of using the complex extension of spacetime and the Gaussian exponential families of probability distributions, and we find a time varying dynamical gravitational constant as a function of Fisher metric together with the corresponding Ryu–Takayanagi formula of such system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Consequently, we obtain a dynamical equation for the entropy in information manifold using Liouville– von Neumann equation from the Hamiltonian of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This Hamiltonian is suggested to be non–Hermitian, which corroborates the approaches that relate non– unitary conformal field theories to information manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This provides some insights on resolving “the problem of time”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' �halshal@scu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='edu 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='13017v1 [gr-qc] 30 Jan 2023 Contents 1 Introduction 2 2 Information and Spacetime Thermodynamics 5 3 Entropy and Riemannian Geometry 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='1 Vector space construction .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 12 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='4 Euclidean structure of the space of observables .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='5 Fisher metric and Kullback-Leibler divergence .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 16 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='6 Hessian structure and Einstein tensor .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='7 Obtaining a pseudo-Riemannian information manifold .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 22 4 Entropy of the Information Manifold 23 5 Discussions and Conclusions 25 6 Appendix 28 1 Introduction One of the main challenges in physics is to find a fundamental dynamics between geom- etry/gravity and quantum matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' And many approaches such as string theory, grav- ity/field correspondence, and loop quantum gravity try to tackle this problem in different ways and frameworks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' check [1] for detailed review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' In this paper, we approach the prob- lem from the perspective of information manifolds and entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' In the last few years, information geometry has earned a great interest in fields like machine learning [2] and deep learning in physics [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The information manifold and entropy concepts, particularly the relative entropy, are extremely useful in understanding many physical patterns that include, but not limited to, quantum computers [4], chemistry [5], biological systems [6] and even economy [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The entropy-area law, corrected and generalized by the outside von Neumann entropy [8–10], is introduced to the quantum information geometry in order to check that geometry obeys the second law of thermodynamics and preserves informa- tion [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Our proposal to approach the gravity/quantum problem arises intuitively from looking at the information paradox in black hole physics from the entanglement entropy perspective [12,13], along with the formally established the second law of thermodynam- ics for black holes Noetherian charges [14–16] and the quantum origin of spacetime and Einstein equations [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The outline of this work is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Right after this passage, and within the introduction section, we summarize the reasons behind correlating the holographic principle with relevant entropy, coarse-grained entropy, Kullback-Leibler divergence, and 2 Fisher information metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' We then investigate in section (2) the corrected entropy-area law by von Neumann entropy in order to both satisfy the second law of thermodynamics and preserve information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' We compute a form of corrected entropy-area law in the light of Liouville-von Neumann equation so that we get a dynamical relation between the quantum Hamiltonian and the time variation of both entropy and area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Later in the same section, we briefly review the gravity/thermodynamics correspondence, developed by Jacobson [17] and by Wald [14] since we will use that later in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' We comment on the relation between the quantum origin of spacetime and the dynamical equation of ex- pansion rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' In section (3), we provide a detailed study of the geometric interpretation of coarse-grained entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' We cover the essential structures of the information geometry, and we apply the thermodynamics/geometry correspondence to derive a quantum infor- mation geometry form of Einstein field equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The Fisher information metric will be introduced to measure how far the cumulant probability density functions are away from each others after being varied with respect to the microscopic variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' In other words, the Fisher metric tells us about the quota of information the microscopic variables carry in the statistical manifolds [18,19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Thus, the cumulant probability density functions become good candidates to count on developing the Fisher metric, instead of using moments of distributions, when we study the correlations between variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' For example, think of the mean square error as a second moment of the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' When it is differentiated with respect to the microscopic value, the varied cumulant probability density measures the level of proficiency in the model-data fitting1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This also appears in Kullback–Leibler (KL) divergence that measures the difference between probability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The last con- cept plays a fundamental role in relating Fisher metric to Shannon/von Neumann entropy, which is what we show later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' And that concept can be seen mathematically as a second derivative with respect to the microscopic variables2 acting on the KL divergence, which is nothing but the inverse of the probability distribution of Shannon/von Neumann entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Such structure is very similar to the mathematical definitions of curvatures in Riemannian manifolds, which is also what we introduce in the same section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Thus, the Fisher metric is in fact a metric from which the curvature structure in the information manifold is ob- tained [21] once proved to be endowed with other properties of Riemannian manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This will lead to reformulate Einstein field equations in the corresponding information manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Additionally, a positive cosmological term in the Einstein equations is obtained in the in- formation manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Later, we relate the components of the field equations to commulant functions and get a more detailed informatic description to the gravitational constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' In section (4), we introduce a dynamical equation of entropy in the information manifold using only quantum information geometry without using any classical components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' It is a new combination of von Neumann master equation, the von Neumann entropy, and the black hole entropy formula, and its generalization to statistical manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Then we use the 1In the statistical jargon it is called score or informant of the likelihood function [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 2In the statistical jargon it is called observed information, and is equal to the mean square error of the same second derivative applied on the cumulant probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 3 RT formula for the statistical manifold to describe its corresponding entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' In section (5), we discuss and comment on the findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The so-far achieved developments in entropic information theory have led to concep- tually rich ideas like entanglement entropy [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This line of thoughts can be traced back to the discovered relation between the area and the black hole entropy in the Bekenstein- Hawking [9,23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Then, the holographic principle, developed by ’t Hooft [24] and Susskind [25], suggests finding a correspondence between the 3D volume and the 2D area, which leads to the known gauge/gravity correspondence, or the AdS/CFT as a quantum field theory with local degrees of freedom [26], and the sufficiently described entropy at the microscopic level in contrary to Bekenstein–Hawking law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Thus, we are allowed to ad- dress the entanglement entropy [27, 28], which describes the quantum information load in the quantum states, as a stored information encoded on the geometric features of the space [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' But since different 3D surface geometries are indeed associated with different entanglement entropies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' the area in Bekenstein-Hawking law is suggested to be replaced by another area law for the extremal surfaces,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' known as Ryu-Takayanagi (RT) surfaces in holography models [12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 13],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' which provides a clue that we can rethink of the entropy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' which is built of microscopic variables,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' as the fundamental underpinnings of the spacetime classical geometry upon reintroducing RT surfaces to Wald’s formula for the entropy of black holes [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' But the complicated relations among the microscopic variables, together with the dif- ficulties associated with measuring those variables, are the reasons of why it is always more convenient to express the statistical phenomena corresponding to the microscopic variables in terms of the stochastic variables that are ruled by more fundamental and rela- tively easier-to-measure laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Such reductionism in the description automatically will lead to select some microscopic variables to be relevant and other microscopic variables to be irrelevant [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Relevant variables are all parameterized as functions in time or any other affine parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' We can think of the relation between the microscopic and macroscopic variables like the relation between the speed of gas molecules and the temperature of the whole sample, both can be used to describe different types of thermodynamical energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' These microscopic relevant variables should be averaged, using the relevant density func- tion, such that they define the macroscopic or the stochastic variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' And the averaging process could be done using any density matrix as we discuss in details in subsection (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Additionally, any two different densities, constructed from those relevant variables, are favored over each other according to which of them is capable of defining maximum entropy that corresponds to least amount of data loss due to the unfavorable unavoid- able effects coming from irrelevant information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Such density is known as the canonical coarse-grained density, and its corresponding entropy is known as the relevant entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This is the candidate entropy to describe RT formula of statistical manifolds as discussed in section (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Due to the mathematical difficulties a person might encounter while trying to find the exact states and their corresponding densities, it is suggested to replace the general relevant entropy related to holography with other more specified entropies: the coarse-grained 4 entropy and the relative entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Also, we emphasize that the coarse-grained entropy is “lossy but true” entropy [31] as it depends on the macroscopic variables of the system, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' [32] for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' It is argued that we can coarse-grain any type of quantum entropy, such as entanglement entropy, using observational entropy [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The technique of coarse-graining obtained with help of the projectors acting on Hilbert space has been previously used in information geometry3 [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' It is worth noting that observational entropy is a quantum analogue of the classical Boltzmann entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' More importantly, observational entropy shows the measurement limitations when one tries to get more precise information even if the density state is more precisely known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Thus, the process of coarse-graining is inevitable because even pure states span over more than one macrostate in the phase space due to superposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Yet, observational entropy is bounded from below by von Neumann entropy, and equals to the later if the former satisfies the coarse-graining conditions in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' [33], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' after several consecutive coarse-graining processes, one can end up having a fine-grained entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Generally for finite-dimensional systems, the observational entropy can be expressed as a relative entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' More precisely, observational entropy can take the form of Kullback-Leibler divergence, which is a type of relative entropy, from which we obtain Fisher information metric in statistical information manifolds4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 2 Information and Spacetime Thermodynamics In order to obtain an entropy-area law that respects the second law of thermodynamics and preserves information, we need the full entropy of a black hole to contain both the entropy that represents what is inside the horizon and the entropy of the quantum matter outside the horizon [11,35,36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This means the Bekenstein generalized entropy law [8–10] would take the following form SBH = AH 4Gℏ + Smatter, (1) where SBH is the full entropy of the black hole, AH is the area of black hole horizon and Smatter is the von Neumann entropy of the matter outside the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The constants are Planck constant ℏ and Newton’s gravitational constant G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' On one side, von Neumann entropy for a quantum-mechanical system described by a density matrix ρ is given by Smatter = − Tr(ρ ln ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (2) On the other side, the time-evolution equation of the density matrix ρ is given by Liouville- von Neumann equation [37], dρ dt = 1 iℏ[H, ρ], (3) 3Balian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' [19] did not restrict the physical observables to be the Hilbert space projectors but Safránek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' did [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 4It is argued that semiclassical coarse-graining of holographic states, as realized in tensor networks, results in a flow in spacetimes approaching RT surface [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 5 where H is the Hamiltonian of the considered quantum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Since the trace operator commutes with the differential time operator, we can in principle write a time evolution equation for the von Neumann entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' For that purpose, we use the straightforward mathematical trick dρ dt = d dt(ρ ln ρ) − dρ dt ln ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (4) Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (4) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (3), we get the quantum time evolution equation as follows iℏ d dt(ρ ln ρ) − iℏdρ dt ln ρ = [H, ρ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (5) We take the trace of both sides and use the fact that trace operator commutes with the differential time operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Then, we substitute Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (2) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (5) to get −iℏ d dt Smatter = Tr � iℏdρ dt ln ρ + [H, ρ] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (6) Thus, an equation for the time evolution of von Neumann entropy is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' For a black hole system, the von Neumann entropy satisfies the second law of thermodynamics only through introducing the coarse-grained entropy as we mentioned in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Therefore, we use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (1) to rewrite the time evolution of entropy as follows −iℏdSBH dt + i 1 4G dAH dt = Tr � iℏdρ dt ln ρ + [H, ρ] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (7) The last equation introduces a a relation between black hole full entropy, black hole horizon area and density matrix of quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The equation depends on the fundamental constants ℏ and G that are suggested to connect gravity and quantum matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' As we observe here, we did not make any additional assumptions to get to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' It does follow from the Liouville-von Neumann equation and direct mathematical manipulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' It appears that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (7) is a quantum/semi-classical form of entropy-area law for the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' It introduces a relation between geometry (Area) and quantum matter (Density Ma- trix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' We will reconsider this relation after we connect entropy as a macroscopic quantum quantity to the area as a geometric quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' But before that, it is worth noting that the two concepts are related in general within the non-dissipative systems, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' systems with dSBH/dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This assumption is valid as SBH is the total corrected entropy of the black hole, and the RT extremal surface, or “island”, is realized [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Such extremal surface realization is related to the existence of flat plateau in the corresponding Page curve [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Therefore, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (7) becomes i 1 4G dAH dt = Tr � iℏdρ dt ln ρ + [H, ρ] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (8) 6 Upon solving the previous equation, either the horizon area would have the fine-grained entropic definition, which is expected as the horizon should contains the information of the entangled particles that fall inside it, or the fine-grained quantum part of the black hole entropy would have a geometric meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Before we discuss the second meaning, which is what we do in section (4), we emphasize that all previous equations are derived assuming that Liouvillian mechanics stems from dρ/dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Also, the Hamilton-Jacobi equation shows that H = − ∂A/∂t, where A is the action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Wald [14] noticed that all what one needs to do is to express the entropy as a function of the density of the state then applies Liouvillian mechanics to solve the Hamilton-Jacobi equation in order to generate the ac- tion that will be extremized to get the conserved quantities together with Euler-Lagrange equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This is why we swiftly review the gravity/thermodynamics correspondence developed by Jacobson [17,39] based on Wald approach [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Assuming Rindler frame of references, Jacobson found that the Einstein equations can be obtained from the entropy-horizon area relation together with the laws of thermody- namics [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' According to Unruh’s radiation [40], the radiation temperature detected by a Rindler observer is directly proportional to the uniform acceleration “a” of that observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Applying the equivalence principle to Rindler transformation affirms the local flatness con- dition [41,42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Thus, each point in the spacetime has its own local Rindler horizons, both past and future, with Killing fields in the null directions of the horizons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The relations between heat flux and the black hole hairs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' the mass and the angular momentum 5, are discussed in [14,43–45] through the Hamiltonian formulation of the first law of ther- modynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' In brief, the heat flow of such system could be defined to obey the averaged null energy condition along a time-like geodesic δQ = −a � TµνkµkνtdtδA = 0, (9) where Tµν is the energy momentum tensor, kµ is the null vector, δA = √γdA is the variation in the congruence cross sectional area of the horizon, and √γ is the determinant of the induced metric of the 2D spatial area element of the horizon element dA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The last equation stems originally from Wald’s formula [39, 46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' And for the null geodesic of kµ that generates the horizon, and upon cancelling the higher order contributions, the Raychaudhuri equation gives θ = −tRµνkµkν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (10) where the affine parameter θ is the rate of change of √γ [47], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' θ = 1 √γ d dt (√γ) (11) which also describes the expansion of δA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The last two equations can be can be combined to give θA dθ dA = −Rµνkµkν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (12) 5Electromagnetic energy is better represented separately inside the density matrix ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 7 As in [17], Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (9) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (10) yield the relations δQ δA = aℏ 2πℓ2 P , (13) which demands that the Einstein equations become Gµν = 2πℓ2 P ℏ Tµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (14) where the Gµν is the Einstein tensor and ℓP is the Planck length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This establishes that Einstein equation is indeed an equation of state as Jacobson emphasizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' It is worth mentioning that another approach of understanding gravity as an entropic force is suggested by Verlinde in [48,49] but we will not review it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Rather, one might say that nothing new in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' But Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (12) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (13) entice us to consider the existence of a new relation between dS = δQ/T and δθ at the microscopic level of ρ considering the von Neumann definition of entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Such (S, θ) relation is indeed a dynamical relation, and we will study that relation in the context of information geometry in section (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' But first, we need to study the meaning of the geometric quantities Gµν and θ in the classical spacetime using Fisher Riemannian metric of information manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' In the next section, we adopt the Fisher metric as it guarantees extending the geometric quantities to any quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Moreover, the observational entropy is realized as the Kullback-Leibler divergence, which is another information quantity from which we can obtain the Fisher metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 3 Entropy and Riemannian Geometry With help of Lie group thermodynamics in the framework of the covariant formalization of geometrized thermodynamics [50,51], we can expand the analysis that resulted in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (8) in the speculative direction of the geometric interpretation of the fine-grained quantum part of the black hole entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Balian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' gauge theory of thermodynamics [19], as an extension of Fisher Riemannian metric of information manifold, guarantees extending the geometric interpretation to any quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' We introduce Balian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Riemann metric in the density matrix space as the Hessian of the fine-grained entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This metric would help identifying the information loss, and hence, could resolve the information loss paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The metric is situated as a canonical function in between the space of states and the space of observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This involves Legendre transforms just like those in Liouvillian mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='1 Vector space construction According to Balian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' [19], it is useful to focus only on the space of the density states as the density state is more suitable for information gathering in comparison with observables 8 themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Instead of using the density state as an explicit parameter to describe the density state space metric, we would rather use the fine-grained entropy to define such metric as the density state could be an incomplete description for the information relative to the observables, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' the quantum relation of the average values of any observable ˆO ⟨ ˆO⟩ = Tr � ˆO ˆρ � , (15) does not define a unique ρ as there are different ρ’s that sufficiently define the observable average value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Besides that the fine-grained entropy function has a global maximum, as we infer from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (2) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (1), the entropy makes the densities satisfying Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (15) equivalent, and hence, the loss in the information of the observables, due to the different ρ’s, is irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' But the density corresponding to maximum entropy ρ0 is more favorable to describe the density space metric as, by construction, it has the minimum information to calculate different types of observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Even for the irrelevant information, ρ0 contains the least of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This is why the entropy S[ρ0], known as the relevant entropy, is one assigned to describe the macroscopic thermal phenomena such as the thermodynamically gravitational quantities in the dissipative systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' But for non-dissipative systems we can use the usual S[ρ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' To elaborate the above discussion, we focus in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (15) on the ˆρ ˆO components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' It is natural to choose the density ˆρ ˆ O to be the density proportional to the eigenprojectors of ˆO in the corresponding sub-Hilbert space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Therefore, all the off-diagonal elements in the general ˆρ will be disregarded, and the left information is stored in the ˆρ ˆ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' By defining the ˆρ0 as the density that has all the relevant information of all observables similar to ˆO, it becomes clearer why S[ρ0] > S[ρ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' We construct a geometric interpretation to the previously mentioned thoughts such that the “variation of the relevant entropy represents a transfer of information between the relevant and the irrelevant variables.” and “dissipation appears as a leakage of the relevant information” [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' With the help of the algebraic structure endowed in the observable space, the geometric construction in the language of manifolds6 would help us in explaining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (7-8) in a purely quantum information way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The observable ˆO has eigenvectors |αk⟩ in the space states, and therefore, it can be written as ⟨ ˆO⟩ = � k ⟨αk|ρ|αk⟩Tr � |αk⟩⟨αk|O � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (16) To define the vector space of the observable ˆO, we set the components and the bases to be oµ := ⟨αk|ρ|αk⟩, (17a) f µ := |αk⟩⟨αk|O, (17b) respectively as the components and the bases of the Liouville vector ⃗O = oµf µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 6In that sense, the density matrix ρ should be seen as a pre-probability [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 9 As the set of all observables ˆO is characterized by the ρ, and ρ itself is an operator that can be written in terms of any set of orthonormal bases {|i⟩}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Then, we can rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (15–17) such that we define the density operator as a vector ⃗ρ with components ρµ := ⟨f µ⟩ = Tr [f µρ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (18) Then, when O becomes ρ, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (15) becomes −→ ⟨ˆρ⟩ := ⃗ρ = eµρµ, (19) where ρµ act as the components of the averaged observable ⟨ˆρ⟩ in its Liouville vector representation −→ ⟨ˆρ⟩, and the basis components for such vector are defined as eµ := |i⟩⟨i| that are dual to f µ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' eµ · f ν = δ ν µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' In such representation, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (15) can be seen as a bilinear relation || ⃗O|| = ⟨ ⃗O, ⃗ρ⟩ := oµρµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (20) ΣP ΣQ t0 ri t0 + δt λ(t) f∗ : ΣQ → ΣP P(ρµ 0) Q(ρµ) P ′(ρµ) ρ1 ρ2 ρ3 1 Figure 1: Density manifold and the evolution of entropy surfaces away from reduced state ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 10 To make the above construction more elaborate, let’s say that ˆO commutes with the spatial observable ˆx, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' the ⃗O is a function in the components of ˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Then, the density ρµ would correspond to the averaged components ⟨xµ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' As the density space is made of all the ⟨xµ⟩ including the irrelevant ones, this means it is valid to embed the function ⃗O as a hypersurface in the space spanned by ρµ after being expressed in terms of the dual basis of ⃗ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This surface is called the the surface of reduced states, and it is extremized at ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' At the same time, it is much easier to describe the density in terms of the microscopic variables ξµ and the stochastic variables Xµ, both as functions parameterized by the spacetime observables xν or their averages, and we will do that later in subsection (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' But for now, we just finished setting the stage up for the debut of the density space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Next, we show this space could be promoted to a density manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='2 Density manifold In the density space, ρ is guaranteed to be a function in time such that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (3) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Geometrically this means that ρ(t), as a point in such space, evolves from the point P(ρ0) on the surface ΣP, where S[ρ] = S[ρ0], to another point on the same ΣP with S[ρ] along some trajectory λ [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This is defined as an exponential function just like how we relate, in Riemannian geometry, a vector to the points belonging to the trajectory that the vector is tangent to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Thus expP : t · ρµ → λρµ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (21) As t ∈ [0, 1], then exp(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='ρµ) = λρµ(1) is the final point P(ρ)(t) at the trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Meanwhile exp(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='ρµ) = λρµ(0) is the initial point P(ρ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The ρµ 0 can take any direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Therefore, its ith component ρi along the tangent of the trajectory is given by ρi 0 ≡ dλ(t) dt ���� t=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (22) Moreover, the ith component of ρi 0 is allowed to be in the direction of any irrelevant basis ρµ as long as it is tangent to the trajectory λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This motivates considering the exponential function (21) as a diffeomorphism between the neighborhood of point P, which belongs to the state space, and the vector ρµ 0, which belongs to the tangent space TPΣ at point P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' If we normalize ρi 0∀i, then the orthonormal basis {ρµ 0} of TPΣ provides the isomorphism E : Rn ∼−→ TPΣ, (23) such that E(ξ1, · · · , ξn)|P = ciρi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The previous manifold-related definitions are valid regardless whether the surface is ex- tremized on not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Therefore, there exists charts ψ on manifold M containing all λ’s and TQΣ, for all Q ∈ M, such that ψ := (exp ·E)−1 : M → Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (24) 11 Also, the exponential map of the density manifold acts similarly to how the known the exponential map exp(tV ) = λV (t) works between any Lie group and its Lie algebra, where V ∈ the algebra, λV (t) ∈ the group, and t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Then, there exists an analytic diffeomorphism in a neighborhood U of V = 0 such that, for the coordinates ξµ ∈ Rn defined by the isomorphism E in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (23), we find that exp(ξiV i) ∈ exp(U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This would define the necessary canonical chart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Moreover, the entropy S[ρ] never loses information as long as the initial point P(ρ0) evolves in a Hamiltonian trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' However, when we disregard the irrelevant informa- tion, we do something similar to the shift and lapse such that the entropy S[ρ] = S[ρ0] evolves in time through the extemized and non-extremized surfaces, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' we can practice push-forward, f∗ : ΣQ → ΣP, (or pull-back) transformations between ΣP and ΣQ, together with applying the rules of Lie derivatives in order to relate the surfaces to each others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Now we are ready to introduce a metric on this information manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='3 Manifold metric We notice that the exponent function in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (21) transfers ρµ to a tangent space with same properties but at different point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Since the logarithmic function is the opposite to the exponent, then, in light of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (20), we can safely say that ln ρ does the same to ρ except that the domain here becomes the dual tangent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This implies that the entropy S[ρ] acts as a bilinear map between the density ρ and the information content (− ln ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' As both ρ and (− ln ρ) are unique geometric vectors, the entropy provides us with a linear isomorphic relationship between the tangents and the dual tangent spaces, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' the duality ρ ≡ � (ln ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (25) is legitimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Therefore, there exists a map G on the ρ space such that G : (ln ρ)µ �→ Gµνρν, (26a) G : (ρ)µ �→ Gµν(ln ρ)ν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (26b) The map G is shown to be symmetric, real, and has positive eigenvalues in the Liouville vector representation [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' It is the best function to play the role of metic in the ρ space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Then, we can use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (26) to demonstrate the entropy as the bilinear function S⟨ρ, ln ρ⟩ := −Gµν(ρ(ξ))ρµρν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (27) The ρ space is dense enough–satisfying the topological features of manifolds–such that we can introduce the infinitesimal change dρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Therefore, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (27) can be redefined infinitesi- mally such that the second differential in the entropy becomes the metric itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' And the ρ space becomes eligible for a promotion to be a Riemannian manifold endowed with the invariant distance −ds2 := d2S⟨ρ, ln ρ⟩ = −Gµν(ρ(ξ))dρµdρν, (28) 12 where the map (26) is explicitly defined as G(eµ, eν) := Gµν(ρ(ξ)) = − d2S dρµdρν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (29) We now can introduce the Hamiltonian in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (7-8) as a superoperator 7 H such that H := H µ ν eµ ⊗ f ν, (30a) H ν µ = ⟨Heµ, f ν⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (30b) If we set O = H, then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (3), or the commutator in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (15), becomes the Liouville superoperator dρ dt = L ρ = −i[H, ρ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (31) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (30) helps defining the components of L as L ν µ = −iTrf ν[H, eµ] = −iTr[f ν, H]eµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (32) Reintroducing the Liouville operator as a superoperator excavates its “super power” such that it manifestly plays the role of the Lie derivatives on the Riemannian ρ manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The Jacobian of transformations between ΣP and ΣQ is given by J := det �∂ρ(t + δt) ∂ρ(t) � = 1 − iL δt (33) or iL µ ν ρν := lim δt→0 ρµ(Q)(t + δt) − ρµ 0(P)(t) δt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (34) It is obvious that, with the above geometric interpretation, we also can introduce the evolution superoperator U := exp[−iL (δt)], (35) which plays a role similar to that of the Lie algebra, or the Killing fields, over the usual Riemannian manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' We find Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (29) leads us to define the dual vector (− ln ρ) as −(ln ρ) := ϱµf µ, (36a) ϱµ := (− ln ρ)µ ≡ ∂S ∂ρµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (36b) Then, we introduce the Legendre transformation S(ϱµ) = S − ⟨ln(ρ)⟩ = S + ϱµρµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (37) 7The prefix super has no Grassmann rings, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' has nothing to do with Supersymmetry or graded algebra in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 13 This transform reintroduces the ρ components to be ρµ := ∂S ∂ϱµ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (38) Therefore, the metric in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (29) can contravariantized as Gµν(ρ(ξ)) := ∂2S ∂ϱµϱν = ∂ρκ ∂ϱµ ∂ρλ ∂ϱν ∂2S ∂ρκ∂ρλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (39) Consequently, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (28) is transformed into d2S = d2S − � d2ϱµρµ + 2dϱµdρµ + ϱµd2ρµ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (40) The metricity ∇G = 0, or the parallel transport along geodesics ∇ρρ = 0, implies that there exists a set of connection coefficients {} on the ρ manifold similar to the Levi-Civita connections on the Einstein manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This means that both metric and connections are related through G(∇eλeµ, eν) := � κ λµ � Gκν(ρ(ξ)), (41a) � κ λµ � = 1 2Gκι(ρ(ξ)) � ∂µGιλ(ρ(ξ))+∂λGιµ(ρ(ξ)) − ∂ιGµλ(ρ(ξ)) � , (41b) which is enough to introduce geodesic equations, Riemann curvature tensor and related other tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Additionally, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (29) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (39) reveal a Hessian structure on the density manifold such that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (41a) can be rearranged to get the corresponding connection coefficients of the first kind � λµν � = −1 2 ∂3S ∂ρλ∂ρµ∂ρν = 1 2 ∂ Gλµ(ρ(ξ)) ∂ρν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (42) In light of Balian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' metric, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (31) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (36a) yield L ν µ ϱν = GµλL λ ν ρν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (43) In light of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (26), the symmetric property of Balian et al metric, and the covariant form of the Liouville superoperator Lµν = Gµλ(ρ(ξ))L λ ν , we differentiate Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (43) such that 0 = 1 2 ∂ Gµλ(ρ(ξ)) ∂ρκ L λ ν ρν + 1 2 (Lµκ + Lκµ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (44) Then, we substitute Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (42) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (31) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (44) such that −idGµν(ρ(ξ)) dt = Lµν + Lνµ , (45) which is the reason why we said before that U , as defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (35), plays a role similar to that of Lie algebra or the Killing fields over the usual Riemannian manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='4 Euclidean structure of the space of observables The density manifold reveals the Euclidean structure in the example we mentioned in sub- section vector space construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' It means for an observable ⃗W there exists a component8 wi such that the orthogonal projection P defines the components wj, j ̸= i, as ⟨⃗W − P ⃗W, ⃗W⟩ = � � ⃗W − ow �j , wi � = δ j i , (46) where the bilinear form is defined according to the map (26), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' we can infer that P is a superoperator, with Greek indices, such that P ⃗W = P µ ν wµf ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (47) Therefore, we can connect this Euclidean space to the Riemannian density manifold by introducing a vielbein structure9 e µ i ≡ (w) µ i such that the Euclidean flat metric corre- sponding to this structure is defined as gij(ρ(ξ)) := Gµν(ρ(ξ))e µ i e ν j , (48a) gjk(ρ(ξ))gki(ρ(ξ)) = δ j i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (48b) In an information manifold this metric plays the same role the spatial spacetime metric γij does in define the expansion parameter in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' It is worth noting that the vielbein acts on the density vector as a projector operator to yield the components of the density vector in the Liouville space, which is another way to define the axes in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' we could start from the projector operator and the vielbein structure backward until we reach the Liouville vector representation of the density operator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' both approaches therefore are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' And if we get back to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (1) and choose a point P ′(t0) ∈ ΣQ along the curve t0, then Pρ(t0) = ρ0(t0), (49a) ρi := ⟨wi⟩ = ⟨wi, ρ⟩ = ⟨wi, ρ0⟩, ∀i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (49b) Now it is safe to infer that the distance on the surfaces Σ in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (1) are given by ds2 Σ = gij(ρ(ξ))dρidρj = gij(ρ(ξ))dρidρj , (50) where gij(ρ(ξ)) could be not equal to gij(ρ(ξ)) in general, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' dρi ̸= gij(ρ(ξ))dρj necessarily as dρi is more like the component oi of ⃗O as defined previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Last important manifold structure can be obtained by combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (22) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (49) such that ⟨wi,r⟩ = 0 , (51) where the vector r = (ρ − ρi 0) is the tangent along the curve t0, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Thus, we have a vector bundle structure, where the base is the surface ΣP and the fibres are the curves t0 + nδt, n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' See Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' [54] for more about the relation with the blurred space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 8The components are those of the measured w of the observable ⃗W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 9Vielbein structure in information manifold is similar to those of the pseudo-Riemannian manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='5 Fisher metric and Kullback-Leibler divergence The previously constructed density manifold has Euclidean signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' So, we target con- structing an information manifold with a Lorentzian signature diag(−1, +1, +1, · · · ), and we achieve this goal in subsection (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' For now, we focus on relating Balian et al metric to Fisher metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' As the vacuum spacetime is in a “continuous” experience of quantum fluctuations, it can be optimized stochastically such that the expectation values of the operators over spacetime, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' the stochastic variables 10, are invariant under coordinate transformations between different frames of references [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This means we can define the density vectors ρµ, which is a function in the classical variables xµ characterizing the space- time itself, as function in the stochastic variables Xµ ≡ Xµ(⟨xµ⟩, σxµ) that are functions in a 2D space of averages Xµ(⟨xµ⟩) and standard deviations σxµ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Then, we may guess that Balian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' metric Gµν (ρ(ξµ)) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (28) could be expressed as an explicit function of ξµ and Xµ variables, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Gµν ≡ Gµν(ξµ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Xµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' In order to check the validity of this guess, we need to check that the probability density stays the same when coordinate transforma- tions are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The best candidate to test this requirement is the Kullback-Leibler divergence DKL = � ξi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='Xi ρ(ξµ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Xµ) ln ρ(ξµ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Xµ) ρ(ξµ + dξµ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Xµ)∆Xi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (52) As the spacetime variables are infinitesimally changing and ρ → dρ, then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (52) becomes DKL = − � M ρ(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' X) � 1 ρ(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' X) ∂ρ(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' X) dξµ dξµ − 1 2 1 ρ2(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' X) ∂ρ(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' X) dξµ ∂ρ(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' X) dξν dξµdξν� dX, (53) where ρ(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' X) ≡ ρ(ξµ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Xµ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (53) is nothing but the Fisher metric ds2 := � i (dρi)2 ρi , (54) which is the same metric in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (28) but as an explicit function in ξµ and Xµ [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Applying Fisher metric, chain rule, and the completeness relation of the density to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (52) yields DKL = � M 1 2Gµν(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' X)dξµdξνdX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (55) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (52) can be seen as DKL ∼ −d2S = d � ∂S ∂ρµ dρµ � = ∂2S ∂ρµ∂ρν ∂ρµ ∂ξκ ∂ρν ∂ξλdξκdξλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (56) 10For more on the properties of the stochastic variables, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 11See Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' [57] for more on the 2D manifold of averages and standard deviations in the exponential families of normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 16 By comparing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (39), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (55), and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (56), and as the Kullback-Leibler divergence is nothing but a modified Shannon entropy, the covariant form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (39) says that Gµν(ϱ(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' X)) = ∂ϱ ∂ξµ ∂ϱ ∂ξν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (57) As we notice, the density is no longer a vector, it is just a function, and the vectors of the new manifold are ∂µ ≡ ∂/∂ξµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' If we strict expressing all densities as as functions in stochastic variables Xµ rather than the canonical ones ξµ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' ρ ≡ ρ(Xµ), then, with help of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (15–20), we suppress Fisher metric density ξµ-dependence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' from now on the Balian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' metric12 Gµν is not the metric we use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' And Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (57) should be improved as ⟨Gµν(ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' X)⟩ = ⟨ ∂ϱ ∂ξµ ∂ϱ ∂ξν ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (58) In order to understand the last result, we need to get back to the Fisher metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Without loss of generality, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (53) describes the relation between the entropy at the reduced state ρ0 and any other state ρ, see the Points P and Q in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Then, in information manifold the probabilistic average of the stochastic Fisher metric gµν plays the same role the spacetime metric does in the Riemannian manifolds, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' gµν ≃ gµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Thus, gµν := ⟨Gµν(ξ, X)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (59) This is crucial for defining Einstein tensor and the analogue of the gravitational constant in the information manifold as we will see by the end of the next subsection (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='6 Hessian structure and Einstein tensor For an information manifold endowed with exponential family of distribution, there exists a potential function φ such that its Hessian defines the metric of that manifold [59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Non- exponential families do the same but in a less straightforward way13, for more on that see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' [62, 63] and references there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' For simplicity we discuss the Hessian structure of ex- ponential families, but our discussion is applicable to non-exponential ones too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' We share with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' [64] constructing Einstein tensor from the Fisher metric in information mani- fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Also, we find Einstein tensor to be endowed with relevant information to construct the energy-momentum tensor from the varying cumulant partition function defined as a scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This is guaranteed naturally since the entropy data of the system underpin the field strength of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Thus, we can consider Einstein equations in information manifold as the equations of coarse-grained states for the original microscopic system of 12We move from the information manifold of Balian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' [19] to the information manifold of Amari [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 13This is mainly because the Bayesian hypothesis [60] related to the Hessian in non-exponential fam- ilies needs variational Bayesian approximations, like Laplace approximation and parameter separation parameterization, to obtain the posterior distribution [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 17 quantum field theories behind the classical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The major difference between this work and the endeavor of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' [64] is that, without assuming any family of exponentials, we find a positively definite cosmological-like background term in the coarse-grained Einstein equa- tions, particularly in the terms containing the derivatives of the Christoffel connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This suggests redefining Einstein tensor in information manifold to become a Lovelock tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Consequently, coarse-graining the states could reveal extra disguised higher or- dered curvature terms in the theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Additionally, in contrary to Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' [64] choice of real probability distributions that suit AdS/CFT, we impose the family of exponentials to be defined as complex probability distributions such that we construct Einstein tensor in a information manifold with Lorentzian signature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This does not change the result that the gravitational constant is indeed dynamical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' However, it changes the sign of the extremal area and the entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This result is consistent with the fact that complex probabilities are associated with non-Hermitian Hamiltonians and their non-unitary transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' [64] admits the problem of controlling the energy scale of the quantum field theory in the information geometric approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' In order to resolve that, we suggest repeating the process coarse-graining until fine-graining is achieved [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Consequently, the information approach would have a renormalization process, and the von Neumann equation will be fine-grained as suggested in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Thus, our modifications may introduce quantum information geometry approach to dS/CFT [65,66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' As shown in Appendix 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='7 of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' [61], the variational Bayesian inferences limited to exponential families, like the KL divergence, can be generalized through the technique of parameter separation parameterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The technique is applicable to both exponential and non-exponential families of distributions by linearly relating ρ to a real valued potential function14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The potential function φ would help defining the metric of the information manifold as shown in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Without loss of generality, the equality ∂µ∂νφ = gµν in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A20) is very similar to the bilinear relation Hess(φ) = ∇∇φ, or the Hessian, in Riemannian geometry [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' We are allowed to do this comparison because of the bundle structure in the density manifold we referred to at the end of subsection (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Remember that ϱ ∈ T∗ PΣ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' the Hessian acts on the potential function φ to get an element in the sections of tangent bundles Hess(φ) ∈ Γ(T∗Σ ⊗ T∗Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Also, the density manifold and the canonical variable manifold, which could be the conventional spacetime, are related through transforming ϱ into ∂µϱ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Then, we can apply a derivative on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A21) to get ∂λgµν = ∂λ∂µ∂νφ = ⟨∂λ(∂µ∂νϱ)⟩ = ⟨∂λ(∂µϱ∂νϱ)⟩ = −⟨∂λϱ∂µϱ∂νϱ⟩, (60) where the tensor relation between the vectors ∂µϱ is suppressed, the last equality comes from ⟨∂µ(∂νϱ)⟩ = ⟨∂µϱ∂νϱ⟩, and the negative sign in last equality comes from the metric definition in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='(27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' More obviously from the behavior of the derivatives, we have ∂λgµν = ⟨∂λ∂µϱ∂νϱ⟩ + ⟨∂µϱ∂λ∂νϱ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (61) 14See function h(y) defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='14 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' [61] 18 Using the symmetric property of gµν, the last two equations lead us to ∂λgµν = 1 2 � ⟨∂λ∂µϱ∂νϱ⟩ + ⟨∂µϱ∂λ∂νϱ⟩ − ⟨∂λϱ∂µϱ∂νϱ⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (62) In light of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (41b), the last result is very enticing to define the Christoffel connection corresponding to the canonical variable manifold as [57,58] Γλ µν = −1 2gλκ∂κ∂µ∂νφ = gλκ� ⟨∂κϱ∂µ∂νϱ⟩ − 1 2⟨∂κϱ∂µϱ∂νϱ⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (63) Next, we use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (63) to calculate the Ricci tensor and the Ricci scalar as functions in the dual density variables and their derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The Ricci tensor is obtained as usual from the contracted Riemann tensor Rµν = gαβRαµβν = ∂αΓα µν − ∂νΓα µα + Γα βαΓβ µν − Γα βνΓβ µα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (64) Meanwhile the Ricci scalar also is obtained as usual for the contacted Ricci tensor R = gµνRµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (65) Then, the Einstein equation in information manifold becomes Gµν = Rµν − 1 2gµνR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (66) As shown in the Appendix, we deconstruct Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (66) into two main pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' We analyze these pieces using the definitions of the metric and Christoffel connections in information manifolds such that the Hessian structure is realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' So, we can reintroduce Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (66) as Gµν = Λgµν + (˜Rµν − 1 2gµν ˜R) + 1 2gµν ˜R, (67a) or ˜Gµν + Λgµν = Gµν − 1 2gµν ˜R, (67b) where ˜Rµν is explained in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 69, and ˜Gµν + Λgµν plays a rule similar to that of Lovelock tensor Aµν = Gµν + Λgµν in Einstein spacetime manifold [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The first new piece obtained from rearranging then deconstructing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (66), which is Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A9-A9′), gives a cosmological-like term in the information manifold as � ∂αΓα µν − ∂νΓα µα � − 1 2gµνgκλ� ∂αΓα κλ − ∂κΓα λα � = 1 2D(D 2 − 1)gµν ≡ 1 2Λgµν , (68) where Λ = D(D/2 − 1) is defined for D ⩾ 2 dimensional manifold as in Lovelock theory of gravity [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This helps our information manifold pass the necessary condition, but not 19 sufficient on its own, to develop an additional Gauss-Bonnet term for the corresponding Riemann tensor that is expressed in its information form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The appearance of D(D/2 − 1) term in the theory is very tempting to study the effects of having higher-curvature terms in the context of holography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' We leave that to be discussed hopefully in a future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' We notice that the cosmological constant we obtained does not depend on the manifold parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' It only depends on the number of dimensions, and it vanishes when we consider D = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This may point to background symmetry behind the cosmological constant, and it may have a relation with vanishing cosmological constant in the context of some M/string theory [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' More importantly, this piece in the ˜Gµν tensor should exist for all kinds of exponential or non-exponential probability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' It neither demands ρ to be exponential, Gaussian or complex one, nor to be in dS or AdS spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' And this is our major difference between this work and the endeavors in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The second piece after deconstructing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (66) appears written in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A14) in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This piece of non differentiated Christoffels, obtained from both Rµν and gµνR, defines a new tensor �Rµν = � Γα βαΓβ µν − Γα βνΓβ µα � − 1 2gµνgκλ� Γα βαΓβ κλ − Γα βκΓβ λα � = 1 4gακgβλ� ⟨∂αϱ∂βϱ∂κϱ⟩⟨∂µϱ∂νϱ∂λϱ⟩ − ⟨∂νϱ∂βϱ∂κϱ⟩⟨∂µϱ∂αϱ∂λϱ⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (69) Without loss of generality 15, we follow the exponential family example in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' We use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A15) and after so Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (69) becomes �Rµν = 1 4gακgβλ �� GαβGκλ − gαβgκλ � + Gκλ � ˙Gαβ⟨X − ⟨x⟩⟩ + 1 2 ¨Gαβ⟨(X − ⟨x⟩)2⟩ + · · · � + Gαβ � ˙Gκλ⟨X − ⟨x⟩⟩ + 1 2 ¨Gκλ⟨(X − ⟨x⟩)2⟩ + · · · �� × ∂µφ∂νφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (70) Or �Rµν = 1 2GD ∂µφ∂νφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (71) where the dynamical entity GD is defined as GD = 1 2gακgβλ �� GαβGκλ − gαβgκλ � + Gκλ � ˙Gαβ⟨X − ⟨x⟩⟩ + 1 2 ¨Gαβ⟨(X − ⟨x⟩)2⟩ + · · · � + Gαβ � ˙Gκλ⟨X − ⟨x⟩⟩ + 1 2 ¨Gκλ⟨(X − ⟨x⟩)2⟩ + · · · �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (72) 15We do not lose generality because the crucial steps in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A20–A21) are guaranteed by the previously mentioned technique of parameter separation parametrization to work even with non-exponential families [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 20 Using the cumulant partition function φ as a classical scalar field in the information man- ifold, we can define a Lagrangian in D = 4 for an effective field theory as L = gµν∂µφ∂νφ, (73) and the corresponding energy-momentum tensor is Tµν = gµλ ∂L ∂(∂λφ)∂νφ − gµνL = 1 G4 � �Rµν − 1 2gµν�R � = 1 G4 �Gµν , (74) where obviously the last equation is Einstein equation with its reduced tensors �Gµν and �Rµν as functions in the stochastic variables X in such information geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Also, we can add the cosmological constant term we obtained before in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Comparing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (74) with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (14), we get G4 ≃ 2πℓ2 P ℏ , (75) which again is nothing but the gravitational constant in the information form for such 4D geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This indicates quantum information geometry induces gravity phenomena, and the gravitational constant is no longer constant besides its ℏ dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This may support studies that implies inducing gravity from quantum mechanics [70–74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Also, It could be connected with realizing space and time as an approximate macroscopic concepts stem fundamentally from quantum field theories [75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Additionally, a varying gravitational constant may give a clue to Dirac’s large numbers hypothesis [76] that also implies varying gravitational constant is based on the simple analysis of dimensionless constants that are provided by nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The results obtained in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (63–68) are discussed in details in the Appendix of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' There, the family of exponentials is not assumed to be either complex, which is what we discuss in the next subsection, or real as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' [64], since we keep everything in the appendix in terms of an arbitrary ϱ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Additionally in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' [64], after the Gaussian exponential was imposed, the “whole” Einstein tensor is equated to a “negative” cosmo- logical constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This is not what we obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Rather, we say that the Einstein tensor Gµν should be split into ∂Γ terms that correspond to a “positively” cosmological constant defined in terms of the dimension D as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (68), and ΓΓ terms that introduce a modi- fied Ricci tensor ˜Rµν as the kinetic term ∂µφ∂νφ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Notice that the Lagrangian L ∼ gµν ˜Rµν ∼ �φ renders the reduced Einstein tensor ˜Gµν upon applying the variational principle with respect to φ, while the Λgµν piece is produced from varying L with re- spect to gµν = ∂µ∂νφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This reminds us with the Y piece in the modified Lagrangian of gravity in the framework of superstrings [77], both share the same Hessian structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' In brief, the Einstein tensor Gµν in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' [64] should rather be treated as a Lovelock tensor Aµν = Gµν + Λgµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Such tensor should be split into a cosmological term Λgµν and a modi- fied Ricci tensor ˜Rµν, and the later can be used to introduce a new modified Einstein tensor ˜Gµν that has no cosmological terms in it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' All that can be obtained without assuming the densities to be expressed as family of exponentials or any other family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 21 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='7 Obtaining a pseudo-Riemannian information manifold A question remains about how to construct an arbitrary (D + 1)-dimensional information geometry with a Lorentzian signature diag(−1, +1, · · · ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' And without loss of generality, we follow the complex16 Gaussian ansatz in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' [78] to get a (1 + 1)-dimensional information manifold with a Lorentzian signature diag(−1, +1) ρ(t) = 1 √ 2πσ exp � −(t − i⟨t⟩)2 2σ � = exp � − ln( √ 2πσ) − t2 2σ2 + it⟨t⟩ σ2 + ⟨t⟩2 2σ2 � , (76) that requires components of the corresponding Fisher metric to be g00 = � d2ξ 1 ρ(ξ) �∂ρ(ξ) ∂ξ0 �2 = −1, (77a) g11 = � d2ξ 1 ρ(ξ) �∂ρ(ξ) ∂ξ1 �2 = 1, (77b) g01 = � d2ξ 1 ρ(ξ) �∂ρ(ξ) ∂ξ0 � �∂ρ(ξ) ∂ξ1 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (77c) As the Gaussian distribution requires defining a complex exponential family using the following distribution ρ(ξµ) = exp [ξµEµ(Xν) − φ(ξν)] , (78) where the function Eµ(Xν) plays the role of any physical property related to the corre- sponding canonical or intensive variables ξν, and φ(ξν) is the cumulant partition function of the Eµ(Xν) states [58], see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A15) and after in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' By comparing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (76) to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (78), it is easy to notice that Eµ := (E0, E1) = (t, t2), (79a) ξµ := (ξ0, ξ1) = �i⟨t⟩ σ2 , − 1 2σ2 � , (79b) φ(ξν) := 1 2 ln � − π ξ1 � − (ξ0)2 4ξ1 = ln �√ 2πσ � + ⟨t⟩2 2σ2 , (79c) which defines the cumulant distribution as ϱ := − ln(ρ(ξ)) ≡ φ(ξν) − ξµEµ(Xν) = ln( √ 2πσ) + t2 2σ2 − it⟨t⟩ σ2 − ⟨t⟩2 2σ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (80) Notice that Eµ(Xν) is a function in t, which is one of the components xµ of the classical spacetime, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' the stochastic variables Xν can be parameterized generally by the classical 16We comment on the consequences of is ansatz in the discussion section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 22 xµ components of the spacetime as we emphasized before in the beginning of subsection (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='5) when we mentioned the coarse-graining the properties of the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Also, as ξµ variables are functions in the averages and the standard deviations of xν, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' ξµ are functions in statistical variables, we can suppress the variable ξ in the metric such that Gµν (ξ, X) ≡ Gµν (X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Moreover, defining the exponential family of probability distributions to be either complex, like how we have just done, or a real, as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' [64], will not change the definition of the cumulant probability distribution17 ϱ = φ(ξν) − ξµEµ(Xν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Therefore, the results in the previous subsection hold correct in both approaches, where our approach is suitable for dS spaces while Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' [64] fits for AdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Constructing arbitrary D-dimensional pseudo-Riemannian information manifold from a classical spacetime is left as an exercise to the reader18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 4 Entropy of the Information Manifold As we have seen in section (2) for non dissipative systems, the time rate change of the en- tropy is directly related to the content of information expressed in Liouville-von Neumann equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' But in the previous section we found that there is a correspondence between the spacetime and the statistical information manifold, this correspondence makes the space- time metric corresponds to Fisher metric even when we compare the Lorentzian signature of both [78,81,82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' As the spatial entropic area A is part of the spacetime, we can construct an area A in the information manifold that corresponds to the A, and the spatial metric of that information area exists in light of our discussion in subsection (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' More about the area A of such blurred spatial space from quantum entanglement contours can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' [54], or after complexifying the exponential distribution families 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' And if we accept the existence of such correspondence, then we can find a equation for the spatial-like ex- pansion rate in the information manifold θ that corresponds to θ as mentioned before in section (2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' θ = 1 A d dt (A) , (81) where, in comparison to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (11) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (48a), the spatial-like expansion rate in the information manifold θ is defined using the determinant of the averaged spatial components of the Fisher metric γ = det(γij) = det(⟨gij⟩), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' θ = 1 √γ d dt (√γ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (82) Since the exponential family is assumed to be either real or complex, the Hamiltonian is expected to be non-Hermitian with non-unitary transformation, and it is expected to be a 17Notice ρ = exp(−βH)/Z gives φ − ϱ = −βH where the energy H is related to the entropy S [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 18This can be obtained in a similar fashion of how AdS is obtained from complexifying dimensions of CFT, see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (9-12) in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' [79], or see the (u, v) thermal coordinates in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' [80] 19A could be developed for AdS as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' [83].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 23 stochastic Hamiltonian in comparison with regular Hamiltonians in regular phase spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Thus, the Hamiltonian should be modified to obey Lindblad master equation [84], which is the most general form of Liouville equation, and we comment on this in the discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Then for a stochastic Hamiltonian H as a function in X and its conjugate momentum, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (8) in the information manifold becomes i 1 4G4 θA = Tr � iℏdρ dt ln ρ + [H, ρ] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (83) And for dissipative systems, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (7) −iℏdSBH dt = −i 1 4G4 θA + Tr � iℏdρ dt ln ρ + [H, ρ]Lb � , (84) which is the entropy of the black hole in the information manifold with no classical com- ponents from the spacetime itself, just information geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Based on Lindblad master equation, the [H, ρ]Lb term is the time evolution of the density under the influence of the interaction Hamiltonian Hi in an open dissipative system [85] dρ(t) dt ≡ [H, ρ]Lb := 1 iℏ[Hi(t), ρ(0)] − 1 ℏ2 � t 0 dt′[Hi(t), [Hi(t′), ρ(t)]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (85) The question now is: what is the meaning of the area A in the information manifold, and how is it related to the spatial extremal area A in the spacetime?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The answer lies in the definition of extremal surface in dS space, which could be compared with that RT surface in the AdS space [86, 87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' In order to get the extremal surfaces in dS space, we must complexify time such that average speed on the timelike surfaces, defined by the ratio of the shortest spatial angular length l of a dS space to the shortest time ϵ, becomes the determining factor of the size of the extermal surface, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' AdS = −πR2 � l cϵ − 1 � , (86) The last equation shows that for some cases, l/ϵ > 1, the area could be negative20, which would lead to negative or even complex valued entropy!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Before the last comment gets “frown upon”, this could be necessary to avoid the disappearance of the spatial surfaces of dS Rindler wedge [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This might not be well-appreciated as it says there could be non- unitary states in CFT, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='e, the corresponding Hamiltonian H could be non-Hermitian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Very recently, it is proved that the non-Hermiticity stems from the fact that the non-unitary CFT, dual to dS, lives on a space-like surface and the time coordinate emerging from an Euclidean CFT [89,90] related to the previously mentioned blurred space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' When we shift to the language of information manifolds, we surprisingly find some “untimely meditations” about the necessity of complexfying the spacetime and probability distributions so that we 20The area could be imaginary for dS spaces with different dimensions, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' [87].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 24 get a Fisher metric as an averaged metric over spacetime fluctuations with a Lorentzian signature [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Consequently, claiming the necessity of non-Hermitian Hamiltonian corre- sponding to the Einstein-Hilbert formulation of GR in information manifold suggests that we can study the dynamics of a Wheeler-deWitt Hamiltonian [91, 92] described by the spatial metric γ on information manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' An example of the pseudo-Hermitian Wheeler- deWitt Hamiltonian is discussed in details in [93], and detailed calculations of such Hamil- tonian in information manifold could be followed from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' [94–96] but they are left for a future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' One last thing to be said about the RT formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' By comparing the approach21 followed in [97], the RT formula in 4D dS becomes SRT = AdS 4G4 = −πR2 4G4 ln( l cϵ) ∼ −πR2 4G4 � l cϵ − 1 � , (87) which is the formula of the entropy of black hole when the horizon coincides with the RT extremal surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' And to get the expression of the entropy of the extremal surface in information manifold, simply replace A → A and G4 → G4, where G4 is the gravitational constant in 4D spacetime as given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' For more details on the leading divergent term in the previous equation and its relation to the holographic entanglement entropy, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 5 Discussions and Conclusions As we have seen, our study suggests reducing the geometrical properties, including space- time itself, to an information geometry language in a way that could evolve the insight on the deep connection between physics and information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' In this work, we studied in details the coarse grained entropy of the black hole that obeys the second law of ther- modynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' We analyzed the entropy-area law corrected by von Neumann entropy of the quantum matter outside its event horizon in order to obey second law of thermo- dynamics and to preserve information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' We constructed the corresponding form of this corrected entropy-area law in quantum information geometric language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Consequently, a corresponding spacetime emerges from the quantum information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' We discussed the link between Wald-Jacobson approaches of thermodynamic/gravity correspondence and Fisher pseudo-Riemannian metric of information manifold that guarantees extending the geomet- ric interpretation to any quantum theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' We formulated Einstein’s field equations in infor- mation geometry forms, and we obtained a modified Ricci tensor that helped constructing a Lagrangian of such theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Also, we used the modified Ricci tensor to introduce the reduced Einstein tensor ˜Gµν, which is directly related to the energy-momentum tensor in such manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The formulated Einstein’s field equations led into two interesting outcomes stemming fundamentally from information geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The first result is finding a quantum 21See Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' [87] for the analysis of RT formula in 2D information manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 25 origin of a positive cosmological constant that is founded on Fisher metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This cosmolog- ical constant resembles those found in Lovelock’s theories in de Sitter background due to complexifying time and the Gaussian exponential families of probability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The second result is a time varying gravitational constant that resembles the idea of Dirac’s large number hypothesis and predicts varying of gravitational constant based on simple analysis of nature constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' We extended our analysis into the information manifold and write down a dynamical equation for the entropy in quantum information manifold using Liouville-von Neumann equation for the quantum system Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' According to our results, the Hamiltonian in the information manifold could be non-Hermitian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The result- ing dynamical equation provides a clue to a direction that could ameliorate the problem of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' It is worth noting that relating Jacobson endeavors to information geometry requires considering non-equilibrium thermodynamics of spacetime and its associated dissipative gravity [99,100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This relation comes from the coarse-graining process and the suggested quantum thermodynamical origin of spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' To achieve fine-graining, the observa- tional entropy will match von Neumann entropy after several consecutive coarse-graining processes [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' And for finite-dimensional systems like spacetime itself, xµ are obviously finite, the observational entropy can be expressed as relative entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Moreover, a non- local heat fluxes is proven to take place in non-equilibrium thermodynamical gravitational systems, for both Einstein’s general relativity and the scalar-tensor modifications, due to their dissipative characters [100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' When applied to Rindler spacetime, The thermal char- acter of such heat flux extends the vacuum thermal state to include the whole Rindler wedge not just the single observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' So, an accelerated observer would access information, due to entanglement entropy, on spacelike slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Consequently, the restricting the neigh- borhood of the Rindler wedge origin, as spacelike slice around it, determines the expansion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' As the time rate change of the expansion coefficient is related to the dissipative energy coupled to the bulk and shear viscosity, then it relates the entropy, both internal at the irreversible level and exchanged at the reversible level, to the Equivalence Principle in such dissipative systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Moreover, the time rate change of the expansion coefficient states a universal relation between viscosity and internal viscous part of the entropy density as found in the AdS/CFT [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' We know that the collapse of a pure state results in a mixed state, which is a process the unitary transformations cannot achieve in irreversible processes due to the problem of preserving the norm of the wave functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Moreover, there is no such realistic quantum system that could be described by a pure state [101,102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' So, we are left with non-unitary transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' As the quantum time-reversible processes governed by Schrodinger evolu- tion equation are always unitary, then there is an obvious contradiction in assuming those processes to comprise the non-reversible macro physics such as the second law of ther- modynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' There is a long debate on the optimal epistemic and/or ontological way to resolve this contradiction, it is usually discussed under the umbrella of wave function jump and dissipative systems, see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' [103] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' One of the remarkable suggestions is that the microscopic variables indeed evolve according to non-unitary processes [104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 26 So, based on the non-unitary irreversible von Neumann’s measurement transitions that render mixed states [105], Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' [106] provides an account on assuming that wave func- tion collapse takes place spontaneously and randomly in space and time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' We know that von Neumann entropy is a function in basis-independent density operators, meanwhile the reversible Shannon entropy, related to Gibbs and Boltzmann ones, is a function in the probability density matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' If von Neumann and Shannon entropies are related, then there must be some stochastic variables that correspond to coarse-graining Liouville equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Classically this is associated with a loss in information, which is discussed in Balian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' endeavor [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' At the quantum level, this is associated with a loss in phase coherence in quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This is suggested to happen in a non-unitary transforma- tion from a pure state to mixed one [106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This guarantees the validity of the second law of thermodynamics despite having Liouville equation to be governed by phase-space Hamiltonians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' However, this does not demand making change to the Schrodinger equation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' rather it says that Liouville equation is physically broken at the quantum level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The price could be replacing the regular Liouville equation with the more general Lindblad master equation of open quantum systems [84], which allows the corresponding Hamiltonian to have non-Hermitian parts [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' These non-unitary quantum processes are found in many high energy systems, such as CFTs with zero and negative central charges that exhibit entanglement entropy [107,108], in condensed matter systems [109], and even in quantum electronics [110].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' A final comment on complexifying time related to what is mentioned in Isham’s report on the problem of time [111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' In the example, if we focus only on the properties of an open quantum system, then the relevant state becomes that of the reduced density matrix obtained by summing over–in another word by tracing out–the states of the surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' If the states of the surroundings are approximately orthogonal–Balian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' assume that too—then the density of quantum system exhibits decoherence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' This is guaranteed to be true as the reduced density matrix is proved to be governed in different examples of Lindblad master equation [112] such as spatial decoherence [113] and quantum Brownian motion [114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Isham emphasizes that the inability to find a satisfactory unitary Hamil- tonian for Wheeler-de Witt equation should not be considered necessarily as a disaster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Rather, this “might reflect something of genuine physical significance”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' For example, in Hartle and Hawking [115] and Vilenkin [116] endeavors, time becomes complex due to the non-unitary evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Aknowledgement The author would like to thank A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Ali for for discussions and comments during the preparation of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 27 6 Appendix This appendix is dedicated for the detailed calculations of the connection terms and their derivatives in Einstein equation in information manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' If we substitute Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (63) in the derivative terms of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (64) we get ∂αΓα µν − ∂νΓα µα = gαβ∂α � ⟨∂µ∂νϱ∂βϱ⟩ − 1 2⟨∂µϱ∂νϱ∂βϱ⟩ � − gαβ∂µ � ⟨∂α∂νϱ∂βϱ⟩ − 1 2⟨∂αϱ∂νϱ∂βϱ⟩ � + ∂α(gαβ) � ⟨∂µ∂νϱ∂βϱ⟩ − 1 2⟨∂µϱ∂νϱ∂βϱ⟩ � − ∂µ(gαβ) � ⟨∂α∂νϱ∂βϱ⟩ − 1 2⟨∂αϱ∂νϱ∂βϱ⟩ � (A1) Also, if we substitute Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (63) in the multiplicative terms of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (64) we get Γα βαΓβ µν − Γα βνΓβ µα = gακgβλ � � ⟨∂α∂βϱ∂κϱ⟩ − 1 2⟨∂αϱ∂βϱ∂κϱ⟩ � × � ⟨∂µ∂νϱ∂λϱ⟩ − 1 2⟨∂µϱ∂νϱ∂λϱ⟩ � − � ⟨∂ν∂βϱ∂κϱ⟩ − 1 2⟨∂νϱ∂βϱ∂κϱ⟩ � × � ⟨∂µ∂αϱ∂λϱ⟩ − 1 2⟨∂µϱ∂αϱ∂λϱ⟩ � � (A2) We distribute the outer derivative applied on first line in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A1), together with exploiting the properties in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (60), to get gαβ∂α � ⟨∂µ∂νϱ∂βϱ⟩ − 1 2⟨∂µϱ∂νϱ∂βϱ⟩ � − gαβ∂µ � ⟨∂α∂νϱ∂βϱ⟩ − 1 2⟨∂αϱ∂νϱ∂βϱ⟩ � = gαβ � ⟨∂µ∂νϱ∂α∂βϱ⟩ − ⟨∂αϱ∂µ∂νϱ∂βϱ⟩ − 1 2 � ⟨∂µ∂αϱ∂νϱ∂βϱ⟩ + ⟨∂µϱ∂α∂νϱ∂βϱ⟩ + ⟨∂µϱ∂νϱ∂α∂βϱ⟩ � − ⟨∂α∂νϱ∂µ∂βϱ⟩ + ⟨∂µϱ∂α∂νϱ∂βϱ⟩ + 1 2 � ⟨∂α∂µϱ∂νϱ∂βϱ⟩ + ⟨∂αϱ∂µ∂νϱ∂βϱ⟩ + ⟨∂αϱ∂νϱ∂µ∂βϱ⟩ �� = gαβ� ⟨∂µ∂νϱ∂α∂βϱ⟩ + 1 2⟨∂µϱ∂α∂νϱ∂βϱ⟩ + 1 2⟨∂µϱ∂β∂νϱ∂αϱ⟩ − ⟨∂µ∂βϱ∂ν∂αϱ⟩ − 1 2⟨∂αϱ∂µ∂νϱ∂βϱ⟩ − 1 2⟨∂µϱ∂α∂βϱ∂νϱ⟩ � (A3) 28 We contract Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A3) using gµν to get gαβ∂α � ⟨∂κ∂κϱ∂βϱ⟩ − 1 2⟨∂κϱ∂κϱ∂βϱ⟩ � − gαβ∂κ � ⟨∂α∂κϱ∂βϱ⟩ − 1 2⟨∂αϱ∂κϱ∂βϱ⟩ � = gαβ� ⟨∂κ∂κϱ∂α∂βϱ⟩ + 1 2⟨∂κϱ∂α∂κϱ∂βϱ⟩ + 1 2⟨∂κϱ∂β∂κϱ∂αϱ⟩ − ⟨∂κ∂βϱ∂κ∂αϱ⟩ − 1 2⟨∂αϱ∂κ∂κϱ∂βϱ⟩ − 1 2⟨∂κϱ∂α∂βϱ∂κϱ⟩ � (A4) Now, we multiply Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A4) by − 1 2gµν then pick terms from the result of such multiplication that match with the terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A3) to show that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' the derivative parts in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (64), corresponds to the cosmological coupling constant term in Lovelock theory of gravity [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The middle steps are obtained using the definition of the Gµν in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A20) and the definition of gµν in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The gµνgµν = GµνGµν = D, where D is the dimension of the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Collect the fourth term in the last two lines of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A3) with − 1 2gµν×the fourth term in the last two lines of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A4) to get −gαβ⟨∂µ∂αϱ∂ν∂βϱ − 1 2gµν∂κ∂αϱ∂κ∂βϱ⟩ = −(1 − D/2)gµν (A5) Collect the second and the third terms in the last two lines of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A3) with − 1 2gµν×the third term in the last two lines of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A4) to get 1 2gαβ⟨∂α∂νϱ∂β∂µϱ + ∂α∂µϱ∂β∂νϱ − gµν∂κ∂αϱ∂κ∂βϱ⟩ = (1 − D/2)gµν (A6) It is obvious that the last two results, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A5) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A6), cancel each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Collect the last term in the last two lines of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A3) with − 1 2gµν×the last term in the last two lines of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A4) to get −1 2gαβ⟨∂α∂βϱ(∂µ∂νϱ − 1 2gµν∂κ∂κϱ)⟩ = −(D/2 − D2/4)gµν (A7) Collect the last term in the last two lines of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A3) with − 1 2gµν×the last term in the last two lines of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A4) to get 1 2gαβ⟨∂α∂βϱ(∂µ∂νϱ − 1 2gµν∂κ∂κϱ)⟩ = (D/2 − D2/4)gµν (A8) It is obvious that the last two results, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A7) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A8), cancel each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' We are left with the fifth term in the last two lines of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A3) and the − 1 2gµν×fifth term in the last two lines of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' We combine both to get −1 2gαβ⟨∂αϱ∂βϱ(∂µ∂νϱ − 1 2gµν∂κ∂κϱ)⟩ = −(D/2 − D2/4)gµν (A9) 29 Therefore, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A9) is the only part that contributes to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A1) � ∂αΓα µν − ∂νΓα µα � − 1 2gµνgκλ� ∂αΓα κλ − ∂κΓα λα � = 1 2D(D 2 − 1)gµν (A9′) where Λ = D(D/2 − 1) is defined for D ⩾ 2 dimensional manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' For the second line in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A1), we use ∂µ(gαβ) = −gακgβλ∂µgκλ = gακgβλ⟨∂µϱ∂κϱ∂λϱ⟩ (A10) so that the second line in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A1) is gακgβλ � ⟨∂αϱ∂κϱ∂λϱ⟩ � ⟨∂µ∂νϱ∂βϱ⟩ − 1 2⟨∂µϱ∂νϱ∂βϱ⟩ � − ⟨∂µϱ∂κϱ∂λϱ⟩ � ⟨∂α∂νϱ∂βϱ⟩ − 1 2⟨∂αϱ∂νϱ∂βϱ⟩ � � (A11) Next, we expand Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The terms with no 1/4 resulting from such expansion are gακgβλ� ⟨∂α∂βϱ∂κϱ⟩⟨∂µ∂νϱ∂λϱ⟩ − 1 2⟨∂α∂βϱ∂κϱ⟩⟨∂µϱ∂νϱ∂λϱ⟩ − 1 2⟨∂αϱ∂βϱ∂κϱ⟩⟨∂µ∂νϱ∂λϱ⟩ − ⟨∂ν∂βϱ∂κϱ⟩⟨∂µ∂αϱ∂λϱ⟩ + 1 2⟨∂ν∂βϱ∂κϱ⟩⟨∂µϱ∂αϱ∂λϱ⟩ + 1 2⟨∂νϱ∂βϱ∂κϱ⟩⟨∂µ∂αϱ∂λϱ⟩ � (A12) For the first line in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A12), exchange β and λ in the first term, change the sign in that term according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Then, collect that term with the third term in the same line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' And for the second line in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A12), exchange κ and α in the first term, change the sign in that term according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Then, collect that term with the third term in the same line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Also for the second and fifth term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A12), change the sign in those terms according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Thus, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A12) becomes − gακgβλ � ⟨∂αϱ∂κϱ∂λϱ⟩ � ⟨∂µ∂νϱ∂βϱ⟩ − 1 2⟨∂µϱ∂νϱ∂βϱ⟩ � − ⟨∂µϱ∂κϱ∂λϱ⟩ � ⟨∂α∂νϱ∂βϱ⟩ − 1 2⟨∂αϱ∂νϱ∂βϱ⟩ � � + 1 2gακgβλ � ⟨∂αϱ∂βϱ∂κϱ⟩⟨∂µϱ∂νϱ∂λϱ⟩ − ⟨∂νϱ∂βϱ∂κϱ⟩⟨∂µϱ∂αϱ∂λϱ⟩ � (A13) We see that the first line in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A13) cancels with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A11), which is obtained from the second line in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='(A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Moreover, we add the second line in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A13) to the terms with 1/4 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Then, we contract the µν indices of the result of such addition, multiply it with − 1 2gµν, and add it to the original terms before the µν contraction such that we introduce a new tensor �Rµν = � Γα βαΓβ µν − Γα βνΓβ µα � − 1 2gµνgκλ� Γα βαΓβ κλ − Γα βκΓβ λα � = 1 4gακgβλ� ⟨∂αϱ∂βϱ∂κϱ⟩⟨∂µϱ∂νϱ∂λϱ⟩ − ⟨∂νϱ∂βϱ∂κϱ⟩⟨∂µϱ∂αϱ∂λϱ⟩ � (A14) 30 Here we follow Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' In order to relate the Fisher metric with the entropy defined as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (29), or Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (39), we know the density means also the relative share of certain energy state E(Xµ) from the total collection of all energy states in the partition function Z [57], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' ρ(ξµ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Xµ) = exp [−βE(Xµ) − ln Z(ξµ)] (A15) which reintroduces the probability distributions to the family of exponentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Then, we can define the density generally as ρ(ξµ) = exp [ξµEµ(Xν) − φ(ξν)] (A16) which is Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (78).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' The corresponding dual density becomes ϱ = − ln ρ = φ(ξν) − ξµEµ (A17) Applying the first and the second derivative with respect to ξµ on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A17) yields ∂µϱ = ∂µφ − Eµ(Xν) (A18) ∂µ∂νϱ = ∂µ∂µφ (A19) In light of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (29), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (39) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (58), the last Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A18–A19) can be rearranged to get ⟨Gµν⟩ = ⟨∂µ∂νϱ⟩ = ∂µ∂νφ = gµν (A20) Since ⟨∂µϱ⟩ = 0, then we apply another differentiation and use Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A20) to get gµν = ⟨∂µ∂νϱ⟩ = ⟨∂µϱ∂νϱ⟩ (A21) despite that ∂µ∂νϱ ̸= ∂µϱ∂νϱ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Moreover, since ⟨Eµ⟩ = ∂µφ , (A22) then the last three equations give gµν = ⟨EµEν⟩ − ⟨Eµ⟩⟨Eν⟩ (A23) = ⟨EµEν⟩ − ∂µφ⟨Eν⟩ = −⟨∂µϱEν⟩ (A24) The last three relations will help us to construct the Christoffel symbol [57,58], from the connections in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (41b), and consequently the Riemann curvature tensor as functions in the density vectors as we will see in a little bit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Now, we substitute Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (57) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A18) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A14), then expand, we obtain �Rµν = 1 4 � D∂µφ∂νφ − gβκ∂µφ⟨EνGβκ⟩ − gαλ∂νφ⟨EµGαλ⟩ + gακgβλ� ⟨EµGαλ⟩⟨EνGβκ⟩ − ⟨∂αϱ∂βϱ∂κϱ⟩⟨EµEν∂λϱ⟩ � + gακgβλ⟨∂αϱ∂βϱ∂κϱ⟩ � ∂µφ⟨Eν∂λϱ⟩ + ∂νφ⟨Eµ∂λϱ⟩ �� (A25) 31 We focus on the last line of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A18) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A20) yield ⟨∂αϱ∂βϱ∂κϱ⟩ = ⟨∂βϱGακ⟩ = ∂βφgακ −⟨EβGακ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' And the terms ⟨∂µϱEν⟩ = −gµν as we infer from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Then, we expand Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A25) to get �Rµν = 1 4gακgβλ� ⟨EµGαβ⟩⟨EνGκλ⟩ − ⟨Eµ⟩⟨Eν⟩gαβgκλ − ⟨∂αϱ∂βϱ∂κϱ⟩⟨EµEν∂λϱ⟩ � (A26) The last term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A26) is negligable as ⟨EµEν∂λϱ⟩ ∼ −⟨∂λ(EµEν)⟩ = −2⟨∂λ[E(µEν)]⟩, and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A19) says that ∂µEν = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Therefore, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A26) becomes �Rµν = 1 4gακgβλ� ⟨EµGαβ⟩⟨EνGκλ⟩ − ⟨Eµ⟩⟨Eν⟩gαβgκλ � (A27) As we defined the stochastic variables Xµ ≡ Xµ(⟨xν⟩, σxν) in the beginning of subsection (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content='5), the same can be done for the stochastic metric Gµν(ξ, X) as we expand it around the ⟨xµ⟩ while we keep σxµ as it is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' So, X ≡ Xµ(⟨x⟩), And the metric becomes Gµν(X) = Gµν(⟨x⟩) + ˙Gµν(⟨x⟩) � X − ⟨x⟩ � + 1 2 ¨Gµν(⟨x⟩) � X − ⟨x⟩ �2 + · · · (A28) where ˙Gµν(⟨x⟩) = lim X→⟨x⟩ ∂ ∂XGµν(X) (A29) and ¨Gµν(⟨x⟩) is the usual second derivative of the above equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Defining Gµν as a function in ⟨x⟩ allows us get ⟨Gµν(⟨x⟩)⟩ = Gµν(⟨x⟩) = Gµν as averaging the average is a redundant process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Now we substitute Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A28) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A27), together with the help of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' (A18-A22) and the approximation ⟨Eµ(X − ⟨x⟩)n⟩ ∼ ∂µφ⟨(X − ⟨x⟩)n⟩, to obtain �Rµν = 1 4gακgβλ �� GαβGκλ − gαβgκλ � + Gκλ � ˙Gαβ⟨X − ⟨x⟩⟩ + 1 2 ¨Gαβ⟨(X − ⟨x⟩)2⟩ + · · · � + Gαβ � ˙Gκλ⟨X − ⟨x⟩⟩ + 1 2 ¨Gκλ⟨(X − ⟨x⟩)2⟩ + · · · �� × ∂µφ∂νφ (A30) Data Availability Statement No Data associated in the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' References [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' Rovelli, “Strings, loops and others: A Critical survey of the present approaches to quantum gravity,” in 15th International Conference on General Relativity and Gravitation (GR15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' 12, 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ntFPT4oBgHgl3EQfKDRL/content/2301.13017v1.pdf'} +page_content=' arXiv:gr-qc/9803024.' metadata={'source': 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mode 100644 index 0000000000000000000000000000000000000000..e0210bf18cff066164f62efec3c924ae5268e56b --- /dev/null +++ b/sNE4T4oBgHgl3EQfVgwh/content/tmp_files/2301.05024v1.pdf.txt @@ -0,0 +1,1409 @@ +arXiv:2301.05024v1 [physics.chem-ph] 4 Jan 2023 +A theory of inductive loops in electrochemical +impedance spectroscopy +Yury Grabovsky∗ +Jacob Guynee† +January 13, 2023 +Abstract +We demonstrate that failure of time-invariance assumption in the modeling of +electrochemical systems by equivalent circuits can lead to the formation of low fre- +quency “inductive loops” that manifest themselves as positive imaginary parts of the +impedance function. Assuming that the properties of the equivalent circuits change +slowly in time we perform an asymptotic analysis and obtain a new integral represen- +tation of the impedance function that reduces to the standard one at high frequencies, +while exhibiting inductive loops at low frequencies. +1 +Introduction +Electrochemical impedance spectroscopy (EIS) is an indispensable tool to describe complex +electrochemical systems in a unified and graphical way. +In this approach any system is +described by a single complex-valued impedance function Z(ω). The basic theory of EIS +[19, 3, 4] says that this function possesses special analytic properties that are most concisely +expressed by the representation +Z(ω) = +1 +iC0ω + +� ∞ +0 +dσ(τ) +1 + iωτ , +� ∞ +0 +dσ(τ) +1 + τ < +∞, +0 < C0 ≤ ∞. +(1) +Here σ is a positive Borel-regular measure on [0, +∞). It is often convenient to approximate +such measures by linear combinations of delta-functions, resulting in rational approximations +of Z(ω) [1]. Such approximations can be interpreted as impedances of electrical circuits made +of resistors and capacitors only. They are called the equivalent circuit models (ECM). One +easily verifiable feature of the representation (1) is the negativity of the imaginary part of +the impedance function. +The experimentally observed arcs in a Nyquist plot of the impedance with positive imag- +inary part in the low frequency part of the spectrum received the unfortunate moniker of +∗Department of Mathematics, Temple University, Philadelphia, PA 19122, USA. +†Department of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA. +1 + +an “inductive loop” [12, 8, 23, 21, 10]. It is generally agreed that magnetic effects play no +significant role in the functioning of most electrochemical systems and devices, and even +when they do, the inductive effects manifest themselves at the high end of the frequency +spectrum [7, 16, 6]. The explanation of inductive loops in the current literature [16, 6] by +means of ECMs with negative resistances and capacitances creates more problems than it +solves. If one permits negative resistances and capacitances, then arbitrary “experimental +data” can be matched by such an impedance function with any degree of precision due to +the Riesz theorem (see e.g. [17, 14, 15]). This is not surprising, since it is the passivity of +the system [22, 9, 20], violated by the the negative resistances and capacitances in ECMs, +that are responsible for the possibility of stable reconstruction of the impedance function. +A far more reasonable explanation of the experimental appearance of low frequency in- +ductive loops is the violation of the time-invariance assumption at time scales commensurate +with the inverses of low frequencies. This explanation is semi-explicit in [21, 16, 18], where +specific electrochemical processes altering the properties of the system on slow time scales +are identified. +When the time-invariance assumption is violated the input-output behavior of the system +can no longer be mathematically described by a well-defined impedance function at low fre- +quencies. In this paper we will show that if one applies a particular impedance measurement +procedure, delivering the correct impedance for any time-invariant system, the inductive +loops could be observed in non time-invariant systems. +The goal of this paper is to extend the classical EIS/ECM approach to electrochemical +systems with slowly “drifting” properties. In Section 3 we demonstrate that a single Voigt el- +ement with drifting properties may exhibit an inductive loop. In Section 4 we use asymptotic +analysis to extend the EIS/ECM approach to non time-invariant systems with slowly vary- +ing properties. The resulting impedance representation formula, though more complicated, +may still be used to reconstruct the impedance function values within the frequency band +containing experimental measurements by performing the classical Kramers-Kronig analysis +[2, 5, 11, 20] for high frequency data, while fitting the additional “drift” parameters using +the inductive loop low frequency data. +The paper is organized as follows. In Section 2 we discuss a measurement technique that +gives a good approximation of the true impedance function for time-invariant systems. We +then show in Section 3 that the same measurement technique for non time-invariant Voigt +elements can produce “inductive loop” data. +Representing a general non time-invariant +electrochemical system by an ECM with non time-invariant Voigt elements connected in +series, gives us a mathematical model that can be analyzed. An asymptotic analysis applied +to each non time-invariant Voigt element leads us in Section 4 to a generalization of the EIS +theory that reduces to classical formulas at high frequencies and is capable of modeling low +frequency inductive loops. +2 + +2 +Impedance measurements +Let us assume that the input signal is the current +I(t) = I0 sin(ωt), +t ∈ [0, T(ω)], +T(ω) = 2πn(ω) +ω +, +(2) +where n(ω) ∈ N should be as large as possible, so that the time T(ω) it takes to make +a measurement is still acceptable to whoever makes the measurements. Of course, this is +an issue only for low frequencies ω. Thus, we can measure the system response only for +frequencies ω ≥ ωmin = 2π/Tmax, where Tmax is the maximal admissible time for making low +frequency measurements. Hence, in the low frequency regime we always choose n(ω) = 1. At +high frequencies we use a different measurement strategy. We choose a fixed measurement +time T0 and set n(ω) =round(T0ω/2π). Hence, for all ω ≥ ωmin we define +n(ω) = max +� +1, round +�T0ω +2π +�� +, +ω ≥ ωmin. +(3) +The Fourier transform of I(t) can be computed explicitly, but is an unwieldy expression, +except at the frequency ω of the input sinusoid: +�I(ω) = I0 +� T(ω) +0 +sin(ωt)e−iωtdt = πI0n +iω += I0T(ω) +2i +. +(4) +For linear, time-invariant (LTI) systems the output voltage U(t) must satisfy �U(ω) = +Z(ω)�I(ω), where the complex factor Z(ω) is called the impedance. +Theoretically, when +I(t) = 0 for t > T(ω), the output voltage is still non-zero. However, in most cases it decays +exponentially fast and can therefore be neglected. In this case we have the approximation +�U(ω) ≈ +� T(ω) +0 +U(t)e−iωtdt. +(5) +Since U(t) is measured, formula (5) can be used to estimate �U(ω) and thus, we have the +formula for estimating the impedance from the experimental data: +Z0(ω) = +2i +T(ω) +� T(ω) +0 +U(t)e−iωtdt. +(6) +If we apply formula (6) to an elementary Voigt circuit, consisting of a resistor R and a +capacitor C connected in parallel, we will not get the correct answer +ZV (ω) = +R +iθω + 1, +θ = RC. +(7) +Figure 1 shows that Z0 is not a very good approximation for ZV , especially in the intermediate +frequency range. The remedy is to understand this discrepancy and then devise a way to +3 + +0.5 +1 +1.5 +Re(Z)/k +0.2 +0.4 +0.6 +0.8 +1 +-Im(Z)/k +Z0 +ZV +Figure 1: Comparison of Z0(ω) and Z(ω) for an elementary Voigt circuit with R = 2kΩ, +C = 200µF. +correct for it. Using the explicit expression for Z0(ω) (which we don’t display here), we +discover that for an elementary Voigt circuit we have +Z0(ω) = ZV (ω) − +� 1 +TωIm(ZV (ω)) + i +T Z′ +V (ω) +� � +1 − e− T +θ +� +, +(8) +where Z′ +V (ω) denotes the derivative of ZV (ω) with respect to ω. Since relation (8) between +the measured and the true impedance function of an elementary Voigt circuit is linear, it +extends to all LTI systems. +We recall that in an LTI system the output (voltage U(t)) depends on the input (current +I(t)) via +U(t) = ρ0I(t) + +� t +−∞ +I(τ)K(t − τ)dτ, +(9) +where the function K(s) is called a memory kernel. +Theorem 2.1. Suppose that the memory kernel decays exponentially: +|K(s)| ≤ R0 +θ e−s/θ, +s > T0. +(10) +for some T0 < T(ω) for all ω. Then +����Z0(ω) − Z(ω) + 1 +TωIm(Z(ω)) + i +T Z′(ω) +���� ≤ R0 +� +1 + θ +T +� +e−T/θ, +(11) +where +Z(ω) = ρ0 + +� ∞ +0 +K(s)e−iωsds +(12) +is the true impedance of the system. +The proof can be found in Appendix A +4 + +Thus, we are lead to a simple method for estimating the true impedance by making +measurements over a longer time T2 = 2πn2(ω)/ω > T1 = 2πn1(ω)/ω. Then, up to an +exponentially small error we must have +Z(ω) ≈ Zexp(ω) = T2Z0(ω; T2) − T1Z0(ω; T1) +T2 − T1 +. +(13) +We still want to require that at high frequencies the time it takes to make a an impedance +measurement is fixed. Let us call it T ′ +0, since we still want n1(ω) to be given by (3). In that +case +nmin +2 +(ω) = n1(ω) + 1 ≤ n2(ω) ≤ max +� +2, round +�T ′ +0ω +2π +�� += nmax +2 +(ω). +(14) +Thus, for measuring very low frequencies we set n1(ω) = 1 and n2(ω) = 2. When ω = 4π/T0 +we have n1(ω) = 2, in which case we must require that n2(ω) ≥ 3. It follows that T ′ +0 ≥ 3T0/2, +and therefore, nmax +2 +(ω) > n1(ω). In practice, the experimenter can choose any integer n2(ω) +satisfying nmin +2 +(ω) ≤ n2(ω) ≤ nmax +2 +(ω), once the constant T ′ +0 ≥ 3T0/2 has been set. +The correction method (13) can be interpreted as a directive to ignore the transient +response over the time interval [0, T1], effectively taking data during time interval [T1, T2]. +Indeed, using formula (6), we have +Zexp(ω) = +2i +T2(ω) − T1(ω) +� T2(ω) +T1(ω) +U(t)e−iωtdt, +T1,2(ω) = 2πn1,2(ω) +ω +. +(15) +One convenient choice is n1 = n(ω) and n2 = 2n(ω), where n(ω) is given by (3), so that the +length of the time interval over which the data is taken is still T(ω) = 2πn(ω)/ω. +Theorem 2.2. Under assumptions of Theorem 2.1 we have the estimate +|Zexp(ω) − Z(ω)| ≤ +T1 +T2 − T1 +R0 +� +1 + θ +T1 +� +e−T1/θ + R0e−T2/θ. +(16) +The proof can be found is in Appendix A. +To see how formula (15) improves the evaluation of the impedance we take the same +Voigt circuit with R = 2kΩ, C = 200µF, shown in Figure 1, and compare Zexp(ω) and +Z(ω) = ZV (ω). The difference between the Nyquist plots of the two functions can no longer +be visualized as in Figure 1, since it is less that 0.01%, according to numerics, when we +choose n1 = n(ω) and n2 = 2n(ω). +3 +Non time-invariant elementary Voigt circuits +In this section we examine elementary Voigt circuits made of a resistor and a capacitor +connected in parallel. Our main assumption is that the parameters R and C of the circuit +do not stay constant, but slowly change in time, instead. We will show that if we apply the +impedance measurement recipe from the previous section to such non time-invariant systems +we may obtain impedance curves with inductive loops. +5 + +3.1 +General theory +Let us start by considering an elementary Voigt circuit consisting of a resistor R = R(t) and +a capacitor C = C(t) connected in parallel. In such a circuit we have the Ohm’s laws for +each of the elements +IR(t) = U(t) +R(t), +IC(t) = C(t) ˙U(t). +Combining this with the Kirchhoff’s law I(t) = IR(t) + IC(t) we obtain the constitutive +relation (i.e. dependence of I(t) on U(t)) in the form +I(t) = U(t) +R(t) + C(t) ˙U(t). +(17) +Since we use the current as the input we need to solve (17) for U(t): +U(t) = +� t +−∞ +I(s) +C(s) exp +� +− +� t +s +dx +θ(x) +� +ds, +θ = RC. +(18) +The quantity θ is called the relaxation time of the Voigt circuit. Formula (15) gives the +experimentally measured “impedance” of a simple Voigt circuit with non time-invariant +elements: +Zexp(ω) = 2i +∆T +� T2 +T1 +e−iωt +� t +0 +sin(ωs) +C(s) +exp +� +− +� t +s +dx +θ(x) +� +dsdt, +∆T = T2 − T1. +(19) +For future reference we also have the following formula for the impedance of a non time- +invariant capacitor and resistor +ZC +exp(ω) = 2i +∆T +� T2 +T1 +e−iωt +� t +0 +sin(ωs) +C(s) dsdt = +2 +∆Tω +� T2 +T1 +(e−iωt − 1)sin(ωt) +C(t) dt, +(20) +ZR +exp(ω) = 2i +∆T +� T2 +T1 +e−iωtR(t) sin(ωt)dt. +(21) +3.2 +Explicit non time-invariant models +Let us now examine a particular model of the time dependence of R(t) and C(t). We assume +that R and C undergo an “exponential drift” from R− (resp. C−) at t = −∞ to R+ (resp. +C+) at t = +∞: +R(t) = aR− + R+et/τ +a + et/τ +, +C(t) = bC− + C+et/τ +b + et/τ +, +a, b > 0. +(22) +The evolution law for R(t) is governed by two parameters: the time scale τ and the “current +position” a > 0 that tells us how far along R(t) is on the way from R− to R+. It is in 1-1 +6 + +correspondence with R(0). The value a = 1 means R(0) is exactly half-way between R+ and +R−. Exact same comments apply to C(t). It is easy to calculate explicitly +− +� t +s +dx +θ(x) = τ +θ+ +�s − t +τ ++ κC ln +�ρCe−s/τ + 1 +ρCe−t/τ + 1 +� ++ κR ln +�ρRe−s/τ + 1 +ρRe−t/τ + 1 +�� +, +(23) +where +κC = (a − ρC)(b − ρC) +ρC(ρC − ρR) +, +κR = (a − ρR)(b − ρR) +ρR(ρR − ρC) +, +ρR = aR− +R+ +, +ρC = bC− +C+ +, +ρR ̸= ρC. +We note a relation +κC + κR = 1 − θ+ +θ− +, +θ± = C±R±. +(24) +If ρR = ρC = ρ, then +− +� t +s +dx +θ(x) = τ +θ+ +�s − t +τ ++ α ln +�ρe−s/τ + 1 +ρe−t/τ + 1 +� ++ +β(e−s/τ − e−t/τ) +(ρe−s/τ + 1)(ρe−t/τ + 1) +� +, +(25) +where +α = 1 − θ+ +θ− +, +β = (a − ρ)(b − ρ) +ρ +. +A different drift model +R(t) = R0e−t/τ + R∞(1 − e−t/τ), +C(t) = C0e−t/τ + C∞(1 − e−t/τ) +(26) +is a limiting case of our model (22) with R+ = R∞, C+ = C∞, and +a, b → 0, +R−, C− → ∞, +aR− → R0 − R∞, +bC− → C0 − C∞. +In this limit, however, the restriction that R− and C− are positive are no longer required, +since R(t) and C(t) are required to be positive only for t ≥ 0. In this drift model we can +still use (23) with +ρR → R0 +R∞ +− 1, +ρC → C0 +C∞ +− 1, +κR → +ρR +ρR − ρC +, +κC → +ρC +ρC − ρR +, +provided ρR ̸= ρC. If ρR = ρC = ρ, then we use formula (25) with α = 1 and β = ρ. +Once, +� t +s θ(x)−1dx has been evaluated, we compute +U(t) = I0 +C+ +� t +0 +sin(ωs) 1 + be−s/τ +ρCe−s/τ + 1 exp +� +− +� t +s +dx +θ(x) +� +ds. +(27) +The experimenatally measured impedance Zexp(ω) is then computed by means of formula +(15). The numerical evaluation of Zexp(ω) presents several challenges, since parameter τ +must be very large, relaxation times θ± are typically very small and ω can range from +10−6Hz to 106Hz, making some integrands highly oscillatory. These issues are addressed in +Appendix B. Figure 2 shows an inductive loop in the Nyquist impedance plot of a non time- +invariant Voigt element, with R(t) and C(t) given by (22), where R+ = 0.4kΩ, R− = 2kΩ, +a = 10, C+ = 1.6mF, C− = 0.8mF, b = 6, τ = 600s. The figure shows that at high +frequencies the impedance is very well captured by the impedance of the time-invariant +Voigt element with R = R(0), and C = C(0). +7 + +0.5 +1 +1.5 +Re(Z)/k +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +-Im(Z)/k +Inductive loop +Instanteneous impedance +Figure 2: An inductive loop in a non time-invariant Voigt element. +4 +Slow parameter drift asymptotics +Assume that 1/τ = ǫ is a small parameter and that ω = ǫΩ, where the rescaled frequency Ω is +assumed to be fixed. We need to find the asymptotics of Z(ǫΩ) as ǫ → 0 and see if ℑ(Z(ǫΩ)) +can indeed be positive. We now assume that R(t) = R0(ǫt) and C(t) = C0(ǫt), where R0(x) +and C0(x) are strictly positive, bounded smooth functions with bounded first derivatives on +R. Our goal is not only to obtain the asymptotics of Z(ω), defined by (6), as ǫ → 0, but +estimate the error between the actual impedance and its asymptotic approximation. We +obtain for the elementary Voigt circuit, using (18) +Zǫ(ω) = 2i +∆T +� T2 +T1 +e−iωt +� t +0 +sin(ωs) +C0(ǫs) exp +� +− +� t +s +dτ +θ(ǫτ) +� +dsdt, +∆T = T2 − T1. +When the drift time scale τ = 1/ǫ is large, the formula for Zǫ(ω) above can be simplified: +Zǫ(ω) = +iΩ +π∆n(ω) +� 2πn2(ω)/Ω +2πn1(ω)/Ω +e−iΩηIm +� R0(η)eiΩη +1 + iωθ0(η) +� +dη + O(ǫ), +(28) +where the order ǫ error O(ǫ) is uniform over the entire frequency spectrum. +Formula (28) shows that if the functions θ0(η) and R0(η) are constants, then the ap- +proximation becomes exact, as it reduces to the classical impedance of the elementary Voigt +circuit. One more simplification is possible in the asymptotics ǫ → 0. Specifically, we can +replace θ0(η) by θ(0) in (28). Our final asymptotics is +Z∗ +ǫ (ω) = Z(ω, Ω) = +iΩ +π∆n(ω) +� 2πn2(ω)/Ω +2πn1(ω)/Ω +e−iΩηIm +� R0(η)eiΩη +1 + iωθ0(0) +� +dη. +(29) +For numerical evaluation of (29) we choose n1 = n(ω), n2 = 2n(ω) and change variables +of integration η = sn/Ω. Then, using the representation Im(z) = (z − z)/2i we obtain the +formula we use in the Matlab code to make Fig. 3: +Z∗ +ε(ω) = 1 +2π +� 4π +2π +R0 +�sn +Ω +� +1 + iωθ0(0)ds − 1 +2π +� 4π +2π +R0 +� sn +Ω +� +e−2ins +1 − iωθ0(0) ds. +(30) +8 + +0.5 +1 +1.5 +Re(Z)/k +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +-Im(Z)/k +Measured impedance +Asymptotic impedance +0.5 +1 +1.5 +Re(Z)/k +-0.04 +-0.03 +-0.02 +-0.01 +0 +0.01 +-Im(Z)/k +Measured impedance +Asymptotic impedance +Figure 3: The quality of the asymptotics of an inductive loop. +For the model (22) we compute +1 +2π +� 4π +2π +R0 +�sn +Ω +� +ds = R+ + (R+ − R−)Ω +2πn +ln +� +1 + a(e− 4πn +Ω − e− 2πn +Ω ) +1 + ae− 2πn +Ω +� +. +In the high frequency regime +R0 +�sn +Ω +� += R0 +�ǫsT0 +2π +� +We can therefore, use a linear approximation R0(x) ≈ bR +mRx, when x ∈ [ǫT0, 2ǫT0], where +the parameters bR and mR are found from the least squares fit. In that case +1 +2π +� 4π +2π +R0 +�sn +Ω +� +e−2insds ≈ imR +2Ω . +Figure 3 shows the Nyquist plot of Z∗ +ǫ (ω) superimposed on the computed “measured” +impedance corresponding to the model (22). +The difference between the two graphs is +not detectable at the scale of the entire Nyquist plot, shown in the left panel of the figure. +The right panel shows the blown-up portion of the plot containing the inductive loop. The +overall relative error of the asymptotic approximation (29) is about 0.1%. In the figure we +used (22) with the same parameter values as in Figure 2. +Now, if we have a general electrochemical system exhibiting parameter drift, then the +measured impedance can be represented as +Zexp(ω) = +iΩ +π(n2(ω) − n1(ω)) +� 2πn2(ω)/Ω +2πn1(ω)/Ω +e−iΩηIm +� +Z(ω, η)eiΩη� +dη + O(ǫ), +(31) +where for each η ∈ R the function ω �→ Z(ω, η) is a classical (instantaneous) impedance +function, provided there is no leading capacitance in the ECM. In the presence of the leading +capacitance, the term +ZC +exp(ω) = +iΩ +π(n2(ω) − n2(ω)) +� 2πn2(ω)/Ω +2πn1(ω)/Ω +(e−iΩη − 1) sin(Ωη) +iωC0(η)dη +must be added to the right-hand side of (31). +9 + +If we approximate the instantaneous impedance function Z(ω, ǫt) by a finite Voigt circuit +ECM [1] +Z(ω, η) = R0(η) + +N +� +j=1 +Rj(η) +1 + iωθ(η), +then, up to a uniform, over the entire spectrum, order ǫ error, we have +Zexp(ω) = +iΩ +π(n2(ω) − n1(ω)) +� 2πn2(ω)/Ω +2πn1(ω)/Ω +e−iΩηIm +� +Z0(ω, η)eiΩη� +dη + O(ǫ), +(32) +where +Z0(ω, η) = R0(η) + +N +� +j=1 +Rj(η) +1 + iωθ(0). +(33) +If Rj(η) in (33) are modeled according to (26), +Rj(η) = Rj(0) + ∆Rj +� +1 − e−µjη� +, +j = 0, . . . , N, +(34) +then the integral in (32) can be computed explicitly: +Zexp(ω) = +N +� +j=0 +�Rj(0) + ∆Rj +1 + iωθj(0) + ∆RjF +�ǫµj +ω , ωθj(0) +�� ++ O(ǫ), +(35) +where θ0 = 0, and +F(W, v) = ie−2πn1W − e−2πn2W +π(n2 − n1)W +v(W + i) − 1 +(1 + v2)(W + 2i). +Typically, one would choose n1(ω) = n(ω) and n2(ω) = 2n(ω), where n(ω) is given by (3). +At low frequencies n(ω) = 1, while at high frequencies 2πn(ω)/ω can be replaced with a +constant T0. +5 +Conclusions and discussion +In this paper we have proposed an explanation of the inductive loop phenomena observed +in experiments. The hypothesis is that the act of the impedance measurement of an elec- +trochemical system might speed up slow processes, such as corrosion and charge diffusion, +altering the properties of the system on time scales commensurate with the inverses of the fre- +quencies at which the impedance is measured. The modeling and analysis of these processes +can lead to specific drift laws that can be used to validate our general theory. +A natural question is whether our theory makes it apparent how the time-dependent +nature of the system causes the occurrence of inductive loops. By way of the answer we refer +to formula (30) and set ω = 0, n(ω) = 1. Then we obtain +Im(Z(ǫΩ)) ≈ 1 +2π +� 4π +2π +R0 +� s +Ω +� +sin(2s)ds +(36) +10 + +While R0(u) is a strictly positive, smooth real function, the sign of the integral in (36) is +the outcome of balancing positive and negative contributions of the oscillatory integrand. +For example, if R0(u) is a decreasing function, then positive contributions will always be +larger than the negative ones and inductive loops will be present. Conversely, if R0(u) is an +increasing function, no inductive loops will be produced. If the slowly oscillating sinusoidal +current input causes a corresponding in-phase oscillation of R0, the inductive loop effect +could be several times stronger. +One open question, not directly addressed in the paper is capturing the time scale τ +over which the system’s properties change. A crude estimate would be the inverse frequency +at which the imaginary part of the impedance hits zero. For example, in the simulation +corresponding to Figure 2 the time scale τ = 600s, while 2π/ω0 = 610s, where Im(Z(ω0)) = +0. Our simulation of the time-dependent elementary Voigt circuit shows that the deviations +from the classical model start at frequencies ω as large as two orders of magnitude over +τ. They become vividly pronounced at frequencies on the order of 1/τ and persist over +frequencies an order of magnitude lower. +Finally, whether or not the inductive loop data can give additional information about the +system beyond the time scale τ depends on whether the drift model (34) is acceptable. In +this case one should be able to use the algorithm in [13] applied to the high frequency data +to compute parameters Rj(0), θj(0) and N. The intermediate and low frequency data can +then be used in a non-linear least squares fit to to estimate drift parameters µj and ∆Rj. +Future research into the inductive loops should address these questions. +Acknowledgments. This material is based upon work supported by the National Sci- +ence Foundation under Grant No. DMS-2005538. +A +Mathematical proofs +A.1 +Proof of Theorem 2.1 and 2.2 +For I(t) = χ[0,T(ω)](t) sin(ωt) we have, making a change of variables s = t − τ, +U(t) = ρ0 sin(ωt) + +� t +0 +K(s) sin(ω(t − s))ds, +0 ≤ t ≤ T(ω). +We can then write +U(t) = ρ0 sin(ωt) + +� ∞ +0 +χ(s,+∞)(t)K(s) sin(ω(t − s))ds, +so that we can substitute this into (6) and switch the order of integration. We obtain +Z0(ω) = ρ0 + +2i +T(ω) +� T(ω) +0 +K(s) +�� T(ω) +s +sin(ω(t − s))e−iωtdt +� +ds. +Computing the inner integral we obtain +Z0(ω) = ρ0 + +� T(ω) +0 +K(s) +� +e−iωs − ωse−iωs − sin(ωs) +2πn +� +ds. +11 + +Using formula (12) for the true impedance, we compute +Z(ω) − +1 +2πn(ω)Im(Z(ω)) − +iω +2πn(ω)Z′(ω) = ρ0 + +� ∞ +0 +K(s) +� +e−iωs + sin(ωs) − ωse−iωs +2πn +� +ds. +Therefore +Z(ω)− Im(Z(ω)) +ωT(ω) +− +i +T(ω)Z′(ω)−Z0(ω) = +� ∞ +T(ω) +K(s) +� +e−iωs + sin(ωs) − ωse−iωs +2πn +� +ds. (37) +For s ≥ T(ω) = 2πn/ω we have ωs ≥ 2πn. Denoting x = ωs and A = 1/(2πn), we estimate +|e−ix + A(sin x − xe−ix)| ≤ |1 − Ax| + A = Ax − 1 + A ≤ Ax, +since Ax ≥ 1 and A ≤ 1/(2π) < 1. Hence, +����Z0(ω) − Z(ω) + Im(Z(ω)) + iωZ′(ω) +2πn(ω) +���� ≤ +1 +T(ω) +� ∞ +T(ω) +s|K(s)|ds. +Using the exponential decay (10) of the memory kernel we obtain the estimate +����Z0(ω) − Z(ω) + Im(Z(ω)) + iωZ′(ω) +2πn(ω) +���� ≤ R0 +� +1 + +θ +T(ω) +� +e− T (ω) +θ , +(38) +proving (11). +Let us now prove Theorem 2.2. We compute, +Zexp(ω) − Z(ω) = +T1 +T2 − T1 +� T2 +T1 +K(s) +� +e−iωs + sin(ωs) − ωse−iωs +2πn1 +� +ds − +� ∞ +T2 +K(s)e−iωsds. +Uisng estimate (38) we obtain +|Zexp(ω) − Z(ω)| ≤ +T1(ω) +T2(ω) − T1(ω)R0 +� +1 + +θ +T1(ω) +� +e− T1(ω) +θ ++ R0e− T2(ω) +θ +. +A.2 +Proof of the asymptotic foormula (31) +Let us analyze the asymptotic behavior of Zǫ(ω). We first change variables in the innermost +integral σ = ǫτ: +Zǫ(ω) = 2i +∆T +� T2 +T1 +e−iωt +� t +0 +sin(ωs) +C0(ǫs) exp +� +−1 +ǫ +� ǫt +ǫs +dσ +θ(σ) +� +dsdt. +Next we change variables ξ = ǫs in the integral with respect to the s variable: +Zǫ(ω) = +2i +∆Tǫ +� T2 +T1 +e−iωt +� tǫ +0 +sin(ωξ/ǫ) +C0(ξ) +exp +� +−1 +ǫ +� ǫt +ξ +dσ +θ(σ) +� +dξdt. +12 + +Finally, we change variables in the outermost integral η = ǫt: +Zǫ(ω) = +2i +∆Tǫ2 +� T2ǫ +T1ǫ +e−iΩη +� η +0 +sin(Ωξ) +C0(ξ) exp +� +−1 +ǫ +� η +ξ +dσ +θ(σ) +� +dξdη, +where Ω = ω/ǫ could be large, when ω is not very small, but could also be of order 1, when +ω is of order ǫ. +The main approximation idea is to realize that +Eǫ(ξ, η) = exp +� +−1 +ǫ +� η +ξ +dσ +θ(σ) +� +is exponentially small when η − ξ is not very small. However, when η ≈ ξ, then θ(σ) ≈ θ(η), +which means that +Eǫ(ξ, η) ≈ E0 +ǫ (ξ, η) = exp +� +−η − ξ +θ(η)ǫ +� +. +To make this approximation quantitative we use the inequality +ex2 − ex1 +x2 − x1 +< emax{x1,x2}, +which is a consequence of convexity of the exponential function. Hence, +|Eǫ(ξ, η) − E0 +ǫ (ξ, η)| ≤ Lθ−1(η − ξ)2 +2ǫ +exp +� +−η − ξ +Mθǫ +� += ǫLB +�η − ξ +ǫ +� +, +where +Mθ = max +t∈R θ(t), +Lθ−1 = max +t∈R +���� +� 1 +θ(t) +�′���� , +B(x) = x2 +2 exp +� +− x +Mθ +� +. +Thus, we can replace Zǫ(ω) with its approximation +Z(1) +ǫ (ω) = +2i +∆Tǫ2 +� T2ǫ +T1ǫ +e−iΩη +� η +0 +sin(Ωξ) +C0(ξ) E0 +ǫ (ξ, η)dξdη. +Moreover, we also have +|Zǫ(ω) − Z(1) +ǫ (ω)| ≤ +2Lθ−1 +mC∆Tǫ +� T2ǫ +T1ǫ +� η +0 +B +�η − ξ +ǫ +� +dξdη, +where +mC = min +t∈R C(t). +Changing variables in the inner integral x = (η − ξ)/ǫ we obtain +|Zǫ(ω) − Z(1) +ǫ (ω)| ≤ 2Lθ−1ǫ +mC +� ∞ +0 +B(x)dx = ǫM3 +θ Lθ−1 +mC +. +13 + +The function E0 +ǫ (ξ, η) is exponentially small when ξ is not very close to η. However, when ξ +is very close to η we can replace C0(ξ) with C0(η). Hence we have +Z(1) +ǫ (ω) ≈ Z(2) +ǫ (ω) = +2i +∆Tǫ2 +� T2ǫ +T1ǫ +e−iΩη +C0(η) +� η +0 +sin(Ωξ)E0 +ǫ (ξ, η)dξdη. +Moreover, +|Z(1) +ǫ (ω) − Z(2) +ǫ (ω)| ≤ 2LC−1 +∆Tǫ2 +� T2ǫ +T1ǫ +� η +0 +(η − ξ) exp +� +−η − ξ +ǫMθ +� +dξdη, +where +LC−1 = max +t∈R +���� +� 1 +C(t) +�′���� . +Changing variables x = (η − ξ)/ǫ in the inner integral we obtain the bound +|Z(1) +ǫ (ω) − Z(2) +ǫ (ω)| ≤ 2LC−1 +∆T +� T2ǫ +T1ǫ +� ∞ +0 +x exp +� +− x +Mθ +� +dxdη = 2LC−1M2 +θ ǫ. +Now, the inner integral in Z(2) +ǫ (ω) can be evaluated explicitly: +� η +0 +sin(Ωξ) exp +� +−η − ξ +θ(η)ǫ +� +dξ = ǫθ sin(Ωη) − ǫ2θ2Ω cos(Ωη) + ǫ2θ2Ωe−η/(ǫθ) +1 + ǫ2θ2Ω2 +. +We observe that the first two terms can be combined nicely, and +Z(2) +ǫ (ω) = +2i +∆Tǫ +� T2ǫ +T1ǫ +e−iΩηIm +� R0(η)eiΩη +1 + iωθ0(η) +� +dη + ∆ǫ(ω) = Z(3) +ǫ (ω) + ∆ǫ(ω), +where +∆ǫ(ω) = 2iΩ +∆T +� T2ǫ +T1ǫ +e−iΩη +θ2e−η/(ǫθ) +C0(η)(1 + ǫ2θ2Ω2)dη. +We estimate +|∆ǫ(ω)| ≤ 2Ωǫ +max +η∈[T1ǫ,T2ǫ] +θ(η)2e−η/ǫθ(η) +C0(η)(1 + ǫ2θ(η)2Ω2) ≤ MRe−T1/Mθ. +Z(3) +ǫ (ω) can be written as a two-scale impedance function (28). +In order to prove a simplified formula (29) we estimate +|Z(3) +ǫ (ω) − Z∗ +ǫ (ω)| ≤ +Ω +π∆n(ω) +MRLθ +1 + m2 +θω2 +� 2πn2(ω)/Ω +2πn1(ω)/Ω +ωηdη = 2πǫ(n1 + n2)MRLθ +1 + m2 +θω2 +. +If we choose n1 = n and n2 = 2n, then in the regime n(ω) = 1 we get the bound +|Z(3) +ǫ (ω) − Z∗ +ǫ (ω)| ≤ 6πǫMRLθ. +14 + +If n(ω) > 1, then n(ω) = T0ω/2π, and we obtain +|Z(3) +ǫ (ω) − Z∗ +ǫ (ω)| ≤ 3ǫT0ωMRLθ +1 + m2 +θω2 +≤ 3ǫT0MRLθ +2mθ +. +Hence, we obtain a uniform in the entire spectrum bound +|Zǫ(ω) − Z∗ +ǫ (ω)| ≤ Mǫ +for some constant M that depends on the parameter functions R0(s) and C0(s), and that +scales like Rθ. +B +Numerical evaluation of the impedance in explicit +drift models +The key to the effective numerical computation of Zexp(ω) for explicit drift models (22) +and (26) is a non-dimensionalization of all quantities. We therefore begin by rescaling the +variable of integration in the definition (19) of Z(ω), t = nˆt/ω. +Z(ω) = +i +I0π +� 4π +2π +U +�nˆt +ω +� +e−inˆtdˆt, +where we have chosen n1 = n(ω), n2 = 2n(ω), and where n(ω) is given by (3). We then +observe that U(t) has the form +U(t) = I0 +C+ +� t +0 +sin(ωs)f +�s +τ +� +e +τ +θ+[g( s +τ )−g( t +τ )]ds, +(39) +where +f(u) = be−u + 1 +ρCe−u + 1, +g(u) = u + κR ln(ρRe−u + 1) + κC ln(ρCe−u + 1), +provided ρC ̸= ρR. If ρC = ρR = ρ, then we use +g(u) = u + α ln(ρe−u + 1) − +β +ρ(ρe−u + 1). +We therefore, rescale the variable of integration s = nˆs/ω in (39): +U +�nˆt +ω +� += I0n +C+ω +� ˆt +0 +sin(nˆs)f +� nˆs +ωτ +� +e +τ +θ+[g( nˆs +ωτ )−g( nˆt +ωτ )]dˆs +Hence, we obtain the formula we use in our Matlab code +Z(ω) = +in +πC+ω +� 4π +2π +e−inˆt +�� ˆt +0 +sin(nˆs)f +� nˆs +ωτ +� +e +τ +θ+ [g( nˆs +ωτ )−g( nˆt +ωτ )]dˆs +� +dˆt. +(40) +15 + +In the frequency band where n(ω)/ω ≈ T0, we approximate, assuming that τ is large, +f +� nˆs +ωτ +� +≈ bf + mf +nˆs +ωτ , +g +� nˆs +ωτ +� +≈ bg + mg +nˆs +ωτ , +where the slopes mf and mg, and intercepts bf and bg are found from linear least squares fits +of f(x) and g(x), x ∈ [T0/τ, 2T0/τ]. Then, all integrals in (40) can be computed explicitly. +Up to exponentially small terms +Z(ω) ≈ +bfR+ +mg + iωθ+ ++ +1 +2ωτ +mfR+ +mg + iωθ+ +� +3T0ω + 3i − +4im2 +g +m2g + ω2θ2 ++ +� +, +n(ω) > 1, +where in the final expression we replaced n(ω) with T0ω/(2π), which is valid exactly in the +high frequency regime. +References +[1] P. Agarwal, M. E. Orazem, and L. H. Garcia-Rubio. Measurement models for elec- +trochemical impedance spectroscopy: I. demonstration of applicability. Journal of the +Electrochemical Society, 139(7):1917, 1992. +[2] P. Agarwal, M. E. Orazem, and L. H. Garcia-Rubio. Application of measurement models +to impedance spectroscopy: III. Evaluation of consistency with the Kramers-Kronig +relations. Journal of the Electrochemical Society, 142(12):4159, 1995. +[3] A. Bard and L. Faulkner. Electrochemical Methods; Fundamentals and Applications. +Wiley Interscience, Hoboken, NJ, 2nd edition, 2000. +[4] E. Barsoukov and J. R. Macdonald, editors. Impedance spectroscopy: theory, experiment, +and applications. John Wiley & Sons Inc., 2nd edition, 2005. +[5] B. A. Boukamp. A linear Kronig-Kramers transform test for immittance data validation. +Journal of the electrochemical society, 142(6):1885, 1995. +[6] B. A. Boukamp. Distribution (function) of relaxation times, successor to complex non- +linear least squares analysis of electrochemical impedance spectroscopy? +Journal of +Physics: Energy, 2(4):042001, 2020. +[7] H. Brandst¨atter, I. Hanzu, and M. Wilkening. Myth and reality about the origin of +inductive loops in impedance spectra of lithium-ion electrodes—a critical experimental +approach. Electrochimica acta, 207:218–223, 2016. +[8] H.-M. Cho, Y. J. Park, and H.-C. Shin. Semiempirical analysis of time-dependent ele- +mentary polarizations in electrochemical cells. Journal of The Electrochemical Society, +157(1):A8–A18, 2009. +16 + +[9] C. Coelho, J. Phillips, and L. Silveira. Passive constrained rational approximation algo- +rithm using nevanlinna-pick interpolation. In Proceedings of the conference on Design, +automation and test in Europe, page 923. IEEE Computer Society, 2002. +[10] S. Cruz-Manzo and P. Greenwood. Low frequency inductive loop in eis measurements of +an open-cathode polymer electrolyte fuel cell stack. impedance of water vapour diffusion +in the cathode catalyst layer. Journal of Electroanalytical Chemistry, 900:115733, 2021. +[11] A. Dienstfrey and L. Greengard. Analytic continuation, singular-value expansions, and +Kramers-Kronig analysis. Inverse Problems, 17(5):1307, 2001. +[12] J. S. Gnanaraj, R. W. Thompson, S. N. Iaconatti, J. F. DiCarlo, and K. M. Abraham. +Formation and growth of surface films on graphitic anode materials for li-ion batteries. +Electrochemical and solid-state letters, 8(2):A128–A132, 2005. +[13] Y. Grabovsky. +Reconstructing Stieltjes functions from their approximate values: a +search for a needle in a haystack. SIAM J. Appl. Math., 82(4), 2022. +[14] Y. Grabovsky and N. Hovsepyan. On feasibility of extrapolation of the complex electro- +magnetic permittivity function using Kramers-Kronig relations. SIAM J. Math Anal., +53(6):6993–7023, 2021. +[15] Y. Grabovsky and N. Hovsepyan. Optimal error estimates for analytic continuation in +the upper half-plane. Comm Pure Appl Math, 2021. to appear. +[16] D. Klotz. Negative capacitance or inductive loop?–a general assessment of a common +low frequency impedance feature. Electrochemistry Communications, 98:58–62, 2019. +[17] J. R. Partington et al. Interpolation, identification, and sampling. Number 17 in London +Mathematical Society monographs (new series). Oxford University Press, 1997. +[18] A. Schiefer, M. Heinzmann, and A. Weber. Inductive low-frequency processes in pemfc- +impedance spectra. Fuel Cells, 20(4):499–506, 2020. +[19] J. Scully, D. Silverman, and M. Kendig, editors. Electrochemical Impedance: Analysis +and Interpretation. ASTM, 1993. +[20] A. Srivastava. Causality and passivity: From electromagnetism and network theory to +metamaterials. Mechanics of Materials, 154:103710, 2021. +[21] S. Taibl, G. Fafilek, and J. Fleig. Impedance spectra of fe-doped srtio 3 thin films upon +bias voltage: inductive loops as a trace of ion motion. Nanoscale, 8(29):13954–13966, +2016. +[22] M. Wohlers and E. Beltrami. Distribution theory as the basis of generalized passive- +network analysis. IEEE Transactions on Circuit Theory, 12(2):164–170, 1965. +17 + +[23] Q.-C. Zhuang, X.-Y. Qiu, S.-D. Xu, Y.-H. Qiang, and S. Su. +Diagnosis of electro- +chemical impedance spectroscopy in lithium-ion batteries. Lithium Ion Batteries—New +Developments, 8:189–227, 2012. +18 + diff --git a/sNE4T4oBgHgl3EQfVgwh/content/tmp_files/load_file.txt b/sNE4T4oBgHgl3EQfVgwh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cc0e5dd056b6dcbee6427853f3d3f42c4c456347 --- /dev/null +++ b/sNE4T4oBgHgl3EQfVgwh/content/tmp_files/load_file.txt @@ -0,0 +1,410 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf,len=409 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='05024v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='chem-ph] 4 Jan 2023 A theory of inductive loops in electrochemical impedance spectroscopy Yury Grabovsky∗ Jacob Guynee† January 13, 2023 Abstract We demonstrate that failure of time-invariance assumption in the modeling of electrochemical systems by equivalent circuits can lead to the formation of low fre- quency “inductive loops” that manifest themselves as positive imaginary parts of the impedance function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Assuming that the properties of the equivalent circuits change slowly in time we perform an asymptotic analysis and obtain a new integral represen- tation of the impedance function that reduces to the standard one at high frequencies, while exhibiting inductive loops at low frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' 1 Introduction Electrochemical impedance spectroscopy (EIS) is an indispensable tool to describe complex electrochemical systems in a unified and graphical way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' In this approach any system is described by a single complex-valued impedance function Z(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' The basic theory of EIS [19, 3, 4] says that this function possesses special analytic properties that are most concisely expressed by the representation Z(ω) = 1 iC0ω + � ∞ 0 dσ(τ) 1 + iωτ , � ∞ 0 dσ(τ) 1 + τ < +∞, 0 < C0 ≤ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' (1) Here σ is a positive Borel-regular measure on [0, +∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' It is often convenient to approximate such measures by linear combinations of delta-functions, resulting in rational approximations of Z(ω) [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Such approximations can be interpreted as impedances of electrical circuits made of resistors and capacitors only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' They are called the equivalent circuit models (ECM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' One easily verifiable feature of the representation (1) is the negativity of the imaginary part of the impedance function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' The experimentally observed arcs in a Nyquist plot of the impedance with positive imag- inary part in the low frequency part of the spectrum received the unfortunate moniker of ∗Department of Mathematics, Temple University, Philadelphia, PA 19122, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' †Department of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' 1 an “inductive loop” [12, 8, 23, 21, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' It is generally agreed that magnetic effects play no significant role in the functioning of most electrochemical systems and devices, and even when they do, the inductive effects manifest themselves at the high end of the frequency spectrum [7, 16, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' The explanation of inductive loops in the current literature [16, 6] by means of ECMs with negative resistances and capacitances creates more problems than it solves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' If one permits negative resistances and capacitances, then arbitrary “experimental data” can be matched by such an impedance function with any degree of precision due to the Riesz theorem (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' [17, 14, 15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' This is not surprising, since it is the passivity of the system [22, 9, 20], violated by the the negative resistances and capacitances in ECMs, that are responsible for the possibility of stable reconstruction of the impedance function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' A far more reasonable explanation of the experimental appearance of low frequency in- ductive loops is the violation of the time-invariance assumption at time scales commensurate with the inverses of low frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' This explanation is semi-explicit in [21, 16, 18], where specific electrochemical processes altering the properties of the system on slow time scales are identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' When the time-invariance assumption is violated the input-output behavior of the system can no longer be mathematically described by a well-defined impedance function at low fre- quencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' In this paper we will show that if one applies a particular impedance measurement procedure, delivering the correct impedance for any time-invariant system, the inductive loops could be observed in non time-invariant systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' The goal of this paper is to extend the classical EIS/ECM approach to electrochemical systems with slowly “drifting” properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' In Section 3 we demonstrate that a single Voigt el- ement with drifting properties may exhibit an inductive loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' In Section 4 we use asymptotic analysis to extend the EIS/ECM approach to non time-invariant systems with slowly vary- ing properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' The resulting impedance representation formula, though more complicated, may still be used to reconstruct the impedance function values within the frequency band containing experimental measurements by performing the classical Kramers-Kronig analysis [2, 5, 11, 20] for high frequency data, while fitting the additional “drift” parameters using the inductive loop low frequency data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' In Section 2 we discuss a measurement technique that gives a good approximation of the true impedance function for time-invariant systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' We then show in Section 3 that the same measurement technique for non time-invariant Voigt elements can produce “inductive loop” data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Representing a general non time-invariant electrochemical system by an ECM with non time-invariant Voigt elements connected in series, gives us a mathematical model that can be analyzed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' An asymptotic analysis applied to each non time-invariant Voigt element leads us in Section 4 to a generalization of the EIS theory that reduces to classical formulas at high frequencies and is capable of modeling low frequency inductive loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' 2 2 Impedance measurements Let us assume that the input signal is the current I(t) = I0 sin(ωt), t ∈ [0, T(ω)], T(ω) = 2πn(ω) ω , (2) where n(ω) ∈ N should be as large as possible, so that the time T(ω) it takes to make a measurement is still acceptable to whoever makes the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Of course, this is an issue only for low frequencies ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Thus, we can measure the system response only for frequencies ω ≥ ωmin = 2π/Tmax, where Tmax is the maximal admissible time for making low frequency measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Hence, in the low frequency regime we always choose n(ω) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' At high frequencies we use a different measurement strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' We choose a fixed measurement time T0 and set n(ω) =round(T0ω/2π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Hence, for all ω ≥ ωmin we define n(ω) = max � 1, round �T0ω 2π �� , ω ≥ ωmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' (3) The Fourier transform of I(t) can be computed explicitly, but is an unwieldy expression, except at the frequency ω of the input sinusoid: �I(ω) = I0 � T(ω) 0 sin(ωt)e−iωtdt = πI0n iω = I0T(ω) 2i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' (4) For linear, time-invariant (LTI) systems the output voltage U(t) must satisfy �U(ω) = Z(ω)�I(ω), where the complex factor Z(ω) is called the impedance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Theoretically, when I(t) = 0 for t > T(ω), the output voltage is still non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' However, in most cases it decays exponentially fast and can therefore be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' In this case we have the approximation �U(ω) ≈ � T(ω) 0 U(t)e−iωtdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' (5) Since U(t) is measured, formula (5) can be used to estimate �U(ω) and thus, we have the formula for estimating the impedance from the experimental data: Z0(ω) = 2i T(ω) � T(ω) 0 U(t)e−iωtdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' (6) If we apply formula (6) to an elementary Voigt circuit, consisting of a resistor R and a capacitor C connected in parallel, we will not get the correct answer ZV (ω) = R iθω + 1, θ = RC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' (7) Figure 1 shows that Z0 is not a very good approximation for ZV , especially in the intermediate frequency range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' The remedy is to understand this discrepancy and then devise a way to 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='5 Re(Z)/k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='8 1 Im(Z)/k Z0 ZV Figure 1: Comparison of Z0(ω) and Z(ω) for an elementary Voigt circuit with R = 2kΩ, C = 200µF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' correct for it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Using the explicit expression for Z0(ω) (which we don’t display here), we discover that for an elementary Voigt circuit we have Z0(ω) = ZV (ω) − � 1 TωIm(ZV (ω)) + i T Z′ V (ω) � � 1 − e− T θ � , (8) where Z′ V (ω) denotes the derivative of ZV (ω) with respect to ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Since relation (8) between the measured and the true impedance function of an elementary Voigt circuit is linear, it extends to all LTI systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' We recall that in an LTI system the output (voltage U(t)) depends on the input (current I(t)) via U(t) = ρ0I(t) + � t −∞ I(τ)K(t − τ)dτ, (9) where the function K(s) is called a memory kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Suppose that the memory kernel decays exponentially: |K(s)| ≤ R0 θ e−s/θ, s > T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' (10) for some T0 < T(ω) for all ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Then ����Z0(ω) − Z(ω) + 1 TωIm(Z(ω)) + i T Z′(ω) ���� ≤ R0 � 1 + θ T � e−T/θ, (11) where Z(ω) = ρ0 + � ∞ 0 K(s)e−iωsds (12) is the true impedance of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' The proof can be found in Appendix A 4 Thus, we are lead to a simple method for estimating the true impedance by making measurements over a longer time T2 = 2πn2(ω)/ω > T1 = 2πn1(ω)/ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Then, up to an exponentially small error we must have Z(ω) ≈ Zexp(ω) = T2Z0(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' T2) − T1Z0(ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' T1) T2 − T1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' (13) We still want to require that at high frequencies the time it takes to make a an impedance measurement is fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Let us call it T ′ 0, since we still want n1(ω) to be given by (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' In that case nmin 2 (ω) = n1(ω) + 1 ≤ n2(ω) ≤ max � 2, round �T ′ 0ω 2π �� = nmax 2 (ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' (14) Thus, for measuring very low frequencies we set n1(ω) = 1 and n2(ω) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' When ω = 4π/T0 we have n1(ω) = 2, in which case we must require that n2(ω) ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' It follows that T ′ 0 ≥ 3T0/2, and therefore, nmax 2 (ω) > n1(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' In practice, the experimenter can choose any integer n2(ω) satisfying nmin 2 (ω) ≤ n2(ω) ≤ nmax 2 (ω), once the constant T ′ 0 ≥ 3T0/2 has been set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' The correction method (13) can be interpreted as a directive to ignore the transient response over the time interval [0, T1], effectively taking data during time interval [T1, T2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Indeed, using formula (6), we have Zexp(ω) = 2i T2(ω) − T1(ω) � T2(ω) T1(ω) U(t)e−iωtdt, T1,2(ω) = 2πn1,2(ω) ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' (15) One convenient choice is n1 = n(ω) and n2 = 2n(ω), where n(ω) is given by (3), so that the length of the time interval over which the data is taken is still T(ω) = 2πn(ω)/ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Under assumptions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='1 we have the estimate |Zexp(ω) − Z(ω)| ≤ T1 T2 − T1 R0 � 1 + θ T1 � e−T1/θ + R0e−T2/θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' (16) The proof can be found is in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' To see how formula (15) improves the evaluation of the impedance we take the same Voigt circuit with R = 2kΩ, C = 200µF, shown in Figure 1, and compare Zexp(ω) and Z(ω) = ZV (ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' The difference between the Nyquist plots of the two functions can no longer be visualized as in Figure 1, since it is less that 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='01%, according to numerics, when we choose n1 = n(ω) and n2 = 2n(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' 3 Non time-invariant elementary Voigt circuits In this section we examine elementary Voigt circuits made of a resistor and a capacitor connected in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Our main assumption is that the parameters R and C of the circuit do not stay constant, but slowly change in time, instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' We will show that if we apply the impedance measurement recipe from the previous section to such non time-invariant systems we may obtain impedance curves with inductive loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='1 General theory Let us start by considering an elementary Voigt circuit consisting of a resistor R = R(t) and a capacitor C = C(t) connected in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' In such a circuit we have the Ohm’s laws for each of the elements IR(t) = U(t) R(t), IC(t) = C(t) ˙U(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Combining this with the Kirchhoff’s law I(t) = IR(t) + IC(t) we obtain the constitutive relation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' dependence of I(t) on U(t)) in the form I(t) = U(t) R(t) + C(t) ˙U(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' (17) Since we use the current as the input we need to solve (17) for U(t): U(t) = � t −∞ I(s) C(s) exp � − � t s dx θ(x) � ds, θ = RC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' (18) The quantity θ is called the relaxation time of the Voigt circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Formula (15) gives the experimentally measured “impedance” of a simple Voigt circuit with non time-invariant elements: Zexp(ω) = 2i ∆T � T2 T1 e−iωt � t 0 sin(ωs) C(s) exp � − � t s dx θ(x) � dsdt, ∆T = T2 − T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' (19) For future reference we also have the following formula for the impedance of a non time- invariant capacitor and resistor ZC exp(ω) = 2i ∆T � T2 T1 e−iωt � t 0 sin(ωs) C(s) dsdt = 2 ∆Tω � T2 T1 (e−iωt − 1)sin(ωt) C(t) dt, (20) ZR exp(ω) = 2i ∆T � T2 T1 e−iωtR(t) sin(ωt)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' (21) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='2 Explicit non time-invariant models Let us now examine a particular model of the time dependence of R(t) and C(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' We assume that R and C undergo an “exponential drift” from R− (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' C−) at t = −∞ to R+ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' C+) at t = +∞: R(t) = aR− + R+et/τ a + et/τ , C(t) = bC− + C+et/τ b + et/τ , a, b > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' (22) The evolution law for R(t) is governed by two parameters: the time scale τ and the “current position” a > 0 that tells us how far along R(t) is on the way from R− to R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' It is in 1-1 6 correspondence with R(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' The value a = 1 means R(0) is exactly half-way between R+ and R−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Exact same comments apply to C(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' It is easy to calculate explicitly − � t s dx θ(x) = τ θ+ �s − t τ + κC ln �ρCe−s/τ + 1 ρCe−t/τ + 1 � + κR ln �ρRe−s/τ + 1 ρRe−t/τ + 1 �� , (23) where κC = (a − ρC)(b − ρC) ρC(ρC − ρR) , κR = (a − ρR)(b − ρR) ρR(ρR − ρC) , ρR = aR− R+ , ρC = bC− C+ , ρR ̸= ρC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' We note a relation κC + κR = 1 − θ+ θ− , θ± = C±R±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' (24) If ρR = ρC = ρ, then − � t s dx θ(x) = τ θ+ �s − t τ + α ln �ρe−s/τ + 1 ρe−t/τ + 1 � + β(e−s/τ − e−t/τ) (ρe−s/τ + 1)(ρe−t/τ + 1) � , (25) where α = 1 − θ+ θ− , β = (a − ρ)(b − ρ) ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' A different drift model R(t) = R0e−t/τ + R∞(1 − e−t/τ), C(t) = C0e−t/τ + C∞(1 − e−t/τ) (26) is a limiting case of our model (22) with R+ = R∞, C+ = C∞, and a, b → 0, R−, C− → ∞, aR− → R0 − R∞, bC− → C0 − C∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' In this limit, however, the restriction that R− and C− are positive are no longer required, since R(t) and C(t) are required to be positive only for t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' In this drift model we can still use (23) with ρR → R0 R∞ − 1, ρC → C0 C∞ − 1, κR → ρR ρR − ρC , κC → ρC ρC − ρR , provided ρR ̸= ρC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' If ρR = ρC = ρ, then we use formula (25) with α = 1 and β = ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Once, � t s θ(x)−1dx has been evaluated, we compute U(t) = I0 C+ � t 0 sin(ωs) 1 + be−s/τ ρCe−s/τ + 1 exp � − � t s dx θ(x) � ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' (27) The experimenatally measured impedance Zexp(ω) is then computed by means of formula (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' The numerical evaluation of Zexp(ω) presents several challenges, since parameter τ must be very large, relaxation times θ± are typically very small and ω can range from 10−6Hz to 106Hz, making some integrands highly oscillatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' These issues are addressed in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Figure 2 shows an inductive loop in the Nyquist impedance plot of a non time- invariant Voigt element, with R(t) and C(t) given by (22), where R+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='4kΩ, R− = 2kΩ, a = 10, C+ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='6mF, C− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='8mF, b = 6, τ = 600s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' The figure shows that at high frequencies the impedance is very well captured by the impedance of the time-invariant Voigt element with R = R(0), and C = C(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='5 Re(Z)/k 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='2 Im(Z)/k Inductive loop Instanteneous impedance Figure 2: An inductive loop in a non time-invariant Voigt element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' 4 Slow parameter drift asymptotics Assume that 1/τ = ǫ is a small parameter and that ω = ǫΩ, where the rescaled frequency Ω is assumed to be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' We need to find the asymptotics of Z(ǫΩ) as ǫ → 0 and see if ℑ(Z(ǫΩ)) can indeed be positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' We now assume that R(t) = R0(ǫt) and C(t) = C0(ǫt), where R0(x) and C0(x) are strictly positive, bounded smooth functions with bounded first derivatives on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Our goal is not only to obtain the asymptotics of Z(ω), defined by (6), as ǫ → 0, but estimate the error between the actual impedance and its asymptotic approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' We obtain for the elementary Voigt circuit, using (18) Zǫ(ω) = 2i ∆T � T2 T1 e−iωt � t 0 sin(ωs) C0(ǫs) exp � − � t s dτ θ(ǫτ) � dsdt, ∆T = T2 − T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' When the drift time scale τ = 1/ǫ is large, the formula for Zǫ(ω) above can be simplified: Zǫ(ω) = iΩ π∆n(ω) � 2πn2(ω)/Ω 2πn1(ω)/Ω e−iΩηIm � R0(η)eiΩη 1 + iωθ0(η) � dη + O(ǫ), (28) where the order ǫ error O(ǫ) is uniform over the entire frequency spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Formula (28) shows that if the functions θ0(η) and R0(η) are constants, then the ap- proximation becomes exact, as it reduces to the classical impedance of the elementary Voigt circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' One more simplification is possible in the asymptotics ǫ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Specifically, we can replace θ0(η) by θ(0) in (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Our final asymptotics is Z∗ ǫ (ω) = Z(ω, Ω) = iΩ π∆n(ω) � 2πn2(ω)/Ω 2πn1(ω)/Ω e−iΩηIm � R0(η)eiΩη 1 + iωθ0(0) � dη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' (29) For numerical evaluation of (29) we choose n1 = n(ω), n2 = 2n(ω) and change variables of integration η = sn/Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Then, using the representation Im(z) = (z − z)/2i we obtain the formula we use in the Matlab code to make Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' 3: Z∗ ε(ω) = 1 2π � 4π 2π R0 �sn Ω � 1 + iωθ0(0)ds − 1 2π � 4π 2π R0 � sn Ω � e−2ins 1 − iωθ0(0) ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' (30) 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='5 Re(Z)/k 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='2 Im(Z)/k Measured impedance Asymptotic impedance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='5 Re(Z)/k 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='01 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='01 Im(Z)/k Measured impedance Asymptotic impedance Figure 3: The quality of the asymptotics of an inductive loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' For the model (22) we compute 1 2π � 4π 2π R0 �sn Ω � ds = R+ + (R+ − R−)Ω 2πn ln � 1 + a(e− 4πn Ω − e− 2πn Ω ) 1 + ae− 2πn Ω � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' In the high frequency regime R0 �sn Ω � = R0 �ǫsT0 2π � We can therefore, use a linear approximation R0(x) ≈ bR +mRx, when x ∈ [ǫT0, 2ǫT0], where the parameters bR and mR are found from the least squares fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' In that case 1 2π � 4π 2π R0 �sn Ω � e−2insds ≈ imR 2Ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Figure 3 shows the Nyquist plot of Z∗ ǫ (ω) superimposed on the computed “measured” impedance corresponding to the model (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' The difference between the two graphs is not detectable at the scale of the entire Nyquist plot, shown in the left panel of the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' The right panel shows the blown-up portion of the plot containing the inductive loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' The overall relative error of the asymptotic approximation (29) is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' In the figure we used (22) with the same parameter values as in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Now, if we have a general electrochemical system exhibiting parameter drift, then the measured impedance can be represented as Zexp(ω) = iΩ π(n2(ω) − n1(ω)) � 2πn2(ω)/Ω 2πn1(ω)/Ω e−iΩηIm � Z(ω, η)eiΩη� dη + O(ǫ), (31) where for each η ∈ R the function ω �→ Z(ω, η) is a classical (instantaneous) impedance function, provided there is no leading capacitance in the ECM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' In the presence of the leading capacitance, the term ZC exp(ω) = iΩ π(n2(ω) − n2(ω)) � 2πn2(ω)/Ω 2πn1(ω)/Ω (e−iΩη − 1) sin(Ωη) iωC0(η)dη must be added to the right-hand side of (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' 9 If we approximate the instantaneous impedance function Z(ω, ǫt) by a finite Voigt circuit ECM [1] Z(ω, η) = R0(η) + N � j=1 Rj(η) 1 + iωθ(η), then, up to a uniform, over the entire spectrum, order ǫ error, we have Zexp(ω) = iΩ π(n2(ω) − n1(ω)) � 2πn2(ω)/Ω 2πn1(ω)/Ω e−iΩηIm � Z0(ω, η)eiΩη� dη + O(ǫ), (32) where Z0(ω, η) = R0(η) + N � j=1 Rj(η) 1 + iωθ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' (33) If Rj(η) in (33) are modeled according to (26), Rj(η) = Rj(0) + ∆Rj � 1 − e−µjη� , j = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' , N, (34) then the integral in (32) can be computed explicitly: Zexp(ω) = N � j=0 �Rj(0) + ∆Rj 1 + iωθj(0) + ∆RjF �ǫµj ω , ωθj(0) �� + O(ǫ), (35) where θ0 = 0, and F(W, v) = ie−2πn1W − e−2πn2W π(n2 − n1)W v(W + i) − 1 (1 + v2)(W + 2i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Typically, one would choose n1(ω) = n(ω) and n2(ω) = 2n(ω), where n(ω) is given by (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' At low frequencies n(ω) = 1, while at high frequencies 2πn(ω)/ω can be replaced with a constant T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' 5 Conclusions and discussion In this paper we have proposed an explanation of the inductive loop phenomena observed in experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' The hypothesis is that the act of the impedance measurement of an elec- trochemical system might speed up slow processes, such as corrosion and charge diffusion, altering the properties of the system on time scales commensurate with the inverses of the fre- quencies at which the impedance is measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' The modeling and analysis of these processes can lead to specific drift laws that can be used to validate our general theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' A natural question is whether our theory makes it apparent how the time-dependent nature of the system causes the occurrence of inductive loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' By way of the answer we refer to formula (30) and set ω = 0, n(ω) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Then we obtain Im(Z(ǫΩ)) ≈ 1 2π � 4π 2π R0 � s Ω � sin(2s)ds (36) 10 While R0(u) is a strictly positive, smooth real function, the sign of the integral in (36) is the outcome of balancing positive and negative contributions of the oscillatory integrand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' For example, if R0(u) is a decreasing function, then positive contributions will always be larger than the negative ones and inductive loops will be present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Conversely, if R0(u) is an increasing function, no inductive loops will be produced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' If the slowly oscillating sinusoidal current input causes a corresponding in-phase oscillation of R0, the inductive loop effect could be several times stronger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' One open question, not directly addressed in the paper is capturing the time scale τ over which the system’s properties change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' A crude estimate would be the inverse frequency at which the imaginary part of the impedance hits zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' For example, in the simulation corresponding to Figure 2 the time scale τ = 600s, while 2π/ω0 = 610s, where Im(Z(ω0)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Our simulation of the time-dependent elementary Voigt circuit shows that the deviations from the classical model start at frequencies ω as large as two orders of magnitude over τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' They become vividly pronounced at frequencies on the order of 1/τ and persist over frequencies an order of magnitude lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Finally, whether or not the inductive loop data can give additional information about the system beyond the time scale τ depends on whether the drift model (34) is acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' In this case one should be able to use the algorithm in [13] applied to the high frequency data to compute parameters Rj(0), θj(0) and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' The intermediate and low frequency data can then be used in a non-linear least squares fit to to estimate drift parameters µj and ∆Rj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Future research into the inductive loops should address these questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' This material is based upon work supported by the National Sci- ence Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' DMS-2005538.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' A Mathematical proofs A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='1 Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='2 For I(t) = χ[0,T(ω)](t) sin(ωt) we have, making a change of variables s = t − τ, U(t) = ρ0 sin(ωt) + � t 0 K(s) sin(ω(t − s))ds, 0 ≤ t ≤ T(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' We can then write U(t) = ρ0 sin(ωt) + � ∞ 0 χ(s,+∞)(t)K(s) sin(ω(t − s))ds, so that we can substitute this into (6) and switch the order of integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' We obtain Z0(ω) = ρ0 + 2i T(ω) � T(ω) 0 K(s) �� T(ω) s sin(ω(t − s))e−iωtdt � ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Computing the inner integral we obtain Z0(ω) = ρ0 + � T(ω) 0 K(s) � e−iωs − ωse−iωs − sin(ωs) 2πn � ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' 11 Using formula (12) for the true impedance, we compute Z(ω) − 1 2πn(ω)Im(Z(ω)) − iω 2πn(ω)Z′(ω) = ρ0 + � ∞ 0 K(s) � e−iωs + sin(ωs) − ωse−iωs 2πn � ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Therefore Z(ω)− Im(Z(ω)) ωT(ω) − i T(ω)Z′(ω)−Z0(ω) = � ∞ T(ω) K(s) � e−iωs + sin(ωs) − ωse−iωs 2πn � ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' (37) For s ≥ T(ω) = 2πn/ω we have ωs ≥ 2πn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Denoting x = ωs and A = 1/(2πn), we estimate |e−ix + A(sin x − xe−ix)| ≤ |1 − Ax| + A = Ax − 1 + A ≤ Ax, since Ax ≥ 1 and A ≤ 1/(2π) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Hence, ����Z0(ω) − Z(ω) + Im(Z(ω)) + iωZ′(ω) 2πn(ω) ���� ≤ 1 T(ω) � ∞ T(ω) s|K(s)|ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Using the exponential decay (10) of the memory kernel we obtain the estimate ����Z0(ω) − Z(ω) + Im(Z(ω)) + iωZ′(ω) 2πn(ω) ���� ≤ R0 � 1 + θ T(ω) � e− T (ω) θ , (38) proving (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Let us now prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' We compute, Zexp(ω) − Z(ω) = T1 T2 − T1 � T2 T1 K(s) � e−iωs + sin(ωs) − ωse−iωs 2πn1 � ds − � ∞ T2 K(s)e−iωsds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Uisng estimate (38) we obtain |Zexp(ω) − Z(ω)| ≤ T1(ω) T2(ω) − T1(ω)R0 � 1 + θ T1(ω) � e− T1(ω) θ + R0e− T2(ω) θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content='2 Proof of the asymptotic foormula (31) Let us analyze the asymptotic behavior of Zǫ(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' We first change variables in the innermost integral σ = ǫτ: Zǫ(ω) = 2i ∆T � T2 T1 e−iωt � t 0 sin(ωs) C0(ǫs) exp � −1 ǫ � ǫt ǫs dσ θ(σ) � dsdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Next we change variables ξ = ǫs in the integral with respect to the s variable: Zǫ(ω) = 2i ∆Tǫ � T2 T1 e−iωt � tǫ 0 sin(ωξ/ǫ) C0(ξ) exp � −1 ǫ � ǫt ξ dσ θ(σ) � dξdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' 12 Finally, we change variables in the outermost integral η = ǫt: Zǫ(ω) = 2i ∆Tǫ2 � T2ǫ T1ǫ e−iΩη � η 0 sin(Ωξ) C0(ξ) exp � −1 ǫ � η ξ dσ θ(σ) � dξdη, where Ω = ω/ǫ could be large, when ω is not very small, but could also be of order 1, when ω is of order ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' The main approximation idea is to realize that Eǫ(ξ, η) = exp � −1 ǫ � η ξ dσ θ(σ) � is exponentially small when η − ξ is not very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' However, when η ≈ ξ, then θ(σ) ≈ θ(η), which means that Eǫ(ξ, η) ≈ E0 ǫ (ξ, η) = exp � −η − ξ θ(η)ǫ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' To make this approximation quantitative we use the inequality ex2 − ex1 x2 − x1 < emax{x1,x2}, which is a consequence of convexity of the exponential function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Hence, |Eǫ(ξ, η) − E0 ǫ (ξ, η)| ≤ Lθ−1(η − ξ)2 2ǫ exp � −η − ξ Mθǫ � = ǫLB �η − ξ ǫ � , where Mθ = max t∈R θ(t), Lθ−1 = max t∈R ���� � 1 θ(t) �′���� , B(x) = x2 2 exp � − x Mθ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Thus, we can replace Zǫ(ω) with its approximation Z(1) ǫ (ω) = 2i ∆Tǫ2 � T2ǫ T1ǫ e−iΩη � η 0 sin(Ωξ) C0(ξ) E0 ǫ (ξ, η)dξdη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Moreover, we also have |Zǫ(ω) − Z(1) ǫ (ω)| ≤ 2Lθ−1 mC∆Tǫ � T2ǫ T1ǫ � η 0 B �η − ξ ǫ � dξdη, where mC = min t∈R C(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Changing variables in the inner integral x = (η − ξ)/ǫ we obtain |Zǫ(ω) − Z(1) ǫ (ω)| ≤ 2Lθ−1ǫ mC � ∞ 0 B(x)dx = ǫM3 θ Lθ−1 mC .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' 13 The function E0 ǫ (ξ, η) is exponentially small when ξ is not very close to η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' However, when ξ is very close to η we can replace C0(ξ) with C0(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Hence we have Z(1) ǫ (ω) ≈ Z(2) ǫ (ω) = 2i ∆Tǫ2 � T2ǫ T1ǫ e−iΩη C0(η) � η 0 sin(Ωξ)E0 ǫ (ξ, η)dξdη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Moreover, |Z(1) ǫ (ω) − Z(2) ǫ (ω)| ≤ 2LC−1 ∆Tǫ2 � T2ǫ T1ǫ � η 0 (η − ξ) exp � −η − ξ ǫMθ � dξdη, where LC−1 = max t∈R ���� � 1 C(t) �′���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Changing variables x = (η − ξ)/ǫ in the inner integral we obtain the bound |Z(1) ǫ (ω) − Z(2) ǫ (ω)| ≤ 2LC−1 ∆T � T2ǫ T1ǫ � ∞ 0 x exp � − x Mθ � dxdη = 2LC−1M2 θ ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Now, the inner integral in Z(2) ǫ (ω) can be evaluated explicitly: � η 0 sin(Ωξ) exp � −η − ξ θ(η)ǫ � dξ = ǫθ sin(Ωη) − ǫ2θ2Ω cos(Ωη) + ǫ2θ2Ωe−η/(ǫθ) 1 + ǫ2θ2Ω2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' We observe that the first two terms can be combined nicely, and Z(2) ǫ (ω) = 2i ∆Tǫ � T2ǫ T1ǫ e−iΩηIm � R0(η)eiΩη 1 + iωθ0(η) � dη + ∆ǫ(ω) = Z(3) ǫ (ω) + ∆ǫ(ω), where ∆ǫ(ω) = 2iΩ ∆T � T2ǫ T1ǫ e−iΩη θ2e−η/(ǫθ) C0(η)(1 + ǫ2θ2Ω2)dη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' We estimate |∆ǫ(ω)| ≤ 2Ωǫ max η∈[T1ǫ,T2ǫ] θ(η)2e−η/ǫθ(η) C0(η)(1 + ǫ2θ(η)2Ω2) ≤ MRe−T1/Mθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Z(3) ǫ (ω) can be written as a two-scale impedance function (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' In order to prove a simplified formula (29) we estimate |Z(3) ǫ (ω) − Z∗ ǫ (ω)| ≤ Ω π∆n(ω) MRLθ 1 + m2 θω2 � 2πn2(ω)/Ω 2πn1(ω)/Ω ωηdη = 2πǫ(n1 + n2)MRLθ 1 + m2 θω2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' If we choose n1 = n and n2 = 2n, then in the regime n(ω) = 1 we get the bound |Z(3) ǫ (ω) − Z∗ ǫ (ω)| ≤ 6πǫMRLθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' 14 If n(ω) > 1, then n(ω) = T0ω/2π, and we obtain |Z(3) ǫ (ω) − Z∗ ǫ (ω)| ≤ 3ǫT0ωMRLθ 1 + m2 θω2 ≤ 3ǫT0MRLθ 2mθ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Hence, we obtain a uniform in the entire spectrum bound |Zǫ(ω) − Z∗ ǫ (ω)| ≤ Mǫ for some constant M that depends on the parameter functions R0(s) and C0(s), and that scales like Rθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' B Numerical evaluation of the impedance in explicit drift models The key to the effective numerical computation of Zexp(ω) for explicit drift models (22) and (26) is a non-dimensionalization of all quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' We therefore begin by rescaling the variable of integration in the definition (19) of Z(ω), t = nˆt/ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Z(ω) = i I0π � 4π 2π U �nˆt ω � e−inˆtdˆt, where we have chosen n1 = n(ω), n2 = 2n(ω), and where n(ω) is given by (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' We then observe that U(t) has the form U(t) = I0 C+ � t 0 sin(ωs)f �s τ � e τ θ+[g( s τ )−g( t τ )]ds, (39) where f(u) = be−u + 1 ρCe−u + 1, g(u) = u + κR ln(ρRe−u + 1) + κC ln(ρCe−u + 1), provided ρC ̸= ρR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' If ρC = ρR = ρ, then we use g(u) = u + α ln(ρe−u + 1) − β ρ(ρe−u + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' We therefore, rescale the variable of integration s = nˆs/ω in (39): U �nˆt ω � = I0n C+ω � ˆt 0 sin(nˆs)f � nˆs ωτ � e τ θ+[g( nˆs ωτ )−g( nˆt ωτ )]dˆs Hence, we obtain the formula we use in our Matlab code Z(ω) = in πC+ω � 4π 2π e−inˆt �� ˆt 0 sin(nˆs)f � nˆs ωτ � e τ θ+ [g( nˆs ωτ )−g( nˆt ωτ )]dˆs � dˆt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' (40) 15 In the frequency band where n(ω)/ω ≈ T0, we approximate, assuming that τ is large, f � nˆs ωτ � ≈ bf + mf nˆs ωτ , g � nˆs ωτ � ≈ bg + mg nˆs ωτ , where the slopes mf and mg, and intercepts bf and bg are found from linear least squares fits of f(x) and g(x), x ∈ [T0/τ, 2T0/τ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Then, all integrals in (40) can be computed explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Up to exponentially small terms Z(ω) ≈ bfR+ mg + iωθ+ + 1 2ωτ mfR+ mg + iωθ+ � 3T0ω + 3i − 4im2 g m2g + ω2θ2 + � , n(ω) > 1, where in the final expression we replaced n(ω) with T0ω/(2π), which is valid exactly in the high frequency regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' References [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Agarwal, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Orazem, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Garcia-Rubio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Measurement models for elec- trochemical impedance spectroscopy: I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' demonstration of applicability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Journal of the Electrochemical Society, 139(7):1917, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' [2] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Agarwal, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Orazem, and L.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' Lithium Ion Batteries—New Developments, 8:189–227, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} +page_content=' 18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNE4T4oBgHgl3EQfVgwh/content/2301.05024v1.pdf'} diff --git a/ttE0T4oBgHgl3EQfbQCD/content/tmp_files/2301.02346v1.pdf.txt b/ttE0T4oBgHgl3EQfbQCD/content/tmp_files/2301.02346v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..47b8f8d4c47e5417e24ce83e0ac71870d89201b8 --- /dev/null +++ b/ttE0T4oBgHgl3EQfbQCD/content/tmp_files/2301.02346v1.pdf.txt @@ -0,0 +1,2359 @@ +MNRAS 000, 000–000 (2021) +Preprint 9 January 2023 +Compiled using MNRAS LATEX style file v3.0 +The Gaia view of the Cepheus OB2 association +M´at´e Szil´agyi1,2,3⋆, M´aria Kun1,2, P´eter ´Abrah´am1,2,3, G´abor Marton1,2,3 +1Konkoly Thege Mikl´os Astronomical Institute, Research Centre for Astronomy and Earth Sciences, E¨otv¨os Lor´and Research Network, +H-1121 Budapest, Konkoly-Thege Mikl´os ´ut 15–17, Hungary +2CSFK, MTA Centre of Excellence, H-1121 Budapest, Konkoly Thege Mikl´os ´ut 15–17, Hungary +3ELTE E¨otv¨os Lor´and University, Institute of Physics, P´azm´any P´eter s´et´any 1/A, 1117 Budapest, Hungary +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +OB associations, birthplaces of the most luminous stars, are key objects for understanding the formation of high-mass +stars and their effects on their environments. The aim of this work is to explore the structure and kinematics of the +Cepheus OB2 association and characterize the history of star formation in the region – in particular, the role of the +Cepheus Bubble, surrounding Cepheus OB2. Based on Gaia DR3 data we study the spatial and age distribution +and kinematics of young stars in the region. We select candidate pre-main-sequence stars in the MG vs. GBP−GRP +colour–magnitude diagram, and using a clustering algorithm, we identify 13 stellar groups belonging to Cep OB2. +Four groups, consisting of 10–13 Myr old low- and intermediate-mass stars, are located in the interior of the bubble, +and are part of the oldest subsystem of the association Cep OB2a. Younger groups are found on the periphery. The +tangential velocities suggest that some groups on the periphery were born in an expanding system of star-forming +clouds, whereas others have been formed due to the collision of their parent cloud with the expanding bubble. +Key words: stars: pre-main-sequence – stars: formation – ISM: clouds – ISM: individual objects : Cepheus Bubble – +The Galaxy: open clusters and associations: individual: Cepheus OB2 +1 INTRODUCTION +OB associations are unbound, dispersing groups of young, +high luminosity stars, birthplaces of the high-mass stars of +our Galaxy (Blaauw 1964; de Zeeuw et al. 1999; Wright +2020). High-mass stars have a strong impact on the structure, +physics and chemistry of the interstellar medium, thus their +astrophysical importance is enormous. Studying the proper- +ties of associations is important for understanding their for- +mation and evolution. Dimensions of OB associations are 10– +100 parsecs, and star formation in them may proceed for tens +of million years. Most of them consist of substructures of dif- +ferent ages, including clusters of higher densities. Substruc- +tures of different ages are spatially and kinematically sep- +arated. Several associations are surrounded by supershells, +created by the interaction of expanding ionization fronts, stel- +lar winds, and supernova explosions of short-lived, high-mass +stars with the surrounding interstellar medium (e.g. Bruh- +weiler et al. 1980). Due to their 100–1000 pc dimensions su- +pershells are able to propagate star formation across the in- +terstellar medium on very large scales (McCray & Kafatos +1987). +The new astrometric and photometric data from the Gaia +space telescope (Gaia Collaboration et al. 2016) can an- +swer several long-standing questions related to the struc- +ture, origin, and evolution of OB associations. These ques- +⋆ E-mail: szilagyi.mate@csfk.org +tions include the expansion of associations and propagating +star formation. The low space density of stars in association +(< 0.1 M⊙ pc−3) could reasonably be explained by the ex- +pansion of originally dense, compact systems. Gaia data have +shown that not all of the observed structure can be explained +by expansion (e. g. Wright et al. 2016; Wright & Mamajek +2018; Kounkel & Covey 2019). Melnik & Dambis (2020) found +expansion in five of the 28 examined associations. Kinematic +studies by Lim et al. (2019, 2021, 2022) suggest that the +formation of OB associations may result from structure for- +mation driven by supersonic turbulence, rather than from the +dynamical evolution of individual embedded clusters. The ob- +served age differences of substructures can be explained by +star formation in the gas compressed by the expanding ion- +ization front and stellar wind bubble or supernova shock. +Supershells may trigger star formation via several mecha- +nisms: the expanding shock front may compress pre-existing +clouds, or accumulate the ambient, low-density gas into a +thin, dense layer, or may collide with ambient molecular +clouds and with other bubbles (Ehlerov´a & Palouˇs 2002). +These different mechanisms may result in new stellar pop- +ulations which differ from each other in structure, velocity +compared to the energy source, and star-forming time scales. +Supershells, associated with nearby OB associations give an +opportunity to study in detail various scenarios of propagat- +ing star formation. Precise parallaxes and proper motions, +available in the Gaia data, allow us to study the spatial and +© 2021 The Authors +arXiv:2301.02346v1 [astro-ph.SR] 6 Jan 2023 + +2 +Szil´agyi et al. +kinematic substructures of OB associations, and explore the +role of various types of triggered star formation. +The subject of the present work is the Cepheus OB2 asso- +ciation, discovered by Ambartsumian (1949). The luminous +stars defining the association occupy an area of some 10◦×10◦ +around the Galactic position (l, b) = (102.1, +4.6) (Kun, Kiss +& Balog 2008). At a mean distance of 900 pc (Contreras et al. +2002) this angular size corresponds to a diameter of some +150 pc. Simonson (1968) identified 74 members of Cep OB2 +based on spectroscopy and UBV photometry. Further mem- +bers were identified based on HIPPARCOS data (de Zeeuw +et al. 1999). Binary frequency among the high-mass stars of +Cep OB2 was examined by Peter et al. (2012). They estab- +lished that the multiplicity of massive (M ≥ 10 M⊙) stars +seems to be significantly higher than that of intermediate- +mass stars. Two open clusters, the ∼4 Myr old Trumpler 37, +embedded in the H ii region IC 1396, and the ∼ 12 Myr old +NGC 7160 are the historical subsystems of Cep OB2 (Sicilia- +Aguilar et al. 2005, 2006). Trumpler 37 itself is composed of +several subgroups of different ages and structure, indicative +of star formation triggered by interactions of the central O- +type star HD 206267 with ambient clumps of molecular gas +(e.g. Sicilia-Aguilar et al. 2015). +Cep OB2 is associated with the Cepheus Bubble, a super- +shell of some 10◦ in angular diameter. It was identified in +the IRAS 60 and 100 µm images by Kun, Bal´azs & T´oth +(1987). ´Abrah´am, Bal´azs & Kun (2000) studied the struc- +ture and kinematics of the bubble based on the data of the +Leiden/Dwingeloo neutral hydrogen survey. The H i data re- +vealed the expansion of the bubble. Observations of the re- +gion in the 2.6-mm CO line by Patel et al. (1998) revealed an +expanding shell of some 120 pc in diameter, and containing +∼ 4×105 M⊙ gas. The morphology and kinematics suggested +that the bubble was created by the stellar winds of the first +generation of high-mass stars of Cep OB2a during their life- +time of 8–10 Myr, and accelerated by a supernova explosion +∼ 1.7 Myr ago. The star-forming regions along the periphery +of the bubble, such as Sh 2-129, IC 1396, Sh 2-140, L1188, +were probably formed a few million years after the first gen- +eration, by the fragmentation and collapse of the gas, com- +pressed by the expanding stellar wind bubble. The molecular +clouds associated with the bubble are birthplaces of the third +generation of Cep OB2 (see Szegedi-Elek et al. 2019). +We study the Gaia DR3 (Gaia Collaboration et al. 2022) +data of the stars in the area of the Cepheus OB2 and the +Cepheus Bubble. Our goal is to validate and extend the mem- +bership list of Cep OB2, separate stellar populations of vari- +ous ages and velocities, establish their relation to the expand- +ing bubble, and find conclusions on the large-scale structure +of Cep OB2 and on the history of star formation. We de- +fine our initial data set and the method of cluster selection +in Sect. 2. The tools applied to characterize the clusters are +described in Sect. 3. Our results are described in Sect. 4, dis- +cussed in Sect. 5, and briefly summarized in Sect. 6. +2 CEP OB2 MEMBERSHIP BASED ON Gaia +2.1 High-mass members of Cep OB2 +Formation mechanism and environment of high-mass stars +above 10 M⊙ may differ from those of the lower mass +stars (e.g. Tan et al. 2014). The lists of luminous mem- +bers of Cep OB2, published by Simonson (1968), Blaha & +Humphreys (1989), and de Zeeuw et al. (1999) contain 90 +stars above 10 M⊙. In order to compare the distribution +and kinematics of high- and low-mass stars we compiled +our list of massive stars from these (overlapping) tables, +and searched for their Gaia DR3 counterparts within 1′′. +In Gaia DR3, 52 historical members, supergiants and main- +sequence stars earlier than B3 have distances between 800 +and 1000 pc. Out of the 52, 8 stars have RUWE > 1.4, in- +dicating some issue with their astrometry, which could be +caused by binarity (Lindegren 2018). Most of them are in- +deed known binary or multiple systems (e.g. BD+62◦2078, +HD 204827, HD 209744, HD 239743, Fabricius et al. 2002; Pe- +ter et al. 2012). Surface distribution and tangential velocities +of these high-mass stars are displayed in Fig. 1, and listed +in Table 1. Several historical high-mass association members +appear foreground (HD 239712, HD 199661) or background +(HD 235618, +HD 239758, +HIP 109603, +BD+53◦2387, +HD +235783, HD 235795, HD 235813, HD 240010, BD+57◦2615, +BD+53◦2784, µ Cep, HD 239978, HIP 111972) stars in +Gaia DR3. +2.2 Members defined by distances and tangential velocities +2.2.1 Gaia sample +We selected all sources from Gaia DR3 database with: +(i) 96◦ < l < 108◦, +(ii) 2◦ < b < 12◦, +(iii) 800 < d < 1000 pc, +(iv) ϖ/σϖ ≥ 10, +(v) |µ⋆ +α/σµ⋆α| ≥ 5, +(vi) |µδ/σµδ| ≥ 5, +(vii) RUWE ≤ 1.6, +where l and b are the galactic longitude and latitude, d is the +distance from Bailer-Jones et al. (2021), ϖ, µ⋆ +α, µδ and σϖ, +σµ⋆α, σµδ are the parallax, proper motion in right ascension +and declination and their uncertainities, respectively. +We selected the pre-main-sequence stars from the Gaia +sample using their MG vs. GBP−GRP colour-magnitude di- +agram, corrected for interstellar extinction (Fig. 2). We cor- +rected the Gaia colour indices and magnitudes of each star +using the Python implementation1 of the 3D dust maps STIL- +ISM (Lallement et al. 2018). STILISM gives the E(B-V) +colour excess as a function of galactic coordinates and dis- +tance. We transformed the extinction into the Gaia bands +with the coefficients in table 3. of Wang & Chen (2019), as- +suming RV = 3.1. +Following the method described in Zari et al. (2019) we +have used the 10-Myr isochrone from PARSEC (Bressan et al. +2012) to define an area in the de-reddened MG vs. GBP−GRP +colour–magnitude diagram, occupied by young stellar objects +(YSOs). Figure 2 suggests that the stellar sample between +the dashed lines is contaminated with main-sequence stars, +located in the 800–1000 pc distance interval but not related +to the association. To tighten the selection we searched for +spatially and kinematically coherent groups in the sample +selected from the colour–magnitude diagram. +1 https://github.com/edober/dust_maps_3d +MNRAS 000, 000–000 (2021) + +Cepheus OB2 +3 +Table 1. Sample of the high-mass stars described in Sect. 2.1. The full table is available as a supplementary material. +Name +Spectral type +Gaia DR3 source id +Distance +µ⋆ +α +µδ +RVGaia +RVlit +Memb RVref +(pc) +(mas y−1) +(km s−1) +HD 198895 +B1Ve +2183107416725856640 +864+12 +−11 +−2.72 ± 0.02 +−4.91 ± 0.02 +· · · +−23.0 ± 7.4 +1 +7 +HD 199308 +B1.5V +2189911847512999168 +811+16 +−13 +−5.21 ± 0.03 +−4.68 ± 0.02 +· · · +−23.0 ± 4.5 +1 +6 +HD 200857 +B3III +2188891844319403648 +849+16 +−12 +−2.99 ± 0.02 +−4.70 ± 0.02 +· · · +−14.0 ± 4.4 +1 +6 +HD 204150 +B2III +2191963398772774144 +863+20 +−16 +−2.18 ± 0.03 +−2.98 ± 0.03 +· · · +−18.0 ± 12.9 +1;3 +7 +HD 205139 +B1Ib +2191787957952122752 +855+32 +−28 +−1.80 ± 0.06 +−3.75 ± 0.05 +· · · +−14.5 ± 2.9 +1;3 +6 +Membership references: 1 - Simonson (1968), 2 - Blaha & Humphreys (1989), 3 - de Zeeuw et al. (1999). +RV references: 1 - Wilson (1953), 2 - Petrie & Pearce (1961), 3 - Hilditch et al. (1982), 4 - Barbier-Brossat & Figon (2000), 5 - Pourbaix +et al. (2004), 6 - Gontcharov (2006), 7 - Kharchenko et al. (2007), 8 - Boyajian et al. (2007), 9 - de Bruijne & Eilers (2012), 10 - Holgado +et al. (2018), 11 - Katz et al. (2022) +40 km/s +106° +104° +102° +100° +98° +96° +12° +10° +8° +6° +4° +2° +Galactic longitude +Galactic latitude +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +SH 2-140 +NGC 7129 +NGC 7160 +SH 2-129 +IC 1396 +L1188 +840 +860 +880 +900 +920 +940 +Distance [pc] +Figure 1. Members of Cepheus OB2 overplotted on the Planck 857 GHz map of the region. Circles show the members of the stellar groups +found by HDBSCAN, and the white star symbols represent the high-mass (M > 10 M⊙) members of the Cepheus OB2 association from +Table 1. The arrows show the textbfmean tangential velocities of the groups and the OB stars, compared to the LSR, while the colour +bar shows the textbfmean distance of each group. The circle indicates the rough size and position of the Cepheus Bubble. +MNRAS 000, 000–000 (2021) + +4 +Szil´agyi et al. +0 +1 +2 +3 +4 +5 +6 +GBP +GRP [mag] +6 +4 +2 +0 +2 +4 +6 +8 +10 +MG [mag] +100 +101 +102 +Figure 2. De-reddened colour–magnitude diagram of the sources +described in Section 2. The dashed lines from Zari et al. (2019) +border the area defined to select the pre-main-sequence stars. +2.2.2 Search for clusters +We used the Python-implemented package of Hierarchi- +cal Density-Based Spatial Clustering of Applications with +Noise (HDBSCAN) clustering algorithm (McInnes et al. +2017) to find overdensities in our sample obtained in +Sect. 2.2.1. An advantage of this clustering algorithm over +other methods is that HDBSCAN can identify groups with +various densities and arbitrary shapes. Main parameters +of HDBSCAN are minimum cluster size, minimum samples +and cluster selection method. Minimum cluster size de- +fines the minimal number of data points a cluster must +hold at least, while the minimum samples defines how con- +servative a clustering is: the bigger the minimum samples +is, the more points are considered as noise. At default, +cluster selection method uses the Excess of Mass (EOM) +approach to select one or two large and several smaller clus- +ters. For more homogeneous, smaller clusters, we can use the +Leaf method. For a detailed description of the algorithm, see +the website2 of the package. +There are multiple ways for detecting stellar clusters in +the Gaia data: some used ICRS coordinates, parallaxes and +proper motions (e.g. Kounkel & Covey 2019), some used he- +liocentric XYZ coordinates and tangential velocities multi- +plied by a constant (e.g. Kerr et al. 2021). Using the dis- +tance of 900 pc from Contreras et al. (2002) we transformed +the Galactic coordinates l and b of the stars into 2D carte- +sian coordinates compared to the centre of the studied field. +We also calculated the vl,LSR and vb,LSR Galactic tangen- +tial velocity components compared to the Local Standard +of Rest (LSR) using the (UV W)⊙ values from Sch¨onrich +et al. (2010). The fifth dimension is the distance. Due to the +different units we standardized the data in each dimension +by subtracting the mean and dividing with the largest stan- +dard deviation of the dimensions that share the same unit, +2 https://hdbscan.readthedocs.io/en/latest/index.html +which are the distance and vl,LSR respectively. We adopted +minimum cluster size = 25 and minimum samples = 25 +with the Leaf method. +3 CHARACTERIZATION OF THE GROUPS +HDBSCAN found originally 10 groups, consisting altogether +of 874 stars. The final Groups 2 and 4, 5 and 7, and 11 and 12 +were originally merged into three larger groups, respectively, +but their spatial distribution suggested that each of the three +can be split into two smaller subgroups. The Groups 2 and +4 pairs we split at b = 7.8◦. For dividing the Group 5–7 and +11–12 pairs we used two-component Gaussian mixture model +from scikit-learn (Pedregosa et al. 2011) to fit to their +spatial distributions. These actions resulted in 13 groups. We +regard these 13 groups as substructures of Cep OB2, and +examine how their properties reflect the history of star for- +mation in the region. Figure 1 shows the distribution of the +clustered sources in Galactic coordinates, overplotted on the +Planck 857 GHz image of the region. The mean tangential +velocities of the groups, compared to the LSR, are also in- +dicated. The colouring shows the mean distances of groups. +The mean coordinates, distances and velocities of the groups +are shown in Table 2. Known clusters and clouds from the lit- +erature, associated with the individual groups, are indicated. +Surface distribution of stars in individual groups, along with +their tangential velocity vectors compared to the mean veloc- +ity of the groups listed in Table 2, are displayed in Fig. 3a and +Figs. A1a–A12a. Distance histograms and vl,LSR vs. vb,LSR di- +agrams plotted in Figs. 3b and A1bc–A12bc. Table 3, contain- +ing a detailed list of group members, is available in machine- +readable form. +3.1 Radial velocities +Gaia DR3 provides radial velocities for stars with GRVS < +14 mag (Katz et al. 2022). From the 874 group member stars, +191 have radial velocites measured by Gaia. The number of +the stars with available radial velocities for each group are +provided in Table 2. +3.2 Variable stars +Gaia DR3 provides a list containing 24 types of variable +stars identified with machine learning methods (see Rimol- +dini et al. 2022; Eyer et al. 2022, for details). We cross- +matched our group members with it, and found that 355, +∼ 41 % of the member stars are classified as YSO-candidates +in Marton et al. (2022). These stars are overplotted with +red circles in Fig. 3a and Figs. A1a–A12a. Furthermore +60 stars were identified as RS Canum Venaticorum type +variable stars. Additional 17 stars were classified as either +eclipsing binaries (Mowlavi et al. 2022), solar-like variables, +α2 CVn/magnetic chemical peculiar star/rapidly oscillating +Am/Ap star/SX Ari variable stars (Distefano et al. 2022) or +δ Sct/γ Dor/SX Phe. Column best class name in Table 3 +contains these information. +MNRAS 000, 000–000 (2021) + +Cepheus OB2 +5 +800 +900 +1000 +Distance [pc] +0 +2 +4 +6 +(b) +40 +35 +30 +25 +20 +Vl, lsr [km/s] +5 +0 +5 +10 +15 +Vb, lsr [km/s] +(c) +180 +120 +60 +0 +60 +120 +180 +Angle [deg] +0 +2 +4 +6 +8 +(d) +0.0 +0.5 +1.0 +1.5 +G +GRP [mag] +0 +5 +10 +MG [mag] +(e) +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +2.0 +3.0 +4.0 +Age: 2.5 Myr +0.0 +0.5 +1.0 +H +Ks [mag] +0 +1 +2 +J +H [mag] +(f) +2 km/s +105°40' +20' +00' +104°40' +10°40' +20' +00' +9°40' +Galactic longitude +Galactic latitude +(a) +Figure 3. a: Distribution of the members of Group 1, plotted on +the Planck 857 GHz image. The arrows indicate the tangential ve- +locities of the stars compared to the mean tangential velocity of +the group. Gold squares and the red pentagon indicate the Class II +and Class I sources classified by the WISE colour indices, respec- +tively. YSO-candidates from Marton et al. (2022) are overplotted +with red circles. YSOs from Dahm & Hillenbrand (2015) and Kun +et al. (2009) are overplotted with green hexagons. b: Histogram +of distances. c: Tangential velocity components compared to the +LSR. d: Distribution of the angles between the radial vector of each +star from the group centroid and relative tangential velocities (see +a). e: De-reddened MG vs. GBP–GRP colour–absolute magnitude +diagram. The solid line shows the best-fitted PARSEC isochrone. +Dashed lines represent evolutionary tracks from 1 to 20 Myrs of +several stellar masses. For 1.4 M⊙ and below we plotted CIFIST +isochrones (Baraffe et al. 2015). f : De-reddened 2MASS colour– +colour-diagram showing sources with quality criteria ’AAA’. The +plots of the other groups are presented in the Appendix. +3.3 Colour–absolute magnitude diagrams +Figures 3e and A1e–A12e show the M G vs. G−GRP de- +reddened colour–absolute magnitude diagrams (CMD) of +each group. We downloaded PARSEC isochrones (Bressan +et al. 2012) with logarithmic ages from 6 to 7.5 with a step +of 0.1. The CMDs show the best-fitted isochrones, as well as +evolutionary tracks for several initial masses. For 1.4 M⊙ and +below we plotted CIFIST (Baraffe et al. 2015) tracks. +1 +0 +1 +2 +3 +4 +5 +W2 +W3 [mag] +1 +0 +1 +2 +W1 +W2 [mag] +Class I +Class II +transitional disc +discless +Figure 4. AllWISE colour–colour diagram of the group members. +The dashed lines border the areas of Class I and II stars, according +to Koenig & Leisawitz (2014). +3.4 2MASS colour–colour diagrams +We crossmatched our data with the 2MASS (Skrutskie et al. +2006) database. We transformed ICRS coordinates of the +stars from Gaia’s J2016 epoch into J2000 epoch to search +for 2MASS counterparts. Then we searched for coinciding +2MASS sources within 1 arcsec. We calculated the extinctions +in the JHKs bands of 2MASS by the method described in +Sect. 2. De-reddened J−H vs. H −K s colour–colour diagrams +(CCD) are displayed in Figs. 3f and A1f–A12f, showing only +the stars with photometric quality criteria ’AAA’. +3.5 WISE colour–colour diagram +We also searched for counterparts in the AllWISE (Cutri +et al. 2021) catalogue to identify potential disc-bearing stars. +We found counterparts of 733 stars within 1 arcsec. We fol- +lowed the methods described by Koenig & Leisawitz (2014) to +find disc-bearing stars. From these AllWISE sources, 316 ful- +fil the quality criteria essential for constructing their W1−W2 +vs. W2−W3 colour–colour diagram, displayed in Fig. 4. One +star is classified as a Class I source, 58 of the sources are iden- +tified as Class II sources and one star is classified as transi- +tional disc bearing star. Supplemented with 2MASS data we +identified 4 more Class II sources. These stars are plotted +as red pentagons, gold squares, black triangles and blue dia- +monds, respectively in Figs. 3a and A1a–A12a. +3.6 Internal motions of the groups +We examined the internal motions of the substructures of +Cep OB2. Following the method applied by Lim et al. (2019) +we determined the angle between the radial vector of each star +from the centre position of the group and its relative velocity +vector. Figures 3d and A1d–A12d show the histograms of +these angles. Clustering of the angles around 0◦ is indicative +of radial expansion of the group, whereas peaks around ±180◦ +suggest contraction. +MNRAS 000, 000–000 (2021) + +6 +Szil´agyi et al. +Table 2. Mean coordinates, distances, velocities, E(B-V) colour excesses and ages for each stellar group. The last five columns show the number of group member stars, stars with measured +radial velocities, identified as YSO (Marton et al. 2022), RS Canum Venaticorum variable, and other type of variables, listed in Sect. 3.2. The table is available as supplementary material. +Group +Associated object +lmean +bmean +dmean +vl,LSR,mean +vb,LSR,mean +E(B-V) +age +N⋆ +Nrad +NYSO +NRS +Nother +(deg) +(pc) +(km s−1) +(mag) +(Myr) +1 +NGC 7129 +105.36 +9.99 +876 ± 15 +−28.4 ± 2.2 +0.2 ± 2.4 +0.21 +2.5 +33 +6 +16 +3 +1 +2 +SH 2-129, UBC 385 +98.55 +8.04 +829 ± 13 +−30.1 ± 1.7 +3.3 ± 1.3 +0.49 +10.0 +27 +12 +5 +2 +1 +3 +UBC 10a, UBC 167 +102.59 +7.36 +901 ± 6 +−29.0 ± 0.9 +1.9 ± 0.5 +0.43 +10.0 +33 +13 +3 +7 +1 +4 +· · · +98.30 +7.33 +842 ± 13 +−30.3 ± 1.2 +3.4 ± 1.2 +0.48 +15.8 +33 +11 +7 +3 +1 +5 +NGC 7160 +104.01 +6.38 +902 ± 11 +−28.9 ± 1.0 +9.8 ± 3.0 +0.29 +10.0 +64 +19 +7 +7 +4 +6 +UBC 10b +102.76 +5.64 +944 ± 7 +−30.8 ± 1.1 +9.2 ± 1.3 +0.34 +10.0 +26 +8 +1 +2 +0 +7 +[KPR2005] 117 +104.76 +5.63 +906 ± 4 +−26.1 ± 1.0 +0.7 ± 0.9 +0.32 +10.0 +22 +6 +5 +3 +0 +8 +SH 2-140, Pismis–Moreno 1 +106.60 +5.21 +906 ± 5 +−26.4 ± 3.1 +2.4 ± 1.8 +0.61 +10.0 +45 +11 +19 +1 +0 +9 +IC 1396, L1116 +99.87 +5.02 +908 ± 9 +−27.6 ± 1.2 +0.1 ± 1.1 +0.32 +5.0 +30 +9 +13 +3 +0 +10 +UPK 169, Theia 131 +101.52 +4.92 +841 ± 7 +−26.2 ± 1.6 +3.0 ± 1.9 +0.27 +4.0 +39 +15 +5 +0 +0 +11 +Alessi–Teutsch 5 +104.47 +4.18 +873 ± 14 +−27.8 ± 1.9 +−0.2 ± 1.4 +0.37 +7.9 +93 +22 +21 +6 +1 +12 +L1188, [BDS2003] 30 +105.33 +4.00 +872 ± 22 +−25.2 ± 1.8 +−0.5 ± 1.7 +0.6 +7.9 +134 +23 +66 +7 +3 +13 +IC 1396, Trumpler 37 +99.34 +3.78 +904 ± 24 +−33.5 ± 1.9 +−1.5 ± 1.6 +0.35 +5.0 +295 +36 +187 +16 +5 +MNRAS 000, 000–000 (2021) + +Cepheus OB2 +7 +4 THE SUBSTRUCTURES OF CEP OB2 +The historical substructures of Cep OB2 are NGC 7160, sur- +rounded by a number of evolved high-mass stars (Cep OB2a), +and IC 1396 (Cep OB2b) (Simonson & van Someren Greve +1976). These represent two epochs of star formation in the +volume of the association. The Cepheus Bubble has revealed +new probable subgroups and suggested connection between +them. Our search for stellar groups in the Gaia data re- +sulted in further subgroups. Except Group 4 they coincide +with known clusters, but most of them have not been cou- +pled to Cep OB2 in the literature. We describe in this section +the Gaia view of the substructures. +Most of clustered stars have G < 18 mag, only 33 of the 874 +group members are fainter than this limit. The foreground AV +extinctions are between 1–1.8 mag. These data suggest that +the masses of the detected group members are above 0.6 and +0.7 M⊙ in a 5 and 10 Myr old group, respectively. +Group 1, the northernmost group, contains the well-known, +compact young cluster NGC 7129, and a few stars to the +west and north of the cluster. Kun et al. (1987) associated +NGC 7129 with the Cepheus Bubble based on morphologi- +cal considerations. Gaia data, in accordance with the VLBA +parallax published in Reid et al. (2014) confirm that this +cluster is located on the surface of the Cepheus Bubble. +HDBSCAN identified 33 members, 8 of them are classified +as Class II stars, and 1 is identifies as a Class I source. +The colour–magnitude diagram (Fig. 3e) suggests an age of +2.5 Myr. Fig. 3f suggests significant intra-cluster extinction, +which may affect the appearance of the colour–magnitude +diagram. Eleven of the member stars appear in Dahm & Hil- +lenbrand (2015) and one in Kun et al. (2009) as YSOs. They +are overplotted with green hexagons in Fig. 3a. +Groups 2 and 4 are located at the western edge of the +Cepheus Bubble. According to the CMDs in Figs. A1e and +A3e they are 10 and 15.8 Myr old, and most of their mem- +bers are more massive than 1 M⊙. Unlike other groups, the +detected members of Group 2 are aligned evenly along the fit- +ted isochrone: the number of stars above 2 M⊙ is nearly iden- +tical with that in the 1 M⊙ < M < 2 M⊙ interval. Group 2 is +the nearest group with a mean distance of 829±13 pc. It coin- +cides with the cluster UBC 385, identified by Castro-Ginard +et al. (2020), and contains several B8–A0 type stars, classi- +fied as Cep OB2 members by Alksnis (1958). It is the central +cluster of the ring-shaped H ii region Sh 2-129, excited by the +O9.5 IV type (Sota et al. 2014) component of the young triple +system HD 202214 (Balega et al. 2004). Though the distance +of HD 202214, projected at the cluster centre is uncertain, its +spectral type and brightness suggest its membership. None of +the group members were classified as disc-bearing stars. The +neighbouring Group 4 was merged with Group 2 by HDB- +SCAN, but a closer inspection has shown that they differ +slightly in distance. +Groups 3, 5, 6 and 7 are projected inside the Cepheus Bub- +ble. The colour–magnitude diagrams suggest ages of 10 Myr, +thus these groups belong to the first generation of Cep OB2 +(Sicilia-Aguilar et al. 2005). Groups 3, 6, and 7 coincide with +the known clusters UBC 10a, UBC 10b (Castro-Ginard et al. +2018), and [KPR2005] 117 (Kharchenko et al. 2005), respec- +tively. The groups are projected close to each other, but differ +in distance, space velocity and mass distribution. Compared +to the neighbouring groups Groups 5 and 6 have large lat- +itudal tangential velocity components, and high proportion +of low-mass (M ≤ 1 M⊙) members. The angular distribution +of the tangential velocity vectors in Fig. A4d shows the ex- +pansion of Groups 5. These groups lack interstellar matter. +The dark cloud L1178 is located at the south-eastern edge of +the region, but the extinction towards the cloud rises around +700 pc, suggesting that they are not related. +Group 8 is associated with the cluster Pismis–Moreno 1 (Pis- +mis, Moreno & Hasse 1979) that contains HD 211880, the +B0.5V type exciting star of SH 2-140. The colour–magnitude +diagram indicates an age around 10 Myr, in accordance with +earlier estimates (e.g. Cantat-Gaudin et al. 2020). Four stars +were designated as Class II star by the method of Koenig & +Leisawitz (2014). Star formation is on-going in the molecular +cloud bordered by the H ii region (Gutermuth et al. 2009). +Quite a few Class II Spitzer sources, without reliable Gaia +data, are projected within this group (see fig. 1 in Guter- +muth et al. 2009), indicative of either their membership or +the overlapping of the older Pismis–Moreno 1 and the em- +bedded S140 cluster. +Groups 9 and 13 are located at the IC 1396 region. Several +dark clouds of Lynds (1962) are found in the region. The ex- +tinction towards L1086, L1096, L1102 and L1116 rises around +900 pc, indicating that all these clouds may be related to +IC 1396. Group 13 corresponds to the cluster Trumpler 37, +containing the exciting star of IC 1396. We have identified +39 Class II sources and a transitional disc-bearing star in +Group 13. The 295 members include 69 low-mass YSOs iden- +tified by Sicilia-Aguilar et al. (2005), and the intermediate- +mass YSO IRAS 21365+5713 from Contreras et al. (2002). +Figure A12d indicates the expansion of Tr 37. According to +the CMDs, the ages of groups are around 5 Myrs. +Group 10 corresponds to the cluster UPK 169 (Cantat- +Gaudin & Anders 2020). It also appears in Kounkel & Covey +(2019) as part of Theia 131. It is projected halfway between +NGC 7160 and Tr 37, but its distance is some 60 pc smaller. +The mean age of the group members is about 4 Myr. Based +on AllWISE data none of the 39 members have infrared ex- +cess, characteristic of YSO discs. The brightest member of +the group is HD 207538, an O9.7IV spectral-type star (Sota +et al. 2011). +Groups 11 and 12 correspond to the known clusters Alessi– +Teutsch 5 and [BDS2003] 30, respectively (Kharchenko et al. +2005; Bica et al. 2003). Eleven stars of Group 12 were identi- +fied as Class II sources based on AllWISE and 2MASS data. +The most luminous member of Group 11 is the B1IV type +component of the eclipsing binary V* V446 Cep. The star- +forming dark cloud L1188 is associated with Group 12. Ten of +Group 12 members were classified as young stars in Szegedi- +Elek et al. (2019). The CMDs suggest ages of 7.9 Myr for +both groups. +MNRAS 000, 000–000 (2021) + +8 +Szil´agyi et al. +106° +104° +102° +100° +98° +96° +12° +10° +8° +6° +4° +2° +Galactic longitude +Galactic latitude +10 km/s +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +4 +6 +8 +10 +12 +14 +Age [Myr] +Figure 5. 2D structure of Cepheus OB2. Large circles and smaller star symbols represent the groups from Table 2 and the high-mass stars +from Table 1, respectively. Arrows indicate the mean tangential velocities, compared to the mean motion of the system, and the colour +bar indicates group ages. +5 OVERALL VIEW OF CEPHEUS OB2 +The line-of-sight dimension of the system of groups is some +120 pc, similar to the apparent diameter of the Cepheus Bub- +ble. To get an insight into the internal motions of the stars +associated with the Cepheus Bubble we calculated the mean +motion from the velocity data in Table 2. The mean velocities +are vl = −16.13 ± 4.01 km s−1and vb = −5.43 ± 3.87 km s−1. +In Fig. 5 we represent the tangential velocities of the groups +compared to the mean values. The precision of available stel- +lar radial velocities is not sufficient for studying the three- +dimensional velocity structure. +The colour–magnitude diagrams show that star formation +started some 10 million years ago in the studied region. +Groups 3, 5, 6, 7 are located inside the volume bordered by +the Cepheus Bubble, in the volume occupied by the evolved +high-mass members of Cep OB2a. These groups are new com- +ponents of Cep OB2a. The mean distance of this subsystem is +900 pc, the line-of-sight size is about 50 pc, and the dispersion +of the tangential velocities is some 10 km s−1. The tangential +velocities indicate the expansion of the largest group, most +probably due to the disappearance of the the parental cloud. +Expansion of the whole Cep OB2a cannot be demonstrated +by the method applied in Sect. 3.6. The spatial and velocity +structure suggest that each group had its own parent cloud +clump, forming stars independently. Most of the high-mass +stars were not selected by HDBSCAN as group members, the +only exception is the B1 III type HD 208218, the most massive +member of Group 5. The reason may be that the velocities +of the high-mass stars are influenced by their probable mul- +tiplicities. +Groups 1, 2, 4, 8, 9, 12 and 13 line along the apparent +perimeter of the Bubble. Except Groups 2, 4 and 8, these +groups are younger than Cep OB2a. The diversity of their +stellar contents, ages, and velocities suggest a complex history +of star formation. The expansion of the by now extinct H ii +region of Cep OB2a might have affected the formation of +these groups, and/or the evolution of circumstellar discs of +the group members. +MNRAS 000, 000–000 (2021) + +Cepheus OB2 +9 +Table 3. Column description of the table containing the members of +the stellar groups. The list is available as supplementary material. +Column +Description +group +Group identifier +name +Star identifier +source id +Gaia DR3 source id +TMASS +2MASS identifier +AllWISE +AllWISE identifier +ra +RA at J2016 +dec +DEC at J2016 +l +Galactic longitude +b +Galactic latitude +d +Distance from Bailer-Jones et al. (2021) +dmin +16th percentile of distance posterior +from Bailer-Jones et al. (2021) +dmax +84th percentile of distance posterior +from Bailer-Jones et al. (2021) +parallax +Parallax +parallax error +Error of the parallax +pmra +Proper motion in right ascension +pmra error +Error of the proper motion in RA +pmdec +Proper motion in declination +pmdec error +Error of the proper motion in DEC +radial velocity +Radial velocity +radial velocity error +Error of the radial velocity +ruwe +Renormalized unit-weight error +phot g mean mag +Magnitude in G band +phot bp mean mag +Magnitude in GBP band +phot rp mean mag +Magnitude in GRP band +Jmag +Magnitude in 2MASS J band +Hmag +Magnitude in 2MASS H band +Kmag +Magnitude in 2MASS K s band +eb v +E(B-V) from STILISM +allwise class +YSO type by the classification of +Koenig & Leisawitz (2014) +best class name +Variable star type appear in +gaiadr3.vari classifier result +(Rimoldini et al. 2022) +Figure 5 shows that the Groups 11, 12, and 13 move ra- +dially outwards from the centre, suggesting that their parent +clouds were accelerated by the shock wave from the expand- +ing ionization front of the OB stars of Cep OB2a. Each of +these groups is younger than Cep OB2a. Groups 12 and 13 +are associated with large amounts of molecular gas (Szegedi- +Elek et al. 2019; Sicilia-Aguilar et al. 2015, respectively), sites +of active star formation. The colour–magnitude diagrams sug- +gest that Groups 11 and 12 are nearly coeval. The difference +in the proportion of disc-bearing members may result from +the different environments (cf. Dale et al. 2013). +The small groups Group 2 (UBC 385) and 4 are projected +near the westernmost wall of the Cepheus Bubble. Gaia data +have shown that these groups are some 70 pc closer to us +than the central groups. Their ages, suggested by the Gaia +CMDs, are similar to that of Cep OB2a. Group 2 is centred +on the H ii region S129. Since O-type stars and H ii regions +are generally younger than 10 Myr (e.g. Tremblin et al. 2014), +we may speculate that the exciting star HD 202214 was born +later than the lower-mass members of the group. SPH simu- +lations by Dale et al. (2013) show that such a situation may +occur in molecular clouds near ionizing stars. Precise astro- +metric data of the HD 202214 triple system and more data on +the cluster population may clarify the nature of star-forming +processes in the region of S129, and its connection with the +Cepheus Bubble. +Group 8 (Pismis–Moreno 1) is another 10 Myr old group +associated with an H ii region. The main-sequence lifetime +of the B0.5 V type exciting star HD 211880 is compatible +with this age. The molecular cloud bordered by the ion- +ization front contains the S140 embedded cluster (Guter- +muth et al. 2009). This morphology suggests that probably +Group 8 and HD 211880 itself were formed in the same molec- +ular cloud, and star formation propagates radially outwards +from the central Cep OB2a. However, the role of the expand- +ing Cepheus Bubble in the formation of this cluster is un- +likely because of the apparently similar ages of Cep OB2a +and Group 8. +Group 9 is a small group at the outskirts of IC 1396, asso- +ciated with dark clouds, bordering the H ii region. Its age is +similar to that of Tr 37, whereas their tangential velocities, +and probably star formation histories are different. +Group 10 (UPK 169, Cantat-Gaudin & Anders 2020) is +a ∼ 4 Myr old small group, containing the O9 V type star +HD 207538. It is projected inside the Bubble, and its distance +of 841 ± 7 pc suggests association with the near wall. In spite +of its young age, no disc-bearing low-mass star has been de- +tected in this group, probably due to the disruptive radiation +of the hot star. The parent cloud of this group was proba- +bly overrun by the expanding ionization front of Cep OB2a. +Similarly, collision of the expanding bubble with the south- +ernmost edge of an ambient giant molecular cloud, located at +Galactic latitudes 10–13◦ (Grenier et al. 1989), might have +played a role in the formation of Group 1 (NGC 7129). +6 SUMMARY +We have studied the stars from Gaia DR3 in the region +of the Cepheus OB2 association between 96◦ < l < 108◦, +2◦ < b < 12◦ and 800 < d < 1000 pc. We selected candi- +date pre-main-sequence stars using isochrones in the MG vs. +GBP−GRP colour–magnitude diagram. We used HDBSCAN +to find groups in the selected sample, and found 13 stellar +groups, consisting of 874 stars. These 13 groups, located be- +tween 830–940 pc, are subsystems of Cepheus OB2. 355 of the +clustered stars are classified as candidate YSOs by Marton +et al. (2022). Based on WISE data we identified one Class I +source, sixty-two Class II sources and one with transitional +disc, according to the classification of Koenig & Leisawitz +(2014). +We found that star formation in the volume of the as- +sociation started some 10 million years ago. In addition to +the evolved high-mass stars, the oldest subsystem Cep OB2a +contains four groups of low- and intermediate-mass stars. +The tangential velocities suggest the expansion of the group +containing NGC 7160, however the expansion of the whole +Cep OB2a cannot be proved. +In addition to the historical younger subsystem Cep OB2b, +Trumpler 37, several other, 2–8 Myr old groups can be found +in the volume of the association. Most of them might have +been formed under the influence of the expanding bubble. +Spatial and kinematic structure of the region suggest var- +ious means of interaction between the star-forming clouds +MNRAS 000, 000–000 (2021) + +10 +Szil´agyi et al. +and the expanding bubble. The tangential velocities show +that the parent clouds of the young clusters Trumpler 37, +Alessi–Teutsch 5 and [BDS2003] 30 (Groups 11, 12 and 13) +have been accelerated by the expanding ionization front of +Cep OB2a. Other groups reveal places where the expand- +ing ionization front encountered ambient clouds (Groups 1, +8, and 10, aka NGC 7129, S140, and UPK 169). The age of +the cluster Pismis–Moreno 1 (Group 8, S140) suggests that +formation of this group was probably independent of the ex- +panding bubble. The relation of S129 to the Cepheus Bubble +is uncertain, because of the uncertain distance of the exciting +star and the discrepancy of the age of the central cluster and +the exciting star. +ACKNOWLEDGEMENTS +We thank the anonymous referee for the careful and insight- +ful review of our manuscript. We thank Lajos Bal´azs for +his help in statistics. This work has made use of data from +the European Space Agency (ESA) mission Gaia (https: +//www.cosmos.esa.int/gaia), processed by the Gaia Data +Processing and Analysis Consortium (DPAC, https://www. +cosmos.esa.int/web/gaia/dpac/consortium). Funding for +the DPAC has been provided by national institutions, in par- +ticular the institutions participating in the Gaia Multilateral +Agreement. This work was supported by the ESA PRODEX +Contract nr. 4000129910. For this work we have used Astropy +(Astropy Collaboration et al. 2013, 2018), Matplotlib (Hunter +2007), Pandas (McKinney 2010), scikit-learn (Pedregosa et al. +2011), SciPy (Virtanen et al. 2020), TOPCAT (Taylor 2005). +DATA AVAILABILITY +The data of Table 1, 2 and 3 are available in the online sup- +plementary material and at the CDS. +REFERENCES +´Abrah´am P., Bal´azs L. G., Kun M., 2000, A&A, 354, 645 +Alksnis A., 1958, Trudy Astrofiz. Lab. Riga, 7, 33 +Ambartsumian V. 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A., +Blaauw A., 1999, AJ, 117, 354 +This paper has been typeset from a TEX/LATEX file prepared by +the author. +APPENDIX A: FIGURES OF STELLAR GROUPS +In this section, we show the spatial, distance, tangential veloc- +ity distribution, colour–magnitude and colour–colour diagrams of +Groups 2–13. For a detailed description, see Fig. 3. +MNRAS 000, 000–000 (2021) + +12 +Szil´agyi et al. +800 +900 +1000 +Distance [pc] +0 +2 +4 +6 +8 +10 +(b) +40 +35 +30 +25 +20 +Vl, lsr [km/s] +5 +0 +5 +10 +15 +Vb, lsr [km/s] +(c) +180 +120 +60 +0 +60 +120 +180 +Angle [deg] +0 +2 +4 +6 +8 +(d) +0.0 +0.5 +1.0 +1.5 +G +GRP [mag] +0 +5 +10 +MG [mag] +(e) +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +2.0 +3.0 +4.0 +Age: 10.0 Myr +0.0 +0.5 +1.0 +H +Ks [mag] +0 +1 +2 +J +H [mag] +(f) +2 km/s +99°00' 98°45' +30' +15' +00' +8°30' +15' +00' +7°45' +30' +Galactic longitude +Galactic latitude +(a) +Figure A1. Same as Fig. 3 for Group 2. +800 +900 +1000 +Distance [pc] +0 +5 +10 +15 +(b) +40 +35 +30 +25 +20 +Vl, lsr [km/s] +5 +0 +5 +10 +15 +Vb, lsr [km/s] +(c) +180 +120 +60 +0 +60 +120 +180 +Angle [deg] +0 +2 +4 +6 +8 +(d) +0.0 +0.5 +1.0 +1.5 +G +GRP [mag] +0 +5 +10 +MG [mag] +(e) +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +2.0 +3.0 +4.0 +Age: 10.0 Myr +0.0 +0.5 +1.0 +H +Ks [mag] +0 +1 +2 +J +H [mag] +(f) +2 km/s +103°20' +00' +102°40' +20' +00' +8°00' +7°40' +20' +00' +6°40' +Galactic longitude +Galactic latitude +(a) +Figure A2. Same as Fig. 3 for Group 3. +MNRAS 000, 000–000 (2021) + +Cepheus OB2 +13 +800 +900 +1000 +Distance [pc] +0 +5 +10 +(b) +40 +35 +30 +25 +20 +Vl, lsr [km/s] +5 +0 +5 +10 +15 +Vb, lsr [km/s] +(c) +180 +120 +60 +0 +60 +120 +180 +Angle [deg] +0 +2 +4 +6 +8 +(d) +0.0 +0.5 +1.0 +1.5 +G +GRP [mag] +0 +5 +10 +MG [mag] +(e) +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +2.0 +3.0 +4.0 +Age: 15.8 Myr +0.0 +0.5 +1.0 +H +Ks [mag] +0 +1 +2 +J +H [mag] +(f) +2 km/s +99°00' 98°40' +20' +00' +97°40' +7°40' +20' +00' +6°40' +20' +Galactic longitude +Galactic latitude +(a) +Figure A3. Same as Fig. 3 for Group 4. +800 +900 +1000 +Distance [pc] +0 +5 +10 +15 +(b) +40 +35 +30 +25 +20 +Vl, lsr [km/s] +5 +0 +5 +10 +15 +Vb, lsr [km/s] +(c) +180 +120 +60 +0 +60 +120 +180 +Angle [deg] +0 +2 +4 +6 +8 +(d) +0.0 +0.5 +1.0 +1.5 +G +GRP [mag] +0 +5 +10 +MG [mag] +(e) +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +2.0 +3.0 +4.0 +Age: 10.0 Myr +0.0 +0.5 +1.0 +H +Ks [mag] +0 +1 +2 +J +H [mag] +(f) +2 km/s +104°30' +15' +00' +103°45' +30' +7°00' +6°45' +30' +15' +00' +Galactic longitude +Galactic latitude +(a) +Figure A4. Same as Fig. 3 for Group 5. +MNRAS 000, 000–000 (2021) + +14 +Szil´agyi et al. +800 +900 +1000 +Distance [pc] +0 +5 +10 +(b) +40 +35 +30 +25 +20 +Vl, lsr [km/s] +5 +0 +5 +10 +15 +Vb, lsr [km/s] +(c) +180 +120 +60 +0 +60 +120 +180 +Angle [deg] +0 +2 +4 +6 +8 +(d) +0.0 +0.5 +1.0 +1.5 +G +GRP [mag] +0 +5 +10 +MG [mag] +(e) +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +2.0 +3.0 +4.0 +Age: 10.0 Myr +0.0 +0.5 +1.0 +H +Ks [mag] +0 +1 +2 +J +H [mag] +(f) +2 km/s +103°30' +00' +102°30' +00' +6°30' +00' +5°30' +00' +4°30' +Galactic longitude +Galactic latitude +(a) +Figure A5. Same as Fig. 3 for Group 6. +800 +900 +1000 +Distance [pc] +0 +5 +10 +15 +(b) +40 +35 +30 +25 +20 +Vl, lsr [km/s] +5 +0 +5 +10 +15 +Vb, lsr [km/s] +(c) +180 +120 +60 +0 +60 +120 +180 +Angle [deg] +0 +2 +4 +6 +8 +(d) +0.0 +0.5 +1.0 +1.5 +G +GRP [mag] +0 +5 +10 +MG [mag] +(e) +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +2.0 +3.0 +4.0 +Age: 10.0 Myr +0.0 +0.5 +1.0 +H +Ks [mag] +0 +1 +2 +J +H [mag] +(f) +2 km/s +105°20' +00' +104°40' +20' +6°00' +5°40' +20' +00' +Galactic longitude +Galactic latitude +(a) +Figure A6. Same as Fig. 3 for Group 7. +MNRAS 000, 000–000 (2021) + +Cepheus OB2 +15 +800 +900 +1000 +Distance [pc] +0 +10 +20 +30 +(b) +40 +35 +30 +25 +20 +Vl, lsr [km/s] +5 +0 +5 +10 +15 +Vb, lsr [km/s] +(c) +180 +120 +60 +0 +60 +120 +180 +Angle [deg] +0 +2 +4 +6 +8 +(d) +0.0 +0.5 +1.0 +1.5 +G +GRP [mag] +0 +5 +10 +MG [mag] +(e) +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +2.0 +3.0 +4.0 +Age: 10.0 Myr +0.0 +0.5 +1.0 +H +Ks [mag] +0 +1 +2 +J +H [mag] +(f) +2 km/s +107°00' +106°30' +00' +105°30' +6°00' +5°30' +00' +4°30' +Galactic longitude +Galactic latitude +(a) +Figure A7. Same as Fig. 3 for Group 8. The blue diamonds show the Class II sources +identified with 2MASS data. +800 +900 +1000 +Distance [pc] +0 +5 +10 +(b) +40 +35 +30 +25 +20 +Vl, lsr [km/s] +5 +0 +5 +10 +15 +Vb, lsr [km/s] +(c) +180 +120 +60 +0 +60 +120 +180 +Angle [deg] +0 +2 +4 +6 +8 +(d) +0.0 +0.5 +1.0 +1.5 +G +GRP [mag] +0 +5 +10 +MG [mag] +(e) +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +2.0 +3.0 +4.0 +Age: 5.0 Myr +0.0 +0.5 +1.0 +H +Ks [mag] +0 +1 +2 +J +H [mag] +(f) +2 km/s +100°15' +00' +99°45' +30' +5°15' +00' +4°45' +Galactic longitude +Galactic latitude +(a) +Figure A8. Same as Fig. 3 for Group 9. +MNRAS 000, 000–000 (2021) + +16 +Szil´agyi et al. +800 +900 +1000 +Distance [pc] +0 +5 +10 +15 +(b) +40 +35 +30 +25 +20 +Vl, lsr [km/s] +5 +0 +5 +10 +15 +Vb, lsr [km/s] +(c) +180 +120 +60 +0 +60 +120 +180 +Angle [deg] +0 +2 +4 +6 +8 +(d) +0.0 +0.5 +1.0 +1.5 +G +GRP [mag] +0 +5 +10 +MG [mag] +(e) +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +2.0 +3.0 +4.0 +Age: 4.0 Myr +0.0 +0.5 +1.0 +H +Ks [mag] +0 +1 +2 +J +H [mag] +(f) +2 km/s +102°00' 101°30' +00' +6°00' +5°30' +00' +4°30' +Galactic longitude +Galactic latitude +(a) +Figure A9. Same as Fig. 3 for Group 10. +800 +900 +1000 +Distance [pc] +0 +10 +20 +(b) +40 +35 +30 +25 +20 +Vl, lsr [km/s] +5 +0 +5 +10 +15 +Vb, lsr [km/s] +(c) +180 +120 +60 +0 +60 +120 +180 +Angle [deg] +0 +2 +4 +6 +8 +(d) +0.0 +0.5 +1.0 +1.5 +G +GRP [mag] +0 +5 +10 +MG [mag] +(e) +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +2.0 +3.0 +4.0 +Age: 7.9 Myr +0.0 +0.5 +1.0 +H +Ks [mag] +0 +1 +2 +J +H [mag] +(f) +2 km/s +105°00' 104°40' +20' +00' +4°40' +20' +00' +3°40' +Galactic longitude +Galactic latitude +(a) +Figure A10. Same as Fig. 3 for Group 11. +MNRAS 000, 000–000 (2021) + +Cepheus OB2 +17 +800 +900 +1000 +Distance [pc] +0 +10 +20 +(b) +40 +35 +30 +25 +20 +Vl, lsr [km/s] +5 +0 +5 +10 +15 +Vb, lsr [km/s] +(c) +180 +120 +60 +0 +60 +120 +180 +Angle [deg] +0 +5 +10 +15 +(d) +0.0 +0.5 +1.0 +1.5 +G +GRP [mag] +0 +5 +10 +MG [mag] +(e) +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +2.0 +3.0 +4.0 +Age: 7.9 Myr +0.0 +0.5 +1.0 +H +Ks [mag] +0 +1 +2 +J +H [mag] +(f) +2 km/s +106°00'105°40' +20' +00' +104°40' +4°40' +20' +00' +3°40' +20' +Galactic longitude +Galactic latitude +(a) +Figure A11. Same as Fig. 3 for Group 12. YSOs from Szegedi-Elek et al. (2019) are +overplotted with green hexagons. +800 +900 +1000 +Distance [pc] +0 +20 +40 +(b) +40 +35 +30 +25 +20 +Vl, lsr [km/s] +5 +0 +5 +10 +15 +Vb, lsr [km/s] +(c) +180 +120 +60 +0 +60 +120 +180 +Angle [deg] +0 +10 +20 +30 +40 +50 +(d) +0.0 +0.5 +1.0 +1.5 +G +GRP [mag] +0 +5 +10 +MG [mag] +(e) +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +2.0 +3.0 +4.0 +Age: 5.0 Myr +0.0 +0.5 +1.0 +H +Ks [mag] +0 +1 +2 +J +H [mag] +(f) +2 km/s +100°00'99°40' +20' +00' +98°40' +4°40' +20' +00' +3°40' +20' +Galactic longitude +Galactic latitude +(a) +Figure A12. Same as Fig. 3 for Group 13. The black triangle indicate the transitional +disc-bearing source identified by the WISE colour indices. YSOs from Dias et al. (2002) +and Sicilia-Aguilar et al. (2005) are overplotted with green hexagons. +MNRAS 000, 000–000 (2021) + diff --git a/ttE0T4oBgHgl3EQfbQCD/content/tmp_files/load_file.txt b/ttE0T4oBgHgl3EQfbQCD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..10eefb407a9909908cbdabf9195a0a68f2c8a5d5 --- /dev/null +++ b/ttE0T4oBgHgl3EQfbQCD/content/tmp_files/load_file.txt @@ -0,0 +1,1412 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf,len=1411 +page_content='MNRAS 000, 000–000 (2021) Preprint 9 January 2023 Compiled using MNRAS LATEX style file v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 The Gaia view of the Cepheus OB2 association M´at´e Szil´agyi1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='2,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='3 1Konkoly Thege Mikl´os Astronomical Institute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Research Centre for Astronomy and Earth Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' E¨otv¨os Lor´and Research Network,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' H-1121 Budapest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Konkoly-Thege Mikl´os ´ut 15–17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Hungary 2CSFK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' MTA Centre of Excellence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' H-1121 Budapest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Konkoly Thege Mikl´os ´ut 15–17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Hungary 3ELTE E¨otv¨os Lor´and University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Institute of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' P´azm´any P´eter s´et´any 1/A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 1117 Budapest,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Hungary Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' in original form ZZZ ABSTRACT OB associations, birthplaces of the most luminous stars, are key objects for understanding the formation of high-mass stars and their effects on their environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The aim of this work is to explore the structure and kinematics of the Cepheus OB2 association and characterize the history of star formation in the region – in particular, the role of the Cepheus Bubble, surrounding Cepheus OB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Based on Gaia DR3 data we study the spatial and age distribution and kinematics of young stars in the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' We select candidate pre-main-sequence stars in the MG vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' GBP−GRP colour–magnitude diagram, and using a clustering algorithm, we identify 13 stellar groups belonging to Cep OB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Four groups, consisting of 10–13 Myr old low- and intermediate-mass stars, are located in the interior of the bubble, and are part of the oldest subsystem of the association Cep OB2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Younger groups are found on the periphery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The tangential velocities suggest that some groups on the periphery were born in an expanding system of star-forming clouds, whereas others have been formed due to the collision of their parent cloud with the expanding bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Key words: stars: pre-main-sequence – stars: formation – ISM: clouds – ISM: individual objects : Cepheus Bubble – The Galaxy: open clusters and associations: individual: Cepheus OB2 1 INTRODUCTION OB associations are unbound, dispersing groups of young, high luminosity stars, birthplaces of the high-mass stars of our Galaxy (Blaauw 1964;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' de Zeeuw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Wright 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' High-mass stars have a strong impact on the structure, physics and chemistry of the interstellar medium, thus their astrophysical importance is enormous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Studying the proper- ties of associations is important for understanding their for- mation and evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Dimensions of OB associations are 10– 100 parsecs, and star formation in them may proceed for tens of million years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Most of them consist of substructures of dif- ferent ages, including clusters of higher densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Substruc- tures of different ages are spatially and kinematically sep- arated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Several associations are surrounded by supershells, created by the interaction of expanding ionization fronts, stel- lar winds, and supernova explosions of short-lived, high-mass stars with the surrounding interstellar medium (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Bruh- weiler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Due to their 100–1000 pc dimensions su- pershells are able to propagate star formation across the in- terstellar medium on very large scales (McCray & Kafatos 1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The new astrometric and photometric data from the Gaia space telescope (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2016) can an- swer several long-standing questions related to the struc- ture, origin, and evolution of OB associations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' These ques- ⋆ E-mail: szilagyi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='mate@csfk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='org tions include the expansion of associations and propagating star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The low space density of stars in association (< 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='1 M⊙ pc−3) could reasonably be explained by the ex- pansion of originally dense, compact systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Gaia data have shown that not all of the observed structure can be explained by expansion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Wright & Mamajek 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Kounkel & Covey 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Melnik & Dambis (2020) found expansion in five of the 28 examined associations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Kinematic studies by Lim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' (2019, 2021, 2022) suggest that the formation of OB associations may result from structure for- mation driven by supersonic turbulence, rather than from the dynamical evolution of individual embedded clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The ob- served age differences of substructures can be explained by star formation in the gas compressed by the expanding ion- ization front and stellar wind bubble or supernova shock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Supershells may trigger star formation via several mecha- nisms: the expanding shock front may compress pre-existing clouds, or accumulate the ambient, low-density gas into a thin, dense layer, or may collide with ambient molecular clouds and with other bubbles (Ehlerov´a & Palouˇs 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' These different mechanisms may result in new stellar pop- ulations which differ from each other in structure, velocity compared to the energy source, and star-forming time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Supershells, associated with nearby OB associations give an opportunity to study in detail various scenarios of propagat- ing star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Precise parallaxes and proper motions, available in the Gaia data, allow us to study the spatial and © 2021 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='02346v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='SR] 6 Jan 2023 2 Szil´agyi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' kinematic substructures of OB associations, and explore the role of various types of triggered star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The subject of the present work is the Cepheus OB2 asso- ciation, discovered by Ambartsumian (1949).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The luminous stars defining the association occupy an area of some 10◦×10◦ around the Galactic position (l, b) = (102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='1, +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='6) (Kun, Kiss & Balog 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' At a mean distance of 900 pc (Contreras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2002) this angular size corresponds to a diameter of some 150 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Simonson (1968) identified 74 members of Cep OB2 based on spectroscopy and UBV photometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Further mem- bers were identified based on HIPPARCOS data (de Zeeuw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Binary frequency among the high-mass stars of Cep OB2 was examined by Peter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' They estab- lished that the multiplicity of massive (M ≥ 10 M⊙) stars seems to be significantly higher than that of intermediate- mass stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Two open clusters, the ∼4 Myr old Trumpler 37, embedded in the H ii region IC 1396, and the ∼ 12 Myr old NGC 7160 are the historical subsystems of Cep OB2 (Sicilia- Aguilar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2005, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Trumpler 37 itself is composed of several subgroups of different ages and structure, indicative of star formation triggered by interactions of the central O- type star HD 206267 with ambient clumps of molecular gas (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Sicilia-Aguilar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Cep OB2 is associated with the Cepheus Bubble, a super- shell of some 10◦ in angular diameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' It was identified in the IRAS 60 and 100 µm images by Kun, Bal´azs & T´oth (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' ´Abrah´am, Bal´azs & Kun (2000) studied the struc- ture and kinematics of the bubble based on the data of the Leiden/Dwingeloo neutral hydrogen survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The H i data re- vealed the expansion of the bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Observations of the re- gion in the 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='6-mm CO line by Patel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' (1998) revealed an expanding shell of some 120 pc in diameter, and containing ∼ 4×105 M⊙ gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The morphology and kinematics suggested that the bubble was created by the stellar winds of the first generation of high-mass stars of Cep OB2a during their life- time of 8–10 Myr, and accelerated by a supernova explosion ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='7 Myr ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The star-forming regions along the periphery of the bubble, such as Sh 2-129, IC 1396, Sh 2-140, L1188, were probably formed a few million years after the first gen- eration, by the fragmentation and collapse of the gas, com- pressed by the expanding stellar wind bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The molecular clouds associated with the bubble are birthplaces of the third generation of Cep OB2 (see Szegedi-Elek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' We study the Gaia DR3 (Gaia Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2022) data of the stars in the area of the Cepheus OB2 and the Cepheus Bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Our goal is to validate and extend the mem- bership list of Cep OB2, separate stellar populations of vari- ous ages and velocities, establish their relation to the expand- ing bubble, and find conclusions on the large-scale structure of Cep OB2 and on the history of star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' We de- fine our initial data set and the method of cluster selection in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The tools applied to characterize the clusters are described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Our results are described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 4, dis- cussed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 5, and briefly summarized in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2 CEP OB2 MEMBERSHIP BASED ON Gaia 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='1 High-mass members of Cep OB2 Formation mechanism and environment of high-mass stars above 10 M⊙ may differ from those of the lower mass stars (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The lists of luminous mem- bers of Cep OB2, published by Simonson (1968), Blaha & Humphreys (1989), and de Zeeuw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' (1999) contain 90 stars above 10 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' In order to compare the distribution and kinematics of high- and low-mass stars we compiled our list of massive stars from these (overlapping) tables, and searched for their Gaia DR3 counterparts within 1′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' In Gaia DR3, 52 historical members, supergiants and main- sequence stars earlier than B3 have distances between 800 and 1000 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Out of the 52, 8 stars have RUWE > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='4, in- dicating some issue with their astrometry, which could be caused by binarity (Lindegren 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Most of them are in- deed known binary or multiple systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' BD+62◦2078, HD 204827, HD 209744, HD 239743, Fabricius et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Pe- ter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Surface distribution and tangential velocities of these high-mass stars are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 1, and listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Several historical high-mass association members appear foreground (HD 239712, HD 199661) or background (HD 235618, HD 239758, HIP 109603, BD+53◦2387, HD 235783, HD 235795, HD 235813, HD 240010, BD+57◦2615, BD+53◦2784, µ Cep, HD 239978, HIP 111972) stars in Gaia DR3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='2 Members defined by distances and tangential velocities 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='1 Gaia sample We selected all sources from Gaia DR3 database with: (i) 96◦ < l < 108◦, (ii) 2◦ < b < 12◦, (iii) 800 < d < 1000 pc, (iv) ϖ/σϖ ≥ 10, (v) |µ⋆ α/σµ⋆α| ≥ 5, (vi) |µδ/σµδ| ≥ 5, (vii) RUWE ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='6, where l and b are the galactic longitude and latitude, d is the distance from Bailer-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' (2021), ϖ, µ⋆ α, µδ and σϖ, σµ⋆α, σµδ are the parallax, proper motion in right ascension and declination and their uncertainities, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' We selected the pre-main-sequence stars from the Gaia sample using their MG vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' GBP−GRP colour-magnitude di- agram, corrected for interstellar extinction (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' We cor- rected the Gaia colour indices and magnitudes of each star using the Python implementation1 of the 3D dust maps STIL- ISM (Lallement et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' STILISM gives the E(B-V) colour excess as a function of galactic coordinates and dis- tance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' We transformed the extinction into the Gaia bands with the coefficients in table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' of Wang & Chen (2019), as- suming RV = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Following the method described in Zari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' (2019) we have used the 10-Myr isochrone from PARSEC (Bressan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2012) to define an area in the de-reddened MG vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' GBP−GRP colour–magnitude diagram, occupied by young stellar objects (YSOs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Figure 2 suggests that the stellar sample between the dashed lines is contaminated with main-sequence stars, located in the 800–1000 pc distance interval but not related to the association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' To tighten the selection we searched for spatially and kinematically coherent groups in the sample selected from the colour–magnitude diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 1 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='com/edober/dust_maps_3d MNRAS 000, 000–000 (2021) Cepheus OB2 3 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Sample of the high-mass stars described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The full table is available as a supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Name Spectral type Gaia DR3 source id Distance µ⋆ α µδ RVGaia RVlit Memb RVref (pc) (mas y−1) (km s−1) HD 198895 B1Ve 2183107416725856640 864+12 −11 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='02 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='02 · · −23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 ± 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='4 1 7 HD 199308 B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5V 2189911847512999168 811+16 −13 −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='21 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='03 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='68 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='02 · · −23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 1 6 HD 200857 B3III 2188891844319403648 849+16 −12 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='02 −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='02 · · −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='4 1 6 HD 204150 B2III 2191963398772774144 863+20 −16 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='03 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='03 · · −18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 ± 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='9 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='3 7 HD 205139 B1Ib 2191787957952122752 855+32 −28 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='06 −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='75 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='05 · · −14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='9 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='3 6 Membership references: 1 - Simonson (1968), 2 - Blaha & Humphreys (1989), 3 - de Zeeuw et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' RV references: 1 - Wilson (1953), 2 - Petrie & Pearce (1961), 3 - Hilditch et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' (1982), 4 - Barbier-Brossat & Figon (2000), 5 - Pourbaix et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' (2004), 6 - Gontcharov (2006), 7 - Kharchenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' (2007), 8 - Boyajian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' (2007), 9 - de Bruijne & Eilers (2012), 10 - Holgado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' (2018), 11 - Katz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' (2022) 40 km/s 106° 104° 102° 100° 98° 96° 12° 10° 8° 6° 4° 2° Galactic longitude Galactic latitude 1 2 3 4 5 6 7 8 9 10 11 12 13 SH 2-140 NGC 7129 NGC 7160 SH 2-129 IC 1396 L1188 840 860 880 900 920 940 Distance [pc] Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Members of Cepheus OB2 overplotted on the Planck 857 GHz map of the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Circles show the members of the stellar groups found by HDBSCAN, and the white star symbols represent the high-mass (M > 10 M⊙) members of the Cepheus OB2 association from Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The arrows show the textbfmean tangential velocities of the groups and the OB stars, compared to the LSR, while the colour bar shows the textbfmean distance of each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The circle indicates the rough size and position of the Cepheus Bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' MNRAS 000, 000–000 (2021) 4 Szil´agyi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 0 1 2 3 4 5 6 GBP GRP [mag] 6 4 2 0 2 4 6 8 10 MG [mag] 100 101 102 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' De-reddened colour–magnitude diagram of the sources described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The dashed lines from Zari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' (2019) border the area defined to select the pre-main-sequence stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='2 Search for clusters We used the Python-implemented package of Hierarchi- cal Density-Based Spatial Clustering of Applications with Noise (HDBSCAN) clustering algorithm (McInnes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2017) to find overdensities in our sample obtained in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' An advantage of this clustering algorithm over other methods is that HDBSCAN can identify groups with various densities and arbitrary shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Main parameters of HDBSCAN are minimum cluster size, minimum samples and cluster selection method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Minimum cluster size de- fines the minimal number of data points a cluster must hold at least, while the minimum samples defines how con- servative a clustering is: the bigger the minimum samples is, the more points are considered as noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' At default, cluster selection method uses the Excess of Mass (EOM) approach to select one or two large and several smaller clus- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' For more homogeneous, smaller clusters, we can use the Leaf method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' For a detailed description of the algorithm, see the website2 of the package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' There are multiple ways for detecting stellar clusters in the Gaia data: some used ICRS coordinates, parallaxes and proper motions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Kounkel & Covey 2019), some used he- liocentric XYZ coordinates and tangential velocities multi- plied by a constant (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Kerr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Using the dis- tance of 900 pc from Contreras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' (2002) we transformed the Galactic coordinates l and b of the stars into 2D carte- sian coordinates compared to the centre of the studied field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' We also calculated the vl,LSR and vb,LSR Galactic tangen- tial velocity components compared to the Local Standard of Rest (LSR) using the (UV W)⊙ values from Sch¨onrich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The fifth dimension is the distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Due to the different units we standardized the data in each dimension by subtracting the mean and dividing with the largest stan- dard deviation of the dimensions that share the same unit, 2 https://hdbscan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='readthedocs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='io/en/latest/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='html which are the distance and vl,LSR respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' We adopted minimum cluster size = 25 and minimum samples = 25 with the Leaf method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 3 CHARACTERIZATION OF THE GROUPS HDBSCAN found originally 10 groups, consisting altogether of 874 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The final Groups 2 and 4, 5 and 7, and 11 and 12 were originally merged into three larger groups, respectively, but their spatial distribution suggested that each of the three can be split into two smaller subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The Groups 2 and 4 pairs we split at b = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='8◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' For dividing the Group 5–7 and 11–12 pairs we used two-component Gaussian mixture model from scikit-learn (Pedregosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2011) to fit to their spatial distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' These actions resulted in 13 groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' We regard these 13 groups as substructures of Cep OB2, and examine how their properties reflect the history of star for- mation in the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Figure 1 shows the distribution of the clustered sources in Galactic coordinates, overplotted on the Planck 857 GHz image of the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The mean tangential velocities of the groups, compared to the LSR, are also in- dicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The colouring shows the mean distances of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The mean coordinates, distances and velocities of the groups are shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Known clusters and clouds from the lit- erature, associated with the individual groups, are indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Surface distribution of stars in individual groups, along with their tangential velocity vectors compared to the mean veloc- ity of the groups listed in Table 2, are displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 3a and Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' A1a–A12a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Distance histograms and vl,LSR vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' vb,LSR di- agrams plotted in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 3b and A1bc–A12bc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Table 3, contain- ing a detailed list of group members, is available in machine- readable form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='1 Radial velocities Gaia DR3 provides radial velocities for stars with GRVS < 14 mag (Katz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' From the 874 group member stars, 191 have radial velocites measured by Gaia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The number of the stars with available radial velocities for each group are provided in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='2 Variable stars Gaia DR3 provides a list containing 24 types of variable stars identified with machine learning methods (see Rimol- dini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Eyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2022, for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' We cross- matched our group members with it, and found that 355, ∼ 41 % of the member stars are classified as YSO-candidates in Marton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' These stars are overplotted with red circles in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 3a and Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' A1a–A12a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Furthermore 60 stars were identified as RS Canum Venaticorum type variable stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Additional 17 stars were classified as either eclipsing binaries (Mowlavi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2022), solar-like variables, α2 CVn/magnetic chemical peculiar star/rapidly oscillating Am/Ap star/SX Ari variable stars (Distefano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2022) or δ Sct/γ Dor/SX Phe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Column best class name in Table 3 contains these information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' MNRAS 000, 000–000 (2021) Cepheus OB2 5 800 900 1000 Distance [pc] 0 2 4 6 (b) 40 35 30 25 20 Vl, lsr [km/s] 5 0 5 10 15 Vb, lsr [km/s] (c) 180 120 60 0 60 120 180 Angle [deg] 0 2 4 6 8 (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 G GRP [mag] 0 5 10 MG [mag] (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 Age: 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 Myr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content="0 H Ks [mag] 0 1 2 J H [mag] (f) 2 km/s 105°40' 20' 00' 104°40' 10°40' 20' 00' 9°40' Galactic longitude Galactic latitude (a) Figure 3." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' a: Distribution of the members of Group 1, plotted on the Planck 857 GHz image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The arrows indicate the tangential ve- locities of the stars compared to the mean tangential velocity of the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Gold squares and the red pentagon indicate the Class II and Class I sources classified by the WISE colour indices, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' YSO-candidates from Marton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' (2022) are overplotted with red circles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' YSOs from Dahm & Hillenbrand (2015) and Kun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' (2009) are overplotted with green hexagons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' b: Histogram of distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' c: Tangential velocity components compared to the LSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' d: Distribution of the angles between the radial vector of each star from the group centroid and relative tangential velocities (see a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' e: De-reddened MG vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' GBP–GRP colour–absolute magnitude diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The solid line shows the best-fitted PARSEC isochrone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Dashed lines represent evolutionary tracks from 1 to 20 Myrs of several stellar masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' For 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='4 M⊙ and below we plotted CIFIST isochrones (Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' f : De-reddened 2MASS colour– colour-diagram showing sources with quality criteria ’AAA’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The plots of the other groups are presented in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='3 Colour–absolute magnitude diagrams Figures 3e and A1e–A12e show the M G vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' G−GRP de- reddened colour–absolute magnitude diagrams (CMD) of each group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' We downloaded PARSEC isochrones (Bressan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2012) with logarithmic ages from 6 to 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 with a step of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The CMDs show the best-fitted isochrones, as well as evolutionary tracks for several initial masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' For 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='4 M⊙ and below we plotted CIFIST (Baraffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2015) tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 1 0 1 2 3 4 5 W2 W3 [mag] 1 0 1 2 W1 W2 [mag] Class I Class II transitional disc discless Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' AllWISE colour–colour diagram of the group members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The dashed lines border the areas of Class I and II stars, according to Koenig & Leisawitz (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='4 2MASS colour–colour diagrams We crossmatched our data with the 2MASS (Skrutskie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2006) database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' We transformed ICRS coordinates of the stars from Gaia’s J2016 epoch into J2000 epoch to search for 2MASS counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Then we searched for coinciding 2MASS sources within 1 arcsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' We calculated the extinctions in the JHKs bands of 2MASS by the method described in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' De-reddened J−H vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' H −K s colour–colour diagrams (CCD) are displayed in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 3f and A1f–A12f, showing only the stars with photometric quality criteria ’AAA’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 WISE colour–colour diagram We also searched for counterparts in the AllWISE (Cutri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2021) catalogue to identify potential disc-bearing stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' We found counterparts of 733 stars within 1 arcsec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' We fol- lowed the methods described by Koenig & Leisawitz (2014) to find disc-bearing stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' From these AllWISE sources, 316 ful- fil the quality criteria essential for constructing their W1−W2 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' W2−W3 colour–colour diagram, displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' One star is classified as a Class I source, 58 of the sources are iden- tified as Class II sources and one star is classified as transi- tional disc bearing star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Supplemented with 2MASS data we identified 4 more Class II sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' These stars are plotted as red pentagons, gold squares, black triangles and blue dia- monds, respectively in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 3a and A1a–A12a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='6 Internal motions of the groups We examined the internal motions of the substructures of Cep OB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Following the method applied by Lim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' (2019) we determined the angle between the radial vector of each star from the centre position of the group and its relative velocity vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Figures 3d and A1d–A12d show the histograms of these angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Clustering of the angles around 0◦ is indicative of radial expansion of the group, whereas peaks around ±180◦ suggest contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' MNRAS 000, 000–000 (2021) 6 Szil´agyi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Mean coordinates, distances, velocities, E(B-V) colour excesses and ages for each stellar group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The last five columns show the number of group member stars, stars with measured radial velocities, identified as YSO (Marton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2022), RS Canum Venaticorum variable, and other type of variables, listed in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The table is available as supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Group Associated object lmean bmean dmean vl,LSR,mean vb,LSR,mean E(B-V) age N⋆ Nrad NYSO NRS Nother (deg) (pc) (km s−1) (mag) (Myr) 1 NGC 7129 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='36 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='99 876 ± 15 −28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='4 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='2 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='21 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 33 6 16 3 1 2 SH 2-129, UBC 385 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='55 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='04 829 ± 13 −30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='49 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 27 12 5 2 1 3 UBC 10a, UBC 167 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='59 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='36 901 ± 6 −29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='43 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 33 13 3 7 1 4 · · 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='30 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='33 842 ± 13 −30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='48 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='8 33 11 7 3 1 5 NGC 7160 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='01 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='38 902 ± 11 −28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='8 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='29 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 64 19 7 7 4 6 UBC 10b 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='76 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='64 944 ± 7 −30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='1 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='34 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 26 8 1 2 0 7 [KPR2005] 117 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='76 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='63 906 ± 4 −26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='32 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 22 6 5 3 0 8 SH 2-140, Pismis–Moreno 1 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='60 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='21 906 ± 5 −26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='4 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='61 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 45 11 19 1 0 9 IC 1396, L1116 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='87 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='02 908 ± 9 −27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='32 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 30 9 13 3 0 10 UPK 169, Theia 131 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='52 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='92 841 ± 7 −26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='27 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 39 15 5 0 0 11 Alessi–Teutsch 5 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='47 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='18 873 ± 14 −27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='9 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='37 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='9 93 22 21 6 1 12 L1188, [BDS2003] 30 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='33 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='00 872 ± 22 −25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='2 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='8 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='9 134 23 66 7 3 13 IC 1396, Trumpler 37 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='34 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='78 904 ± 24 −33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='9 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='35 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 295 36 187 16 5 MNRAS 000, 000–000 (2021) Cepheus OB2 7 4 THE SUBSTRUCTURES OF CEP OB2 The historical substructures of Cep OB2 are NGC 7160, sur- rounded by a number of evolved high-mass stars (Cep OB2a), and IC 1396 (Cep OB2b) (Simonson & van Someren Greve 1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' These represent two epochs of star formation in the volume of the association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The Cepheus Bubble has revealed new probable subgroups and suggested connection between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Our search for stellar groups in the Gaia data re- sulted in further subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Except Group 4 they coincide with known clusters, but most of them have not been cou- pled to Cep OB2 in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' We describe in this section the Gaia view of the substructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Most of clustered stars have G < 18 mag, only 33 of the 874 group members are fainter than this limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The foreground AV extinctions are between 1–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='8 mag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' These data suggest that the masses of the detected group members are above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='6 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='7 M⊙ in a 5 and 10 Myr old group, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Group 1, the northernmost group, contains the well-known, compact young cluster NGC 7129, and a few stars to the west and north of the cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Kun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' (1987) associated NGC 7129 with the Cepheus Bubble based on morphologi- cal considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Gaia data, in accordance with the VLBA parallax published in Reid et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' (2014) confirm that this cluster is located on the surface of the Cepheus Bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' HDBSCAN identified 33 members, 8 of them are classified as Class II stars, and 1 is identifies as a Class I source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The colour–magnitude diagram (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 3e) suggests an age of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 3f suggests significant intra-cluster extinction, which may affect the appearance of the colour–magnitude diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Eleven of the member stars appear in Dahm & Hil- lenbrand (2015) and one in Kun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' (2009) as YSOs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' They are overplotted with green hexagons in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Groups 2 and 4 are located at the western edge of the Cepheus Bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' According to the CMDs in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' A1e and A3e they are 10 and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='8 Myr old, and most of their mem- bers are more massive than 1 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Unlike other groups, the detected members of Group 2 are aligned evenly along the fit- ted isochrone: the number of stars above 2 M⊙ is nearly iden- tical with that in the 1 M⊙ < M < 2 M⊙ interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Group 2 is the nearest group with a mean distance of 829±13 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' It coin- cides with the cluster UBC 385, identified by Castro-Ginard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' (2020), and contains several B8–A0 type stars, classi- fied as Cep OB2 members by Alksnis (1958).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' It is the central cluster of the ring-shaped H ii region Sh 2-129, excited by the O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 IV type (Sota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2014) component of the young triple system HD 202214 (Balega et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Though the distance of HD 202214, projected at the cluster centre is uncertain, its spectral type and brightness suggest its membership.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' None of the group members were classified as disc-bearing stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The neighbouring Group 4 was merged with Group 2 by HDB- SCAN, but a closer inspection has shown that they differ slightly in distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Groups 3, 5, 6 and 7 are projected inside the Cepheus Bub- ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The colour–magnitude diagrams suggest ages of 10 Myr, thus these groups belong to the first generation of Cep OB2 (Sicilia-Aguilar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Groups 3, 6, and 7 coincide with the known clusters UBC 10a, UBC 10b (Castro-Ginard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2018), and [KPR2005] 117 (Kharchenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2005), respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The groups are projected close to each other, but differ in distance, space velocity and mass distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Compared to the neighbouring groups Groups 5 and 6 have large lat- itudal tangential velocity components, and high proportion of low-mass (M ≤ 1 M⊙) members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The angular distribution of the tangential velocity vectors in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' A4d shows the ex- pansion of Groups 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' These groups lack interstellar matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The dark cloud L1178 is located at the south-eastern edge of the region, but the extinction towards the cloud rises around 700 pc, suggesting that they are not related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Group 8 is associated with the cluster Pismis–Moreno 1 (Pis- mis, Moreno & Hasse 1979) that contains HD 211880, the B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5V type exciting star of SH 2-140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The colour–magnitude diagram indicates an age around 10 Myr, in accordance with earlier estimates (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Cantat-Gaudin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Four stars were designated as Class II star by the method of Koenig & Leisawitz (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Star formation is on-going in the molecular cloud bordered by the H ii region (Gutermuth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Quite a few Class II Spitzer sources, without reliable Gaia data, are projected within this group (see fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 1 in Guter- muth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2009), indicative of either their membership or the overlapping of the older Pismis–Moreno 1 and the em- bedded S140 cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Groups 9 and 13 are located at the IC 1396 region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Several dark clouds of Lynds (1962) are found in the region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The ex- tinction towards L1086, L1096, L1102 and L1116 rises around 900 pc, indicating that all these clouds may be related to IC 1396.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Group 13 corresponds to the cluster Trumpler 37, containing the exciting star of IC 1396.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' We have identified 39 Class II sources and a transitional disc-bearing star in Group 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The 295 members include 69 low-mass YSOs iden- tified by Sicilia-Aguilar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' (2005), and the intermediate- mass YSO IRAS 21365+5713 from Contreras et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Figure A12d indicates the expansion of Tr 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' According to the CMDs, the ages of groups are around 5 Myrs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Group 10 corresponds to the cluster UPK 169 (Cantat- Gaudin & Anders 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' It also appears in Kounkel & Covey (2019) as part of Theia 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' It is projected halfway between NGC 7160 and Tr 37, but its distance is some 60 pc smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The mean age of the group members is about 4 Myr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Based on AllWISE data none of the 39 members have infrared ex- cess, characteristic of YSO discs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The brightest member of the group is HD 207538, an O9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='7IV spectral-type star (Sota et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Groups 11 and 12 correspond to the known clusters Alessi– Teutsch 5 and [BDS2003] 30, respectively (Kharchenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Bica et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Eleven stars of Group 12 were identi- fied as Class II sources based on AllWISE and 2MASS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The most luminous member of Group 11 is the B1IV type component of the eclipsing binary V* V446 Cep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The star- forming dark cloud L1188 is associated with Group 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Ten of Group 12 members were classified as young stars in Szegedi- Elek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The CMDs suggest ages of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='9 Myr for both groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' MNRAS 000, 000–000 (2021) 8 Szil´agyi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 106° 104° 102° 100° 98° 96° 12° 10° 8° 6° 4° 2° Galactic longitude Galactic latitude 10 km/s 1 2 3 4 5 6 7 8 9 10 11 12 13 4 6 8 10 12 14 Age [Myr] Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2D structure of Cepheus OB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Large circles and smaller star symbols represent the groups from Table 2 and the high-mass stars from Table 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Arrows indicate the mean tangential velocities, compared to the mean motion of the system, and the colour bar indicates group ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 5 OVERALL VIEW OF CEPHEUS OB2 The line-of-sight dimension of the system of groups is some 120 pc, similar to the apparent diameter of the Cepheus Bub- ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' To get an insight into the internal motions of the stars associated with the Cepheus Bubble we calculated the mean motion from the velocity data in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The mean velocities are vl = −16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='13 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='01 km s−1and vb = −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='43 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='87 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 5 we represent the tangential velocities of the groups compared to the mean values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The precision of available stel- lar radial velocities is not sufficient for studying the three- dimensional velocity structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The colour–magnitude diagrams show that star formation started some 10 million years ago in the studied region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Groups 3, 5, 6, 7 are located inside the volume bordered by the Cepheus Bubble, in the volume occupied by the evolved high-mass members of Cep OB2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' These groups are new com- ponents of Cep OB2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The mean distance of this subsystem is 900 pc, the line-of-sight size is about 50 pc, and the dispersion of the tangential velocities is some 10 km s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The tangential velocities indicate the expansion of the largest group, most probably due to the disappearance of the the parental cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Expansion of the whole Cep OB2a cannot be demonstrated by the method applied in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The spatial and velocity structure suggest that each group had its own parent cloud clump, forming stars independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Most of the high-mass stars were not selected by HDBSCAN as group members, the only exception is the B1 III type HD 208218, the most massive member of Group 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The reason may be that the velocities of the high-mass stars are influenced by their probable mul- tiplicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Groups 1, 2, 4, 8, 9, 12 and 13 line along the apparent perimeter of the Bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Except Groups 2, 4 and 8, these groups are younger than Cep OB2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The diversity of their stellar contents, ages, and velocities suggest a complex history of star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The expansion of the by now extinct H ii region of Cep OB2a might have affected the formation of these groups, and/or the evolution of circumstellar discs of the group members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' MNRAS 000, 000–000 (2021) Cepheus OB2 9 Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Column description of the table containing the members of the stellar groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The list is available as supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Column Description group Group identifier name Star identifier source id Gaia DR3 source id TMASS 2MASS identifier AllWISE AllWISE identifier ra RA at J2016 dec DEC at J2016 l Galactic longitude b Galactic latitude d Distance from Bailer-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' (2021) dmin 16th percentile of distance posterior from Bailer-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' (2021) dmax 84th percentile of distance posterior from Bailer-Jones et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' (2021) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='parallax ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='Parallax ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='parallax error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='Error of the parallax ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='pmra ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='Proper motion in right ascension ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='pmra error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='Error of the proper motion in RA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='pmdec ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='Proper motion in declination ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='pmdec error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='Error of the proper motion in DEC ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='radial velocity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='Radial velocity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='radial velocity error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='Error of the radial velocity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='ruwe ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='Renormalized unit-weight error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='phot g mean mag ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='Magnitude in G band ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='phot bp mean mag ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='Magnitude in GBP band ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='phot rp mean mag ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='Magnitude in GRP band ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='Jmag ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='Magnitude in 2MASS J band ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='Hmag ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='Magnitude in 2MASS H band ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='Kmag ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='Magnitude in 2MASS K s band ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='eb v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='E(B-V) from STILISM ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='allwise class ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='YSO type by the classification of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='Koenig & Leisawitz (2014) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='best class name ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='Variable star type appear in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='gaiadr3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='vari classifier result (Rimoldini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2022) Figure 5 shows that the Groups 11, 12, and 13 move ra- dially outwards from the centre, suggesting that their parent clouds were accelerated by the shock wave from the expand- ing ionization front of the OB stars of Cep OB2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Each of these groups is younger than Cep OB2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Groups 12 and 13 are associated with large amounts of molecular gas (Szegedi- Elek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Sicilia-Aguilar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2015, respectively), sites of active star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The colour–magnitude diagrams sug- gest that Groups 11 and 12 are nearly coeval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The difference in the proportion of disc-bearing members may result from the different environments (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Dale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The small groups Group 2 (UBC 385) and 4 are projected near the westernmost wall of the Cepheus Bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Gaia data have shown that these groups are some 70 pc closer to us than the central groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Their ages, suggested by the Gaia CMDs, are similar to that of Cep OB2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Group 2 is centred on the H ii region S129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Since O-type stars and H ii regions are generally younger than 10 Myr (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Tremblin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2014), we may speculate that the exciting star HD 202214 was born later than the lower-mass members of the group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' SPH simu- lations by Dale et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' (2013) show that such a situation may occur in molecular clouds near ionizing stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Precise astro- metric data of the HD 202214 triple system and more data on the cluster population may clarify the nature of star-forming processes in the region of S129, and its connection with the Cepheus Bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Group 8 (Pismis–Moreno 1) is another 10 Myr old group associated with an H ii region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The main-sequence lifetime of the B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 V type exciting star HD 211880 is compatible with this age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The molecular cloud bordered by the ion- ization front contains the S140 embedded cluster (Guter- muth et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' This morphology suggests that probably Group 8 and HD 211880 itself were formed in the same molec- ular cloud, and star formation propagates radially outwards from the central Cep OB2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' However, the role of the expand- ing Cepheus Bubble in the formation of this cluster is un- likely because of the apparently similar ages of Cep OB2a and Group 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Group 9 is a small group at the outskirts of IC 1396, asso- ciated with dark clouds, bordering the H ii region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Its age is similar to that of Tr 37, whereas their tangential velocities, and probably star formation histories are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Group 10 (UPK 169, Cantat-Gaudin & Anders 2020) is a ∼ 4 Myr old small group, containing the O9 V type star HD 207538.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' It is projected inside the Bubble, and its distance of 841 ± 7 pc suggests association with the near wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' In spite of its young age, no disc-bearing low-mass star has been de- tected in this group, probably due to the disruptive radiation of the hot star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The parent cloud of this group was proba- bly overrun by the expanding ionization front of Cep OB2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Similarly, collision of the expanding bubble with the south- ernmost edge of an ambient giant molecular cloud, located at Galactic latitudes 10–13◦ (Grenier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 1989), might have played a role in the formation of Group 1 (NGC 7129).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 6 SUMMARY We have studied the stars from Gaia DR3 in the region of the Cepheus OB2 association between 96◦ < l < 108◦, 2◦ < b < 12◦ and 800 < d < 1000 pc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' We selected candi- date pre-main-sequence stars using isochrones in the MG vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' GBP−GRP colour–magnitude diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' We used HDBSCAN to find groups in the selected sample, and found 13 stellar groups, consisting of 874 stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' These 13 groups, located be- tween 830–940 pc, are subsystems of Cepheus OB2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 355 of the clustered stars are classified as candidate YSOs by Marton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Based on WISE data we identified one Class I source, sixty-two Class II sources and one with transitional disc, according to the classification of Koenig & Leisawitz (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' We found that star formation in the volume of the as- sociation started some 10 million years ago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' In addition to the evolved high-mass stars, the oldest subsystem Cep OB2a contains four groups of low- and intermediate-mass stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The tangential velocities suggest the expansion of the group containing NGC 7160, however the expansion of the whole Cep OB2a cannot be proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' In addition to the historical younger subsystem Cep OB2b, Trumpler 37, several other, 2–8 Myr old groups can be found in the volume of the association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Most of them might have been formed under the influence of the expanding bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Spatial and kinematic structure of the region suggest var- ious means of interaction between the star-forming clouds MNRAS 000, 000–000 (2021) 10 Szil´agyi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' and the expanding bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The tangential velocities show that the parent clouds of the young clusters Trumpler 37, Alessi–Teutsch 5 and [BDS2003] 30 (Groups 11, 12 and 13) have been accelerated by the expanding ionization front of Cep OB2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Other groups reveal places where the expand- ing ionization front encountered ambient clouds (Groups 1, 8, and 10, aka NGC 7129, S140, and UPK 169).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The age of the cluster Pismis–Moreno 1 (Group 8, S140) suggests that formation of this group was probably independent of the ex- panding bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The relation of S129 to the Cepheus Bubble is uncertain, because of the uncertain distance of the exciting star and the discrepancy of the age of the central cluster and the exciting star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We thank the anonymous referee for the careful and insight- ful review of our manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' We thank Lajos Bal´azs for his help in statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' This work has made use of data from the European Space Agency (ESA) mission Gaia (https: //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='int/web/gaia/dpac/consortium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Funding for the DPAC has been provided by national institutions, in par- ticular the institutions participating in the Gaia Multilateral Agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' This work was supported by the ESA PRODEX Contract nr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 4000129910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' For this work we have used Astropy (Astropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2013, 2018), Matplotlib (Hunter 2007), Pandas (McKinney 2010), scikit-learn (Pedregosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2011), SciPy (Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 2020), TOPCAT (Taylor 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} 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+page_content=', de Bruijne J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=', Brown A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=', Blaauw A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=', 1999, AJ, 117, 354 This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' APPENDIX A: FIGURES OF STELLAR GROUPS In this section, we show the spatial, distance, tangential veloc- ity distribution, colour–magnitude and colour–colour diagrams of Groups 2–13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' For a detailed description, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' MNRAS 000, 000–000 (2021) 12 Szil´agyi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 800 900 1000 Distance [pc] 0 2 4 6 8 10 (b) 40 35 30 25 20 Vl, lsr [km/s] 5 0 5 10 15 Vb, lsr [km/s] (c) 180 120 60 0 60 120 180 Angle [deg] 0 2 4 6 8 (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 G GRP [mag] 0 5 10 MG [mag] (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 Age: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 Myr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content="0 H Ks [mag] 0 1 2 J H [mag] (f) 2 km/s 99°00' 98°45' 30' 15' 00' 8°30' 15' 00' 7°45' 30' Galactic longitude Galactic latitude (a) Figure A1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 3 for Group 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 800 900 1000 Distance [pc] 0 5 10 15 (b) 40 35 30 25 20 Vl, lsr [km/s] 5 0 5 10 15 Vb, lsr [km/s] (c) 180 120 60 0 60 120 180 Angle [deg] 0 2 4 6 8 (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 G GRP [mag] 0 5 10 MG [mag] (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='4 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content="0 H Ks [mag] 0 1 2 J H [mag] (f) 2 km/s 103°20' 00' 102°40' 20' 00' 8°00' 7°40' 20' 00' 6°40' Galactic longitude Galactic latitude (a) Figure A2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 3 for Group 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' MNRAS 000, 000–000 (2021) Cepheus OB2 13 800 900 1000 Distance [pc] 0 5 10 (b) 40 35 30 25 20 Vl, lsr [km/s] 5 0 5 10 15 Vb, lsr [km/s] (c) 180 120 60 0 60 120 180 Angle [deg] 0 2 4 6 8 (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content="0 H Ks [mag] 0 1 2 J H [mag] (f) 2 km/s 99°00' 98°40' 20' 00' 97°40' 7°40' 20' 00' 6°40' 20' Galactic longitude Galactic latitude (a) Figure A3." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 3 for Group 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 800 900 1000 Distance [pc] 0 5 10 15 (b) 40 35 30 25 20 Vl, lsr [km/s] 5 0 5 10 15 Vb, lsr [km/s] (c) 180 120 60 0 60 120 180 Angle [deg] 0 2 4 6 8 (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 G GRP [mag] 0 5 10 MG [mag] (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 4.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 3 for Group 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' MNRAS 000, 000–000 (2021) 14 Szil´agyi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 800 900 1000 Distance [pc] 0 5 10 (b) 40 35 30 25 20 Vl, lsr [km/s] 5 0 5 10 15 Vb, lsr [km/s] (c) 180 120 60 0 60 120 180 Angle [deg] 0 2 4 6 8 (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content="0 H Ks [mag] 0 1 2 J H [mag] (f) 2 km/s 103°30' 00' 102°30' 00' 6°30' 00' 5°30' 00' 4°30' Galactic longitude Galactic latitude (a) Figure A5." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 3 for Group 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 800 900 1000 Distance [pc] 0 5 10 15 (b) 40 35 30 25 20 Vl, lsr [km/s] 5 0 5 10 15 Vb, lsr [km/s] (c) 180 120 60 0 60 120 180 Angle [deg] 0 2 4 6 8 (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 G GRP [mag] 0 5 10 MG [mag] (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 Age: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 Myr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content="0 H Ks [mag] 0 1 2 J H [mag] (f) 2 km/s 105°20' 00' 104°40' 20' 6°00' 5°40' 20' 00' Galactic longitude Galactic latitude (a) Figure A6." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 3 for Group 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' MNRAS 000, 000–000 (2021) Cepheus OB2 15 800 900 1000 Distance [pc] 0 10 20 30 (b) 40 35 30 25 20 Vl, lsr [km/s] 5 0 5 10 15 Vb, lsr [km/s] (c) 180 120 60 0 60 120 180 Angle [deg] 0 2 4 6 8 (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 G GRP [mag] 0 5 10 MG [mag] (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 Age: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 Myr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content="0 H Ks [mag] 0 1 2 J H [mag] (f) 2 km/s 107°00' 106°30' 00' 105°30' 6°00' 5°30' 00' 4°30' Galactic longitude Galactic latitude (a) Figure A7." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 3 for Group 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The blue diamonds show the Class II sources identified with 2MASS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 800 900 1000 Distance [pc] 0 5 10 (b) 40 35 30 25 20 Vl, lsr [km/s] 5 0 5 10 15 Vb, lsr [km/s] (c) 180 120 60 0 60 120 180 Angle [deg] 0 2 4 6 8 (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 G GRP [mag] 0 5 10 MG [mag] (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 Age: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 Myr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content="0 H Ks [mag] 0 1 2 J H [mag] (f) 2 km/s 100°15' 00' 99°45' 30' 5°15' 00' 4°45' Galactic longitude Galactic latitude (a) Figure A8." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 3 for Group 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' MNRAS 000, 000–000 (2021) 16 Szil´agyi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 800 900 1000 Distance [pc] 0 5 10 15 (b) 40 35 30 25 20 Vl, lsr [km/s] 5 0 5 10 15 Vb, lsr [km/s] (c) 180 120 60 0 60 120 180 Angle [deg] 0 2 4 6 8 (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 G GRP [mag] 0 5 10 MG [mag] (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 Age: 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 Myr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content="0 H Ks [mag] 0 1 2 J H [mag] (f) 2 km/s 102°00' 101°30' 00' 6°00' 5°30' 00' 4°30' Galactic longitude Galactic latitude (a) Figure A9." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 3 for Group 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 800 900 1000 Distance [pc] 0 10 20 (b) 40 35 30 25 20 Vl, lsr [km/s] 5 0 5 10 15 Vb, lsr [km/s] (c) 180 120 60 0 60 120 180 Angle [deg] 0 2 4 6 8 (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 G GRP [mag] 0 5 10 MG [mag] (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 Age: 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='9 Myr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content="0 H Ks [mag] 0 1 2 J H [mag] (f) 2 km/s 105°00' 104°40' 20' 00' 4°40' 20' 00' 3°40' Galactic longitude Galactic latitude (a) Figure A10." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 3 for Group 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' MNRAS 000, 000–000 (2021) Cepheus OB2 17 800 900 1000 Distance [pc] 0 10 20 (b) 40 35 30 25 20 Vl, lsr [km/s] 5 0 5 10 15 Vb, lsr [km/s] (c) 180 120 60 0 60 120 180 Angle [deg] 0 5 10 15 (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 G GRP [mag] 0 5 10 MG [mag] (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 Age: 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='9 Myr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content="0 H Ks [mag] 0 1 2 J H [mag] (f) 2 km/s 106°00'105°40' 20' 00' 104°40' 4°40' 20' 00' 3°40' 20' Galactic longitude Galactic latitude (a) Figure A11." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 3 for Group 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' YSOs from Szegedi-Elek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' (2019) are overplotted with green hexagons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 800 900 1000 Distance [pc] 0 20 40 (b) 40 35 30 25 20 Vl, lsr [km/s] 5 0 5 10 15 Vb, lsr [km/s] (c) 180 120 60 0 60 120 180 Angle [deg] 0 10 20 30 40 50 (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 G GRP [mag] 0 5 10 MG [mag] (e) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 Age: 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 Myr 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content="0 H Ks [mag] 0 1 2 J H [mag] (f) 2 km/s 100°00'99°40' 20' 00' 98°40' 4°40' 20' 00' 3°40' 20' Galactic longitude Galactic latitude (a) Figure A12." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' Same as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' 3 for Group 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' The black triangle indicate the transitional disc-bearing source identified by the WISE colour indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' YSOs from Dias et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' (2002) and Sicilia-Aguilar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' (2005) are overplotted with green hexagons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} +page_content=' MNRAS 000, 000–000 (2021)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ttE0T4oBgHgl3EQfbQCD/content/2301.02346v1.pdf'} diff --git a/uNA0T4oBgHgl3EQfLv9Z/vector_store/index.pkl b/uNA0T4oBgHgl3EQfLv9Z/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..58063f67acfc4ceb1aef198e3211f35be29d67d0 --- /dev/null +++ b/uNA0T4oBgHgl3EQfLv9Z/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:97b99cad92f63d81de4f1c5898541125c12680d1c1b005535c95801119f9a15d +size 119331 diff --git a/vdAzT4oBgHgl3EQfB_qZ/content/tmp_files/2301.00952v1.pdf.txt b/vdAzT4oBgHgl3EQfB_qZ/content/tmp_files/2301.00952v1.pdf.txt new 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Pandey1,a) +1Graduate Engineer Trainee, Larsen & Toubro Limited, +Faridabad, Haryana, India. +a)Corresponding author: rishav160999@gmail.com +Abstract +In this paper, the production and spectroscopic investigation of +Mercury and Radon isotopes was performed using complete fusion re- +actions neutron evaporation residues and multi-nucleon transfer reac- +tion at the mass-separator MASHA. The MASHA setup is installed on +the beam line of Cyclotron U-400M at Flerov Laboratory of Nuclear +Reactions (FLNR) in Joint Institute for Nuclear Research (JINR), +Dubna, Russia. The isotopes produced in complete fusion reactions +148Sm(40Ar, xn)188−xHg, 166Er(40Ar, xn)206−xRn and multi-nucleon +transfer reaction 48Ca + 242Pu were passed through the magneto- +optical system of MASHA setup with charge state Q=+1 and were +separated on the basis of their mass to charge ratio. For the detection +of these isotopes, a position sensitive Si detector was used. Further, +the experimental data obtained were analysed and spectroscopic in- +vestigations were carried out. +Keywords: Spectroscopic investigation, Complete Fusion Reaction, Multi- +Nucleon Transfer Reaction, Mass separator, MASHA, Position sensitive detector. +1 +INTRODUCTION +The MASHA (Mass Analyzer of Super Heavy Atoms) setup has been designed +as a mass-separator with the resolving power of about 1700, which allows mass +1 +arXiv:2301.00952v1 [nucl-ex] 3 Jan 2023 + +identification of super-heavy nuclides. The setup uses the solid ISOL (Isotope Sep- +aration On-Line) method. The work on the project means the analysis of real data +collected from the experiments of complete fusion reactions neutron evaporation +residues 40Ar + 148Sm → 188−xHg + xn, 40Ar + 166Er → 206−xRn + xn and +multi-nucleon transfer reaction 48Ca + 242Pu using the α-decay chains from the +position sensitive Si detector. +An analysis was done on the experimental data from the above stated reac- +tions to calculate the masses of identified isotopes(which are results of the above +nuclear reactions), their half life, Alpha Branching Ratio (ABR), energy of α-decay +(Ealpha) and their probability to decay with a specific amount of energy. Using +these experimental data, one dimensional α-decay energy spectrum was plotted +and its peak analysis was performed. Further, using the 1D histograms (α-decay +energy spectrum), a heatmap (two dimensional energy-position graph) was ob- +tained for the isotopes. Finally, a comparison was drawn between the theoretical +values and experimental values of Ealpha, and conclusions were made. The former +was obtained from the nuclide chart, while the latter was obtained by performing +experiments at the MASHA facility. +2 +LAYOUT OF MASHA SETUP AND ITS MAIN PARTS +The MASHA setup is shown in Fig.1. It consists of a Target Box with a Hot +Catcher, an ECR Ion Source, Magneto-Optical System (or mass to charge ratio +analyzer), and a Position Sensitive Si Detector (Detection and Control System). It +also consists of an intermediate plane (F1), and for detailed monitoring of isotopes +an additional strip detector has been installed at this middle plane also, apart +from the detector present at focal plane (F2). Each part of the MASHA facility +is explained in detail below. For in-depth knowledge on MASHA setup, one can +refer this paper [1]. +2.1 +TARGET BOX +The target box consists of a rotating disc divided into 6 sectors, which are sputtered +with target material(s) as shown in Fig.2a. The disc rotates with a frequency of +25Hz [1]. The high energetic projectile particle ejected from U-400M cyclotron +collides with the target material present in rotating disc to induce some kind of +nuclear reaction. The products of the nuclear reaction are stopped by the hot +catcher which is discussed below. +2 + +Figure 1: Schematic of the MASHA facility: D1, D2, D3a, D3b are dipole magnets, Q1, Q2, Q3 are +quadrupole lenses, S1, S2 are sextupole lenses. +2.2 +HOT CATCHER +The diagram of hot catcher is shown in Fig.2b. The below setup mainly consists +two components, one is poly-graphene heater and the other one is absorber mate- +rial. The latter is usually made up of thin film of graphite or carbon nanotubes +heated by the former upto a temperature of 1800 − 2000oC. As depicted in figure +the absorber is installed in front of the heater at a distance of 2 mm along the +beam axis of MASHA [2]. The products of the nuclear reaction is stopped by +the absorber material, vaporized to gaseous form and are passed to the ECR ion +source. Fig.3 shows the complete schematic of target box with hot catcher. +(a) Rotating target disc inside the target box. +(b) A thin hot catcher installed just before poly-graphene +heater. +Figure 2 +3 + +Detector +Focal +plane F2 +ECRion +Hot catcher +source +Focal plane F1 +D1 +Q1 +Q2 +Ion +D2 +S2 +beam +Q3 +D3b +S1 +D3aPoligraphene +Thin +heater +hot cateherFigure 3: Complete schematic of target box with hot catcher. +2.3 +ECR ION SOURCE +The ECR (Electron Cyclotron Resonance) ion source with a microwave oscillation +frequency of 2.45 GHz [1–4], acts as an ionization chamber of MASHA spectrome- +ter. It ionizes the atoms of gaseous isotopic products of nuclear reaction to a charge +state Q=+1, and accelerates them to an energy of 38 KeV using three electrode +system [1]. The ionized atoms gets converted to beam and are then separated by +magneto-optical system of the MASHA spectrometer. +2.4 +MAGNETO-OPTICAL SYSTEM +The magneto-optical system separates the beam of ions on the basis of their mass +to charge ratio. The magnetic separation of heavy nuclei is performed using four +dipole magnets(D1, D2, D3a, D3b), three quadrupole lenses (Q1, Q2, Q3) and +two sextupole lenses (S1, S2) as shown in Fig.1 [2]. Once, the heavy nuclei gets +separated they are then detected at different strips of position sensitive Si detector. +2.5 +POSITION SENSITIVE Si DETECTOR +The position sensitive Si detector is a multiple detector system used to detect +the separated heavy nuclei. It is installed at the focal plane (F2) of the MASHA +setup. A clear view of the position sensitive Si detector is shown in Fig.4. The +front detector has a dimension of 240x35 mm2 and it consists of 192 strips. The +upper and lower detector consists of 64 strips each while the left and right lateral +detector consists of 16 strips each. Each strip has a width of 1.25 mm and each +detector has a thickness of 0.3 mm [3–5]. +4 + +Diaphragm +Target +Ion beam +To ECR +ion source +Hot +catcher +Target driverFigure 4: Position sensitive Si detector. 1-front detector, 2-upper detector, 3-lower detector, 4-lateral +detector. +3 +SCIENCE BEHIND THE EXPERIMENT +The test experiments carried out at FLNR are: +1. 40Ar + 148Sm → 188−xHg + xn +2. 40Ar + 166Er → 206−xRn + xn +3. 48Ca + 242Pu → (Any element whose Z varies from 20-114) +The first and second nuclear reactions are complete fusion reactions neutron evap- +oration residues. In such types of reactions the product nucleus formed has no. +of protons exactly equal to no. of protons of projectile particle + no. of protons +of target nucleus. Here, all the nucleons participate in the reaction. While the +third one is a Multi-Nucleon Transfer Reaction (MNTR). In such nuclear reac- +tions different nuclides can be formed whose atomic no. ranges from atomic no. +of projectile particle to sum of atomic nos. of projectile particle + target nucleus. +This means not all nucleons participate in this reaction and can lead to formation +of any possible product. The N/Z ratio of product nucleus can be higher or lesser +than than the optimal ratio required for its stability (i.e. It can be proton rich or +neutron rich). +The U-400M cyclotron installed at FLNR, JINR is used to accelerate projectile +particle (40Ar & 48Ca) to a very high velocity, with an energy ˜240 MeV (for 40Ar ++ 148Sm) and with energy ˜198 MeV (40Ar + 166Er). The high energetic projectile +particle enters into the MASHA setup and induce a nuclear reaction by colliding +with target material sputtered in rotating disc present in target box of MASHA +facility. The products of nuclear reaction are isotopes of Hg (for 40Ar + 148Sm) +5 + +240 mm +2 +mm +5 +3 +4 +3 +4and Rn (for 40Ar + 166Er and 48Ca + 242Pu) which are stopped by the absorber +material of hot catcher. +The absorber material is generally made up of thin film of graphite or carbon +nanotubes which is heated to around 1800−2000oC by means of IR radiations com- +ing out from poly-graphene heater as well as by a direct current passing through +the absorber. This absorber stops the isotopic products of nuclear reaction, vapor- +izes them and their respective atoms diffuses through this absorber material into +the vacuum volume of the hot catcher. Moving along the vacuum pipe, they reach +the ECR ion source [3]. This ECR ion source acts as an ionization chamber of +MASHA setup where the atoms of gaseous isotopic products gets ionized to charge +state Q=+1 and further they are accelerated with the help of three electrode sys- +tem. (The three electrode system consists of one positive electrode, one negative +electrode and one more negative electrode. Hence, an electric field is established +from positive electrode to negative electrode. So, when a charged particle (here, +ion) moves in the direction of electric field, it gets accelerated. +The product isotopes are then separated by their M/Q ratio in the magneto- +optical system of MASHA setup and at last they reach to the focal plane (F2) of +the position sensitive Si detector and are detected at different strip numbers. (i.e. +different isotopes are detected at different strip numbers). +Now, the science is that the separated heavy nuclei undergoes α-decay to pro- +duce daughter nuclei and it’s exactly the alpha particles (with different energies) +given out by both parent nucleus and its daughter nuclei which are detected at +unique strip nos. of position sensitive Si detector. The detector used is a hybrid +pixel detector of the TIMEPIX type, with high resolution and sensitivity which +can detect even a single α or β particle. So, from the experimental data, we plot +α-decay energy spectrum for those strips where an isotope was detected. From +this spectrum (α-decay energy vs. No. of counts) we analyse the prominent peaks +and calculate their α-decay energy (Ea) values. The base peak with maximum +no. of α particles (with constant energy) is our point of interest as it could be +any one of the separated nuclei. Now, using the table of nuclides, we find which +isotope (of product of nuclear reaction) undergoes α-decay with energy very close +to it. That particular isotope will be the one detected at a unique strip number. +Ones, the isotope gets detected, then its mass, ABR, daughter nuclei can easily be +investigated using the table of nuclides. In the same way, one can detect all the +isotopes of an element which is the product of a nuclear reaction. +In this work, a two dimensional energy-position graph (called heatmap) for +all three test experiments has also been analysed. This graph gives a clear un- +derstanding that which isotope is detected at which strip no. and corresponding +to that particular isotope, how many alpha particles (counts) are detected with a +constant energy. This constant energy is the energy of α-decay of that isotope. +6 + +4 +SPECTROSCOPIC INVESTIGATION OF MERCURY ISO- +TOPES USING FULL-FUSION REACTION 148Sm(40Ar, xn)188−xHg +The complete fusion reaction of 148Sm(40Ar, xn)188−xHg was carried out at MASHA +setup. The target material sputtered in rotating disc was 148Sm and the prod- +ucts of the nuclear reaction were isotopes of Hg. However, only the long-lived +isotopes of Hg were detected whose half-life was greater than average separation +time (1.8±0.3 s) used by ISOL method for this reaction [1,3]. +4.0.1 +PRODUCTION OF 180Hg +Figure 5: α-decay energy spectrum of 180Hg and its decay products. +In the α-decay energy spectrum of 180Hg, the most prominent peaks corre- +sponding to parent nucleus and daughter nuclei were analysed. Here, the label +180Hg(Ealpha = 6120KeV, Counts = 57.5, 99.87%, ABR = 0.48) gives a lot of in- +formation. ‘Ealpha = 6120KeV ’ is the energy of α-particle released during the +alpha decay of isotope 180Hg. Counts = 57.5 indicates the no. of α-particles +detected at a strip of detector. ABR = 0.48 is the Alpha Branching Ratio of +180Hg which means ‘The probability of 180Hg to undergo α-decay is 0.48’. It is +to be noted that a nucleus can decay with multiple α-decay energies and for same +reason, here 99.87% means 99.87% of 180Hg decays with energy = 6120 KeV. Now, +coming to the analysis part, in Fig.5 we observe that 180Hg(t 1 +2 = 2.58s) is peaked +at 6120 KeV and its daughter nuclei 176Pt(t 1 +2 = 6.3s), formed due to α-decay of +180Hg and 180Au(t 1 +2 = 8.1s), formed due to Electron Capture (EC) in 180Hg are +peaked at 5750 KeV and 5680 KeV respectively. +4.0.2 +PRODUCTION OF 181Hg +In Fig.6, it is depicted that 181Hg(t 1 +2 = 3.54s) is peaked at 6000 KeV, with prob- +ability of decaying in this energy being 87% and ABR=0.30. While its α-decay +7 + +70 +Alpha Decay Counts +60 +50 +40 +30 +20 +10 +0. +5500 +5600 +5700 +5800 +5900 +6000 +6100 +6200 +Alpha Decay Energy (KeV)Figure 6: α-decay energy spectrum of 181Hg and its decay products. +product 177Pt(t 1 +2 = 11s) is peaked at 5500 KeV, with probability to decay in this +energy being 88.5% and ABR=0.056. +4.0.3 +PRODUCTION OF 182Hg +Figure 7: α-decay energy spectrum of 182Hg and its decay products. +In Fig.7, we can observe the base peak of 182Hg(t 1 +2 = 10.83s) at 5860 KeV. +However, we can also see a peak of 182Hg at 5690 KeV. This is due to the fact +that a radioactive heavy nucleus can decay with multiple α-decay energies with +former being highly probable (99%) and later being less probable (0.57%). It is +also observed that α-decay product of 182Hg is 178Pt(t 1 +2 = 21.1s) which is peaked +at 5450 KeV. +4.0.4 +PRODUCTION OF 183Hg +In Fig.8, the two primary daughters of 183Hg, 183Au(t 1 +2 = 42.8s) and 179Pt(t 1 +2 = +21.1s) formed due to EC and α-decay of parent nucleus are peaked at 5340 KeV, +and 5190 KeV respectively. The parent nucleus 183Hg(t 1 +2 = 9.4s) is peaked at +5890 KeV. +8 + +250 +Alpha Decay Counts +200 +100 +50 - +0. +5400 +5500 +5600 +5700 +5800 +5900 +6000 +6100 +6200 +Alpha Decay Energy (KeV)600 +Alpha Decay Counts +500 +400 +300 +200 +100 +0 +5400 +5500 +5600 +5700 +5800 +5900 +6000 +Alpha Decay Energy (KeV)Figure 8: α-decay energy spectrum of 183Hg and its decay products. +4.0.5 +PRODUCTION OF 184Hg +Figure 9: α-decay energy spectrum of 184Hg and its decay products. +In α-decay energy spectrum of 184Hg shown in Fig.9, the base peak correspond- +ing to 184Hg(t 1 +2 = 30.9s) is obtained at 5530 KeV with probability of decaying +with this energy being 99.44%. While 0.4% is the chance that 184Hg decays with +energy 5360 KeV. 180Pt(t 1 +2 = 56s) being the α-decay product of 184Hg is peaked +at 5160 KeV. +4.0.6 +PRODUCTION OF 185Hg +In Fig.10, it is observed that 185Hg(t 1 +2 = 49.1s) is peaked at 5650 KeV and 5540 +KeV with probability to decay with these energies being 96% and 4% respectively. +The daughter nucleus formed due to EC in 185Hg is 185Au(t 1 +2 = 4.25months), +which is peaked at 5080 KeV with ABR of 0.0026 in α-decay energy spectrum. +Also, the α-decay product of 185Hg, which is 181Pt(t 1 +2 = 52s) was not observed in +this spectrum because of its very low probability to undergo α-decay. +9 + +250 +Alpha Decay Counts +200 +2 150 +100 +50 +0. +5100 +5200 +5300 +5400 +5500 +5600 +Alpha Decay Energy (KeV)700 +Alpha Decay Counts +600 +500 +S +400 +uni +Col +300 +200 +100 +0 +5000 +5100 +5200 +5300 +5400 +5500 +5600 +5700 +5800 +5900 +6000 +Alpha Decay Energy (KeV)Figure 10: α-decay energy spectrum of 185Hg and its decay products. +4.0.7 +HEATMAP OF Hg ISOTOPES +Figure 11: Heatmap of Hg isotopes. +Production of Hg isotopes in complete fusion reaction 148Sm(40Ar, xn)188−xHg +is shown by a two-dimensional graph as shown in Fig.11. Through this energy- +position graph, it’s evident that different isotopes of Hg produced in aforemen- +tioned nuclear reaction are detected at unique strip numbers. However, only those +isotopes of Hg are visible whose half-life is greater than average separation time +(1.8±0.3 s) used by ISOL method for this reaction. Along the Y-axis we see α- +decay energy of each detected isotope and through the color associated with each +10 + +1800 +Alpha Decay Counts +1500 +Counts +1200 +900 +600 +300 +0 +5000 +5100 +5200 +5300 +5400 +5500 +5600 +5700 +Alpha Decay Energy (KeV)Counts +6,97 +975.0 +181Hg +183 +6,57 - +180 +Hg +780,3 + Energy (MeV) +6,17 +585,6 +Alpha Decay I +5,77 +390,9 +5.37 +196,2 +4,97 +i +1,500 +4,57 +100 +导 +0 +140 +180 +Detector Strip Numberisotope, we get the value of no. of counts of α-particles detected by matching it +with the color scale bar on right hand side. +The experimental values of Ealpha is compared with its theoretical values ob- +tained from table of nuclides in Table 1. It is observed that the % change in their +values is even less than 0.3%. So, we conclude that the spectroscopic investigation +performed for Hg isotopes in reaction 148Sm(40Ar, xn)188−xHg is almost accurate. +Table 1: Comparison between theoretical and experimental values of Ealpha(in KeV) of Hg isotopes +produced in reaction 148Sm(40Ar, xn)188−xHg. +Nucleus +Theo. Ealpha +Exp. Ealpha +∆% +180Hg +6119 +6120 +0.016 +181Hg +6006 +6000 +0.099 +182Hg +5867 +5860 +0.119 +183Hg +5904 +5890 +0.230 +184Hg +5535 +5530 +0.090 +185Hg +5653 +5650 +0.053 +5 +SPECTROSCOPIC INVESTIGATION OF RADON ISOTOPES +USING FULL-FUSION REACTION 166Er(40Ar, xn)206−xRn +A complete fusion reaction was performed between high energetic projectile parti- +cle (40Ar) ejected from the window of U-400M cyclotron with an energy ˜198 MeV +and the target material 166Er present in the form of rotating disc in the target +box of MASHA facility. The products of the nuclear reaction were isotopes of Rn +which were detected at focal plane (F2) of the position sensitive Si detector. Fur- +ther, using the experimental data obtained from the detector and control system of +MASHA, their α-decay energy spectrum and energy-position graphs were plotted. +5.0.1 +PRODUCTION OF 201Rn +Figure 12: α-decay energy spectrum of 201Rn and its decay products. +11 + +1800 +Alpha Decay Counts +1500 +1200 +jun +Cou +900. +600. +300 +0. +5500 +5600 +5700 +5800 +5900 +6000 +6100 +6200 +6300 +6400 +6500 +6600 +6700 +6800 +6900 +Alpha Decay Energy (KeV)In Fig.12, we see that the isotope 201Rn(t 1 +2 = 3.8s) is peaked at 6760 KeV. Its +only daughter nucleus formed due to α-decay, 197Po(t 1 +2 = 53.6s) with ABR=0.44 +is peaked at 6280 KeV. However, we can see another 197Po(t 1 +2 = 25.8s) with +ABR=0.84 peaked at 6380 KeV. The probability of occurrence of both 197Po(t 1 +2 = +53.6s) and 197Po(t 1 +2 = 25.8s) are very high with 100% and 99.3% respectively. +5.0.2 +PRODUCTION OF 202Rn +Figure 13: α-decay energy spectrum of 202Rn and its decay products. +As depicted in Fig.13, the two primary daughters of 202Rn, 202At(t 1 +2 = 184s) +and 198Po(t 1 +2 = 1.77months) formed due to EC and α-decay of parent nucleus are +peaked at 6070 KeV, and 6180 KeV respectively. The parent nucleus 202Rn(t 1 +2 = +10.0s) is peaked at 6630 KeV. +5.0.3 +PRODUCTION OF 203Rn +Figure 14: α-decay energy spectrum of 203Rn and its decay products. +During the investigation of 203Rn isotope, it was observed that two different +forms of 203Rn with different t 1 +2 and ABR were detected at same strip of position +12 + +Alpha Decay Counts +2000 + 1500 +luno +8 1000 +500 +5500 +5600 +5700 +5800 +5900 +6000 +6100 +6200 +6300 +6400 +6500 +6600 +6700 +6800 +Alpha Decay Energy (KeV)700 +Alpha Decay Counts +600 +500 +400 +ino +300 +200. +100 +0 +5800 +5900 +6000 +6100 +6200 +6300 +6400 +6500 +6600 +6700 +Alpha Decay Energy (KeV)sensitive detector [see Fig.14]. +203Rn(t 1 +2 = 28s) with ABR=0.8 was peaked at +6550 KeV, while 203Rn(t 1 +2 = 45s) with ABR=0.66 was peaked at 6500 KeV. The +daughter nucleus 199Po(t 1 +2 = 4.17months) formed due to the α-decay emission of +203Rn is peaked at 6060 KeV in α-decay energy spectrum. +5.0.4 +PRODUCTION OF 204Rn +Figure 15: α-decay energy spectrum of 204Rn and its decay products. +From the α-decay energy spectrum of 204Rn, it is noticed that the parent +isotope 204Rn(t 1 +2 = 1.24months) is peaked at 6400 KeV, while its α-decay product +200Po(t 1 +2 = 11.5months) is peaked at 5840 KeV as shown in Fig.15. +5.0.5 +PRODUCTION OF 205Rn +Figure 16: α-decay energy spectrum of 205Rn and its decay products. +From the Fig.16, it is evident that 205Rn(t 1 +2 = 170s) isotope undergoes α- +decay to release 6270 KeV of energy. +The probability that it face α-decay to +produce 201Po(t 1 +2 = 8.9months) with Ealpha = 5760KeV is 0.23. However, this +heavy radioactive nucleus also tries to achieve stability through electron capture, +13 + +Alpha Decay Counts +200 +ounts +150 +100 +50. +5700 +5800 +5900 +6000 +6100 +6200 +6300 +6400 +6500 +Alpha Decay Energy (KeV)45 +Alpha Decay Counts +15 +0 +5700 +5800 +5900 +6000 +6100 +6200 +6300 +6400 +Alpha Decay Energy (KeV)and for same the probability is 0.754 (as seen from table of nuclides). 205At(t 1 +2 = +26.2months) which is the result of EC is peaked at 5910 KeV in α-decay energy +spectrum of 205Rn isotope. +5.0.6 +HEATMAP OF Rn ISOTOPES (166Er(40Ar, xn)206−xRn) +The energy-position graph of Rn isotopes from 201Rn to 205Rn is shown in Fig.17. +These isotopes are the result of nuclear reaction 40Ar + 166Er → 206−xRn + xn +and are separated on the basis of their M/Q ratio in the magneto-optical system +of mass-separator and finally detected at different strip numbers of detector. From +this heatmap, we see Ealpha decreases from 201Rn to 205Rn. We also conclude that +201Rn and 202Rn isotopes are greatly produced in the reaction with high no. of +counts, while 205Rn is produced in relatively less number. The α-decay daughter +nuclei 197Po and 198Po from parent isotopes 201Rn and 202Rn respectively can +also be seen at different strip numbers. +Figure 17: Heatmap of Rn isotopes using complete-fusion reaction 166Er(40Ar, xn)206−xRn. +The comparison between theoretical and experimental values of Ealpha for Rn +isotopes produced in complete fusion reaction 166Er(40Ar, xn)206−xRn is shown in +Table 2. It is noticed here that the % change between these values is less than +0.3%, so our analysis on this subject is nearly accurate. +14 + +Count +7,04 +875,0 +203Rn +201 +204Rn +6,67 - +700,3 +Energy (MeV) +205 +197 +Rn +6,30 - +525,6 +Alpha Decay E +5,92 +350,9 +5,55 +176,2 +5,18 +4,81 +1,500 +0 +4 +0 +8 +100 +120 +140 +160 +180 +Detector Strip NumberTable 2: Comparison between theoretical and experimental values of Ealpha(in KeV) of Rn isotopes +produced in reaction 166Er(40Ar, xn)206−xRn. +Nucleus +Theo. Ealpha +Exp. Ealpha +∆% +201Rn +6773 +6760 +0.192 +202Rn +6639.5 +6630 +0.143 +203Rn(ABR = 0.80) +6549 +6550 +0.015 +203Rn(ABR = 0.66) +6499.3 +6500 +0.011 +204Rn +6418.9 +6400 +0.294 +205Rn +6262 +6270 +0.128 +6 +SPECTROSCOPIC INVESTIGATION OF RADON ISOTOPES +USING MNTR 48Ca + 242Pu +Unlike complete fusion reactions discussed above, a MNTR can have any possible +product nucleus. +However, in the reaction of 48Ca + 242Pu under some fixed +conditions, new neutron-rich Rn isotopes were produced near the neutron N=126 +shell closure configuration, using MNTR. The isotopes produced were identified +first, later their spectroscopic investigations were carried out. However, it was +observed that only those Rn isotopes reached the detector and were identified +which lived at least 35 ms while others decayed in their path. +6.0.1 +PRODUCTION OF 212Rn +Figure 18: α-decay energy spectrum of 212Rn and its decay products. +From the spectrum of 212Rn(t 1 +2 = 23.9months) isotope shown in Fig.18, it is +evident that 99.95% of 212Rn isotopes undergoes α-decay with energy 6250 KeV. +Also, its ABR=1, which means it will only undergo α-decay and no other forms +of decay. 212Rn isotope has N=126 shell configuration, which gives it a very high +stability. Its daughter nucleus 208Po(t 1 +2 = 2.9years) was not observed in α-decay +energy spectrum. +15 + +1200 +Alpha Decay Counts +900 +212Rn(Ealpha=6250KeV, Counts=1138.8, 99.95%, ABR=1.0) +600 +300 +5500 +0 +5600 +5700 +5800 +5900 +6000 +6100 +6200 +6300 +6400 +Alpha Decay Energy (KeV)6.0.2 +PRODUCTION OF 218Rn +The only daughter nucleus of 218Rn due to its α-decay is 214Po(t 1 +2 = 164.3µs), +which is peaked at 7660 KeV. The parent isotope 218Rn(t 1 +2 = 35ms) is peaked at +7110 KeV and 6530 KeV with probability of decaying with these energies being +99.87% and 0.127% respectively. As discussed earlier, the reason behind why we +see two different peaks of 218Rn isotope in α-decay energy spectrum is that the +decay energy of alpha particle is not always fixed. It can have multiple energies, +and there is a fixed probability for a certain energy to get released. Also, we see +that the counts associated with 218Rn isotope and its daughter nucleus is very low +[see Fig.19] because of their low half-life. +Figure 19: α-decay energy spectrum of 218Rn and its decay products. +6.0.3 +PRODUCTION OF 219Rn +Figure 20: α-decay energy spectrum of 219Rn and its decay products. +From the α-decay energy spectrum shown in Fig.20, we see that the parent +isotope, 219Rn(t 1 +2 = 3.96s) is peaked at 6790 KeV, 6600 KeV, 6530 KeV, and +16 + +Alpha Decay Counts +8 +6. +Counts +2 +0 +6400 +6600 +6800 +7000 +7200 +7400 +7600 +7800 +Alpha Decay Energy (KeV)Alpha Decay Counts +500- +400 +Counts +300 +200 - +100 +4000 +4500 +5000 +5500 +6000 +6500 +7000 +7500 +8000 +8500 +9000 +9500 +10000 +Alpha Decay Energy (KeV)6400 KeV with probability to decay with these energies are 79.4%, 12.9%, 0.12%, +and 7.5% respectively. All α-decay energies have the same ABR=1. The α-decay +product of 219Rn is 215Po(t 1 +2 = 1.78ms), and is peaked at 7360 KeV. 211Pb(t 1 +2 = +36.1months) which is the alpha-daughter of 215Po could not be seen in this α- +decay energy spectrum because it only undergoes β− decay. However, its daughter +nucleus, 211Bi(t 1 +2 = 2.14months) is peaked at 6250 KeV, with ABR=0.99 and +probability to decay with this energy is 16.23%. +6.0.4 +HEATMAP OF Rn ISOTOPES (48Ca + 242Pu) +Figure 21: Heatmap of Rn isotopes using MNTR 48Ca + 242Pu. +The energy-position graph of Rn isotopes produced in MNTR 48Ca + 242Pu is +shown in Fig.21. 212Rn and 219Rn isotopes were largely produced in the reaction +with high counts, while 218Rn isotope with few counts is barely visible. This is +because of low half-life of 218Rn(t 1 +2 = 35ms), that major part of it decayed before +reaching the detector. The α-daughter of 219Rn, 215Po was also detected at some +strip number. A huge gap between 212Rn and 218Rn isotopes could be explained +in terms of less half-life of Rn isotopes from A=213-217. Their half-life is even +less than 35 ms which is more or less the average separation time used by ISOL +method for this reaction. So, we conclude that only long-lived Rn isotopes were +detected which lived at least 35 ms [1], while others decayed before reaching the +detector. +17 + +9,98 +Counts +920,0 +215 +9,00. +218 +Rn +219Rn +Energy (MeV) +736,3 +8,02. +552.6 +7,04. +368,9 +6,05. +185.2 +5,07. +4,09 +1,500 +0 +a +3 +4 +5 +0 +110 +3 +140 +Detector Strip NumberTable 3: Comparison between theoretical and experimental values of Ealpha(in KeV) of Rn isotopes +produced in reaction 48Ca + 242Pu. +Nucleus +Theo. Ealpha +Exp. Ealpha +∆% +212Rn +6264 +6250 +0.223 +218Rn +7129.2 +7110 +0.269 +219Rn +6819.1 +6790 +0.427 +The theoretical and experimental values of Ealpha for Rn isotopes produced via +MNTR 48Ca + 242Pu is summarized in Table 3. < 0.3% change in these values is +observed for 212Rn and 218Rn isotopes, while for 219Rn, the % change is < 0.5%. +So, our analysis on this reaction is almost accurate. +7 +RESULTS AND CONCLUSIONS +In this entire work, the production and spectroscopic investigation of Hg and +Rn isotopes was performed using full fusion reactions 148Sm(40Ar, xn)188−xHg, +166Er(40Ar, xn)206−xRn and multi-nucleon transfer reaction 48Ca + 242Pu. The +final product in all these reactions were isotopes of Hg and Rn. The experimen- +tal data obtained from the MASHA setup were analysed and 1D α-decay energy +spectrum graphs were plotted for those strips of detector which had detected any +isotopic product of nuclear reaction. Further, this 1D histograms were used to +plot a 2D energy-position graph, separately for Hg and Rn isotopes. The masses +of super-heavy nuclei which were detected at different strips of Si based Position +Sensitive Detector (PSD), have also been identified. Using 1D histograms and nu- +clide chart, the values of Ealpha, ABR, Counts, and the probability to decay with +a specific amount of energy were calculated for all isotopic products of nuclear +reactions studied in this paper. +8 +ACKNOWLEDGEMENTS +The author expresses his gratitude to Mr. +Viacheslav Vedeneev, Flerov Labo- +ratory of Nuclear Reactions, Joint Institute for Nuclear Research for providing +experimental data obtained from the MASHA setup. The author is also grateful +to his parents for motivating him throughout his research work. +9 +STATEMENTS AND DECLARATIONS +The author has no competing interests to declare that are relevant to the content +of this article. +18 + +References +[1] V. Y. Vedeneev, A. Rodin, L. Krupa, A. Belozerov, E. Chernysheva, +S. Dmitriev, A. Gulyaev, A. Gulyaeva, D. Kamas, J. Kliman, et al., The cur- +rent status of the masha setup, Hyperfine Interactions 238 (1) (2017) 1–14. +[2] M. Mamatova, A. Seitkali, E. Kudaibergenova, A. Rodin, L. Krupa, E. Cherny- +sheva, V. Vedeneev, A. Novoselov, A. Podshibyakin, V. Salamatin, et al., Study +of production stability of radon and mercury isotopes in complete fusion reac- +tions at the mass-separator masha by “solid hot catcher” technique, in: AIP +Conference Proceedings, Vol. 2163, AIP Publishing LLC, 2019, p. 070002. +[3] A. +Rodin, +A. +Belozerov, +E. +Chernysheva, +S. +Dmitriev, +A. +Gulyaev, +A. Gulyaeva, M. Itkis, J. Kliman, N. Kondratiev, L. Krupa, et al., Separa- +tion efficiency of the masha facility for short-lived mercury isotopes, Hyperfine +Interactions 227 (1) (2014) 209–221. +[4] A. Rodin, A. Belozerov, D. Vanin, V. Y. Vedeneyev, A. Gulyaev, A. Gulyaeva, +S. Dmitriev, M. Itkis, J. Kliman, N. Kondratiev, et al., Masha separator on +the heavy ion beam for determining masses and nuclear physical properties +of isotopes of heavy and superheavy elements, Instruments and Experimental +Techniques 57 (4) (2014) 386–393. +[5] A. Rodin, E. Chernysheva, S. Dmitriev, A. Gulyaev, D. Kamas, J. Kliman, +L. Krupa, A. Novoselov, Y. T. Oganessian, A. Op´ıchal, et al., Features of +the solid-state isol method for fusion evaporation reactions induced by heavy +ions, in: Exotic Nuclei: Proceedings of the International Symposium on Exotic +Nuclei, World Scientific, 2020, pp. 437–443. +19 + diff --git a/vdAzT4oBgHgl3EQfB_qZ/content/tmp_files/load_file.txt b/vdAzT4oBgHgl3EQfB_qZ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6b9ddee835af3ae868f642b94495d859a03d0fc0 --- /dev/null +++ b/vdAzT4oBgHgl3EQfB_qZ/content/tmp_files/load_file.txt @@ -0,0 +1,468 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf,len=467 +page_content='Production and spectroscopic investigation of Mercury and Radon isotopes produced in complete fusion reaction and multi-nucleon transfer reaction at MASHA facility R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Pandey1,a) 1Graduate Engineer Trainee, Larsen & Toubro Limited, Faridabad, Haryana, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' a)Corresponding author: rishav160999@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='com Abstract In this paper, the production and spectroscopic investigation of Mercury and Radon isotopes was performed using complete fusion re- actions neutron evaporation residues and multi-nucleon transfer reac- tion at the mass-separator MASHA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The MASHA setup is installed on the beam line of Cyclotron U-400M at Flerov Laboratory of Nuclear Reactions (FLNR) in Joint Institute for Nuclear Research (JINR), Dubna, Russia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The isotopes produced in complete fusion reactions 148Sm(40Ar, xn)188−xHg, 166Er(40Ar, xn)206−xRn and multi-nucleon transfer reaction 48Ca + 242Pu were passed through the magneto- optical system of MASHA setup with charge state Q=+1 and were separated on the basis of their mass to charge ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' For the detection of these isotopes, a position sensitive Si detector was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Further, the experimental data obtained were analysed and spectroscopic in- vestigations were carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Keywords: Spectroscopic investigation, Complete Fusion Reaction, Multi- Nucleon Transfer Reaction, Mass separator, MASHA, Position sensitive detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 1 INTRODUCTION The MASHA (Mass Analyzer of Super Heavy Atoms) setup has been designed as a mass-separator with the resolving power of about 1700, which allows mass 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='00952v1 [nucl-ex] 3 Jan 2023 identification of super-heavy nuclides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The setup uses the solid ISOL (Isotope Sep- aration On-Line) method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The work on the project means the analysis of real data collected from the experiments of complete fusion reactions neutron evaporation residues 40Ar + 148Sm → 188−xHg + xn, 40Ar + 166Er → 206−xRn + xn and multi-nucleon transfer reaction 48Ca + 242Pu using the α-decay chains from the position sensitive Si detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' An analysis was done on the experimental data from the above stated reac- tions to calculate the masses of identified isotopes(which are results of the above nuclear reactions), their half life, Alpha Branching Ratio (ABR), energy of α-decay (Ealpha) and their probability to decay with a specific amount of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Using these experimental data, one dimensional α-decay energy spectrum was plotted and its peak analysis was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Further, using the 1D histograms (α-decay energy spectrum), a heatmap (two dimensional energy-position graph) was ob- tained for the isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Finally, a comparison was drawn between the theoretical values and experimental values of Ealpha, and conclusions were made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The former was obtained from the nuclide chart, while the latter was obtained by performing experiments at the MASHA facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 2 LAYOUT OF MASHA SETUP AND ITS MAIN PARTS The MASHA setup is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' It consists of a Target Box with a Hot Catcher, an ECR Ion Source, Magneto-Optical System (or mass to charge ratio analyzer), and a Position Sensitive Si Detector (Detection and Control System).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' It also consists of an intermediate plane (F1), and for detailed monitoring of isotopes an additional strip detector has been installed at this middle plane also, apart from the detector present at focal plane (F2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Each part of the MASHA facility is explained in detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' For in-depth knowledge on MASHA setup, one can refer this paper [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='1 TARGET BOX The target box consists of a rotating disc divided into 6 sectors, which are sputtered with target material(s) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The disc rotates with a frequency of 25Hz [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The high energetic projectile particle ejected from U-400M cyclotron collides with the target material present in rotating disc to induce some kind of nuclear reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The products of the nuclear reaction are stopped by the hot catcher which is discussed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 2 Figure 1: Schematic of the MASHA facility: D1, D2, D3a, D3b are dipole magnets, Q1, Q2, Q3 are quadrupole lenses, S1, S2 are sextupole lenses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='2 HOT CATCHER The diagram of hot catcher is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The below setup mainly consists two components, one is poly-graphene heater and the other one is absorber mate- rial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The latter is usually made up of thin film of graphite or carbon nanotubes heated by the former upto a temperature of 1800 − 2000oC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' As depicted in figure the absorber is installed in front of the heater at a distance of 2 mm along the beam axis of MASHA [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The products of the nuclear reaction is stopped by the absorber material, vaporized to gaseous form and are passed to the ECR ion source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='3 shows the complete schematic of target box with hot catcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' (a) Rotating target disc inside the target box.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' (b) A thin hot catcher installed just before poly-graphene heater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Figure 2 3 Detector Focal plane F2 ECRion Hot catcher source Focal plane F1 D1 Q1 Q2 Ion D2 S2 beam Q3 D3b S1 D3aPoligraphene Thin heater hot cateherFigure 3: Complete schematic of target box with hot catcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='3 ECR ION SOURCE The ECR (Electron Cyclotron Resonance) ion source with a microwave oscillation frequency of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='45 GHz [1–4], acts as an ionization chamber of MASHA spectrome- ter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' It ionizes the atoms of gaseous isotopic products of nuclear reaction to a charge state Q=+1, and accelerates them to an energy of 38 KeV using three electrode system [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The ionized atoms gets converted to beam and are then separated by magneto-optical system of the MASHA spectrometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='4 MAGNETO-OPTICAL SYSTEM The magneto-optical system separates the beam of ions on the basis of their mass to charge ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The magnetic separation of heavy nuclei is performed using four dipole magnets(D1, D2, D3a, D3b), three quadrupole lenses (Q1, Q2, Q3) and two sextupole lenses (S1, S2) as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='1 [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Once, the heavy nuclei gets separated they are then detected at different strips of position sensitive Si detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='5 POSITION SENSITIVE Si DETECTOR The position sensitive Si detector is a multiple detector system used to detect the separated heavy nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' It is installed at the focal plane (F2) of the MASHA setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' A clear view of the position sensitive Si detector is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The front detector has a dimension of 240x35 mm2 and it consists of 192 strips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The upper and lower detector consists of 64 strips each while the left and right lateral detector consists of 16 strips each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Each strip has a width of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='25 mm and each detector has a thickness of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='3 mm [3–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 4 Diaphragm Target Ion beam To ECR ion source Hot catcher Target driverFigure 4: Position sensitive Si detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 1-front detector, 2-upper detector, 3-lower detector, 4-lateral detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 3 SCIENCE BEHIND THE EXPERIMENT The test experiments carried out at FLNR are: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 40Ar + 148Sm → 188−xHg + xn 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 40Ar + 166Er → 206−xRn + xn 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 48Ca + 242Pu → (Any element whose Z varies from 20-114) The first and second nuclear reactions are complete fusion reactions neutron evap- oration residues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' In such types of reactions the product nucleus formed has no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' of protons exactly equal to no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' of protons of projectile particle + no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' of protons of target nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Here, all the nucleons participate in the reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' While the third one is a Multi-Nucleon Transfer Reaction (MNTR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' In such nuclear reac- tions different nuclides can be formed whose atomic no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' ranges from atomic no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' of projectile particle to sum of atomic nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' of projectile particle + target nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' This means not all nucleons participate in this reaction and can lead to formation of any possible product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The N/Z ratio of product nucleus can be higher or lesser than than the optimal ratio required for its stability (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' It can be proton rich or neutron rich).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The U-400M cyclotron installed at FLNR, JINR is used to accelerate projectile particle (40Ar & 48Ca) to a very high velocity, with an energy ˜240 MeV (for 40Ar + 148Sm) and with energy ˜198 MeV (40Ar + 166Er).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The high energetic projectile particle enters into the MASHA setup and induce a nuclear reaction by colliding with target material sputtered in rotating disc present in target box of MASHA facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The products of nuclear reaction are isotopes of Hg (for 40Ar + 148Sm) 5 240 mm 2 mm 5 3 4 3 4and Rn (for 40Ar + 166Er and 48Ca + 242Pu) which are stopped by the absorber material of hot catcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The absorber material is generally made up of thin film of graphite or carbon nanotubes which is heated to around 1800−2000oC by means of IR radiations com- ing out from poly-graphene heater as well as by a direct current passing through the absorber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' This absorber stops the isotopic products of nuclear reaction, vapor- izes them and their respective atoms diffuses through this absorber material into the vacuum volume of the hot catcher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Moving along the vacuum pipe, they reach the ECR ion source [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' This ECR ion source acts as an ionization chamber of MASHA setup where the atoms of gaseous isotopic products gets ionized to charge state Q=+1 and further they are accelerated with the help of three electrode sys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' (The three electrode system consists of one positive electrode, one negative electrode and one more negative electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Hence, an electric field is established from positive electrode to negative electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' So, when a charged particle (here, ion) moves in the direction of electric field, it gets accelerated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The product isotopes are then separated by their M/Q ratio in the magneto- optical system of MASHA setup and at last they reach to the focal plane (F2) of the position sensitive Si detector and are detected at different strip numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' different isotopes are detected at different strip numbers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Now, the science is that the separated heavy nuclei undergoes α-decay to pro- duce daughter nuclei and it’s exactly the alpha particles (with different energies) given out by both parent nucleus and its daughter nuclei which are detected at unique strip nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' of position sensitive Si detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The detector used is a hybrid pixel detector of the TIMEPIX type, with high resolution and sensitivity which can detect even a single α or β particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' So, from the experimental data, we plot α-decay energy spectrum for those strips where an isotope was detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' From this spectrum (α-decay energy vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' of counts) we analyse the prominent peaks and calculate their α-decay energy (Ea) values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The base peak with maximum no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' of α particles (with constant energy) is our point of interest as it could be any one of the separated nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Now, using the table of nuclides, we find which isotope (of product of nuclear reaction) undergoes α-decay with energy very close to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' That particular isotope will be the one detected at a unique strip number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Ones, the isotope gets detected, then its mass, ABR, daughter nuclei can easily be investigated using the table of nuclides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' In the same way, one can detect all the isotopes of an element which is the product of a nuclear reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' In this work, a two dimensional energy-position graph (called heatmap) for all three test experiments has also been analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' This graph gives a clear un- derstanding that which isotope is detected at which strip no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' and corresponding to that particular isotope, how many alpha particles (counts) are detected with a constant energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' This constant energy is the energy of α-decay of that isotope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 6 4 SPECTROSCOPIC INVESTIGATION OF MERCURY ISO- TOPES USING FULL-FUSION REACTION 148Sm(40Ar, xn)188−xHg The complete fusion reaction of 148Sm(40Ar, xn)188−xHg was carried out at MASHA setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The target material sputtered in rotating disc was 148Sm and the prod- ucts of the nuclear reaction were isotopes of Hg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' However, only the long-lived isotopes of Hg were detected whose half-life was greater than average separation time (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='3 s) used by ISOL method for this reaction [1,3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='1 PRODUCTION OF 180Hg Figure 5: α-decay energy spectrum of 180Hg and its decay products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' In the α-decay energy spectrum of 180Hg, the most prominent peaks corre- sponding to parent nucleus and daughter nuclei were analysed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Here, the label 180Hg(Ealpha = 6120KeV, Counts = 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='5, 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='87%, ABR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='48) gives a lot of in- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' ‘Ealpha = 6120KeV ’ is the energy of α-particle released during the alpha decay of isotope 180Hg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Counts = 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='5 indicates the no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' of α-particles detected at a strip of detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' ABR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='48 is the Alpha Branching Ratio of 180Hg which means ‘The probability of 180Hg to undergo α-decay is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='48’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' It is to be noted that a nucleus can decay with multiple α-decay energies and for same reason, here 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='87% means 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='87% of 180Hg decays with energy = 6120 KeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Now, coming to the analysis part, in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='5 we observe that 180Hg(t 1 2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='58s) is peaked at 6120 KeV and its daughter nuclei 176Pt(t 1 2 = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='3s), formed due to α-decay of 180Hg and 180Au(t 1 2 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='1s), formed due to Electron Capture (EC) in 180Hg are peaked at 5750 KeV and 5680 KeV respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='2 PRODUCTION OF 181Hg In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='6, it is depicted that 181Hg(t 1 2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='54s) is peaked at 6000 KeV, with prob- ability of decaying in this energy being 87% and ABR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' While its α-decay 7 70 Alpha Decay Counts 60 50 40 30 20 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 5500 5600 5700 5800 5900 6000 6100 6200 Alpha Decay Energy (KeV)Figure 6: α-decay energy spectrum of 181Hg and its decay products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' product 177Pt(t 1 2 = 11s) is peaked at 5500 KeV, with probability to decay in this energy being 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='5% and ABR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='056.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='3 PRODUCTION OF 182Hg Figure 7: α-decay energy spectrum of 182Hg and its decay products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='7, we can observe the base peak of 182Hg(t 1 2 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='83s) at 5860 KeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' However, we can also see a peak of 182Hg at 5690 KeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' This is due to the fact that a radioactive heavy nucleus can decay with multiple α-decay energies with former being highly probable (99%) and later being less probable (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='57%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' It is also observed that α-decay product of 182Hg is 178Pt(t 1 2 = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='1s) which is peaked at 5450 KeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='4 PRODUCTION OF 183Hg In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='8, the two primary daughters of 183Hg, 183Au(t 1 2 = 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='8s) and 179Pt(t 1 2 = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='1s) formed due to EC and α-decay of parent nucleus are peaked at 5340 KeV, and 5190 KeV respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The parent nucleus 183Hg(t 1 2 = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='4s) is peaked at 5890 KeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 8 250 Alpha Decay Counts 200 100 50 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 5400 5500 5600 5700 5800 5900 6000 6100 6200 Alpha Decay Energy (KeV)600 Alpha Decay Counts 500 400 300 200 100 0 5400 5500 5600 5700 5800 5900 6000 Alpha Decay Energy (KeV)Figure 8: α-decay energy spectrum of 183Hg and its decay products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='5 PRODUCTION OF 184Hg Figure 9: α-decay energy spectrum of 184Hg and its decay products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' In α-decay energy spectrum of 184Hg shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='9, the base peak correspond- ing to 184Hg(t 1 2 = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='9s) is obtained at 5530 KeV with probability of decaying with this energy being 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='44%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' While 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='4% is the chance that 184Hg decays with energy 5360 KeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 180Pt(t 1 2 = 56s) being the α-decay product of 184Hg is peaked at 5160 KeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='6 PRODUCTION OF 185Hg In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='10, it is observed that 185Hg(t 1 2 = 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='1s) is peaked at 5650 KeV and 5540 KeV with probability to decay with these energies being 96% and 4% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The daughter nucleus formed due to EC in 185Hg is 185Au(t 1 2 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='25months), which is peaked at 5080 KeV with ABR of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='0026 in α-decay energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Also, the α-decay product of 185Hg, which is 181Pt(t 1 2 = 52s) was not observed in this spectrum because of its very low probability to undergo α-decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 9 250 Alpha Decay Counts 200 2 150 100 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 5100 5200 5300 5400 5500 5600 Alpha Decay Energy (KeV)700 Alpha Decay Counts 600 500 S 400 uni Col 300 200 100 0 5000 5100 5200 5300 5400 5500 5600 5700 5800 5900 6000 Alpha Decay Energy (KeV)Figure 10: α-decay energy spectrum of 185Hg and its decay products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='7 HEATMAP OF Hg ISOTOPES Figure 11: Heatmap of Hg isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Production of Hg isotopes in complete fusion reaction 148Sm(40Ar, xn)188−xHg is shown by a two-dimensional graph as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Through this energy- position graph, it’s evident that different isotopes of Hg produced in aforemen- tioned nuclear reaction are detected at unique strip numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' However, only those isotopes of Hg are visible whose half-life is greater than average separation time (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='8±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='3 s) used by ISOL method for this reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Along the Y-axis we see α- decay energy of each detected isotope and through the color associated with each 10 1800 Alpha Decay Counts 1500 Counts 1200 900 600 300 0 5000 5100 5200 5300 5400 5500 5600 5700 Alpha Decay Energy (KeV)Counts 6,97 975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='0 181Hg 183 6,57 - 180 Hg 780,3 Energy (MeV) 6,17 585,6 Alpha Decay I 5,77 390,9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='37 196,2 4,97 i 1,500 4,57 100 导 0 140 180 Detector Strip Numberisotope, we get the value of no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' of counts of α-particles detected by matching it with the color scale bar on right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The experimental values of Ealpha is compared with its theoretical values ob- tained from table of nuclides in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' It is observed that the % change in their values is even less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='3%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' So, we conclude that the spectroscopic investigation performed for Hg isotopes in reaction 148Sm(40Ar, xn)188−xHg is almost accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Table 1: Comparison between theoretical and experimental values of Ealpha(in KeV) of Hg isotopes produced in reaction 148Sm(40Ar, xn)188−xHg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Nucleus Theo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Ealpha Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Ealpha ∆% 180Hg 6119 6120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='016 181Hg 6006 6000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='099 182Hg 5867 5860 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='119 183Hg 5904 5890 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='230 184Hg 5535 5530 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='090 185Hg 5653 5650 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='053 5 SPECTROSCOPIC INVESTIGATION OF RADON ISOTOPES USING FULL-FUSION REACTION 166Er(40Ar, xn)206−xRn A complete fusion reaction was performed between high energetic projectile parti- cle (40Ar) ejected from the window of U-400M cyclotron with an energy ˜198 MeV and the target material 166Er present in the form of rotating disc in the target box of MASHA facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The products of the nuclear reaction were isotopes of Rn which were detected at focal plane (F2) of the position sensitive Si detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Fur- ther, using the experimental data obtained from the detector and control system of MASHA, their α-decay energy spectrum and energy-position graphs were plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='1 PRODUCTION OF 201Rn Figure 12: α-decay energy spectrum of 201Rn and its decay products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 11 1800 Alpha Decay Counts 1500 1200 jun Cou 900.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 300 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 5500 5600 5700 5800 5900 6000 6100 6200 6300 6400 6500 6600 6700 6800 6900 Alpha Decay Energy (KeV)In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='12, we see that the isotope 201Rn(t 1 2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='8s) is peaked at 6760 KeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Its only daughter nucleus formed due to α-decay, 197Po(t 1 2 = 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='6s) with ABR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='44 is peaked at 6280 KeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' However, we can see another 197Po(t 1 2 = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='8s) with ABR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='84 peaked at 6380 KeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The probability of occurrence of both 197Po(t 1 2 = 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='6s) and 197Po(t 1 2 = 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='8s) are very high with 100% and 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='3% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='2 PRODUCTION OF 202Rn Figure 13: α-decay energy spectrum of 202Rn and its decay products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='13, the two primary daughters of 202Rn, 202At(t 1 2 = 184s) and 198Po(t 1 2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='77months) formed due to EC and α-decay of parent nucleus are peaked at 6070 KeV, and 6180 KeV respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The parent nucleus 202Rn(t 1 2 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='0s) is peaked at 6630 KeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='3 PRODUCTION OF 203Rn Figure 14: α-decay energy spectrum of 203Rn and its decay products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' During the investigation of 203Rn isotope, it was observed that two different forms of 203Rn with different t 1 2 and ABR were detected at same strip of position 12 Alpha Decay Counts 2000 1500 luno 8 1000 500 5500 5600 5700 5800 5900 6000 6100 6200 6300 6400 6500 6600 6700 6800 Alpha Decay Energy (KeV)700 Alpha Decay Counts 600 500 400 ino 300 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 100 0 5800 5900 6000 6100 6200 6300 6400 6500 6600 6700 Alpha Decay Energy (KeV)sensitive detector [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 203Rn(t 1 2 = 28s) with ABR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='8 was peaked at 6550 KeV, while 203Rn(t 1 2 = 45s) with ABR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='66 was peaked at 6500 KeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The daughter nucleus 199Po(t 1 2 = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='17months) formed due to the α-decay emission of 203Rn is peaked at 6060 KeV in α-decay energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='4 PRODUCTION OF 204Rn Figure 15: α-decay energy spectrum of 204Rn and its decay products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' From the α-decay energy spectrum of 204Rn, it is noticed that the parent isotope 204Rn(t 1 2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='24months) is peaked at 6400 KeV, while its α-decay product 200Po(t 1 2 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='5months) is peaked at 5840 KeV as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='5 PRODUCTION OF 205Rn Figure 16: α-decay energy spectrum of 205Rn and its decay products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' From the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='16, it is evident that 205Rn(t 1 2 = 170s) isotope undergoes α- decay to release 6270 KeV of energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The probability that it face α-decay to produce 201Po(t 1 2 = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='9months) with Ealpha = 5760KeV is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' However, this heavy radioactive nucleus also tries to achieve stability through electron capture, 13 Alpha Decay Counts 200 ounts 150 100 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 5700 5800 5900 6000 6100 6200 6300 6400 6500 Alpha Decay Energy (KeV)45 Alpha Decay Counts 15 0 5700 5800 5900 6000 6100 6200 6300 6400 Alpha Decay Energy (KeV)and for same the probability is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='754 (as seen from table of nuclides).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 205At(t 1 2 = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='2months) which is the result of EC is peaked at 5910 KeV in α-decay energy spectrum of 205Rn isotope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='6 HEATMAP OF Rn ISOTOPES (166Er(40Ar, xn)206−xRn) The energy-position graph of Rn isotopes from 201Rn to 205Rn is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' These isotopes are the result of nuclear reaction 40Ar + 166Er → 206−xRn + xn and are separated on the basis of their M/Q ratio in the magneto-optical system of mass-separator and finally detected at different strip numbers of detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' From this heatmap, we see Ealpha decreases from 201Rn to 205Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' We also conclude that 201Rn and 202Rn isotopes are greatly produced in the reaction with high no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' of counts, while 205Rn is produced in relatively less number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The α-decay daughter nuclei 197Po and 198Po from parent isotopes 201Rn and 202Rn respectively can also be seen at different strip numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Figure 17: Heatmap of Rn isotopes using complete-fusion reaction 166Er(40Ar, xn)206−xRn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The comparison between theoretical and experimental values of Ealpha for Rn isotopes produced in complete fusion reaction 166Er(40Ar, xn)206−xRn is shown in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' It is noticed here that the % change between these values is less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='3%, so our analysis on this subject is nearly accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 14 Count 7,04 875,0 203Rn 201 204Rn 6,67 - 700,3 Energy (MeV) 205 197 Rn 6,30 - 525,6 Alpha Decay E 5,92 350,9 5,55 176,2 5,18 4,81 1,500 0 4 0 8 100 120 140 160 180 Detector Strip NumberTable 2: Comparison between theoretical and experimental values of Ealpha(in KeV) of Rn isotopes produced in reaction 166Er(40Ar, xn)206−xRn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Nucleus Theo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Ealpha Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Ealpha ∆% 201Rn 6773 6760 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='192 202Rn 6639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='5 6630 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='143 203Rn(ABR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='80) 6549 6550 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='015 203Rn(ABR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='66) 6499.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='3 6500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='011 204Rn 6418.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='9 6400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='294 205Rn 6262 6270 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='128 6 SPECTROSCOPIC INVESTIGATION OF RADON ISOTOPES USING MNTR 48Ca + 242Pu Unlike complete fusion reactions discussed above, a MNTR can have any possible product nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' However, in the reaction of 48Ca + 242Pu under some fixed conditions, new neutron-rich Rn isotopes were produced near the neutron N=126 shell closure configuration, using MNTR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The isotopes produced were identified first, later their spectroscopic investigations were carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' However, it was observed that only those Rn isotopes reached the detector and were identified which lived at least 35 ms while others decayed in their path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='1 PRODUCTION OF 212Rn Figure 18: α-decay energy spectrum of 212Rn and its decay products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' From the spectrum of 212Rn(t 1 2 = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='9months) isotope shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='18, it is evident that 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='95% of 212Rn isotopes undergoes α-decay with energy 6250 KeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Also, its ABR=1, which means it will only undergo α-decay and no other forms of decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 212Rn isotope has N=126 shell configuration, which gives it a very high stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Its daughter nucleus 208Po(t 1 2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='9years) was not observed in α-decay energy spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 15 1200 Alpha Decay Counts 900 212Rn(Ealpha=6250KeV, Counts=1138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='8, 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='95%, ABR=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='0) 600 300 5500 0 5600 5700 5800 5900 6000 6100 6200 6300 6400 Alpha Decay Energy (KeV)6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='2 PRODUCTION OF 218Rn The only daughter nucleus of 218Rn due to its α-decay is 214Po(t 1 2 = 164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='3µs), which is peaked at 7660 KeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The parent isotope 218Rn(t 1 2 = 35ms) is peaked at 7110 KeV and 6530 KeV with probability of decaying with these energies being 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='87% and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='127% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' As discussed earlier, the reason behind why we see two different peaks of 218Rn isotope in α-decay energy spectrum is that the decay energy of alpha particle is not always fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' It can have multiple energies, and there is a fixed probability for a certain energy to get released.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Also, we see that the counts associated with 218Rn isotope and its daughter nucleus is very low [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='19] because of their low half-life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Figure 19: α-decay energy spectrum of 218Rn and its decay products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='3 PRODUCTION OF 219Rn Figure 20: α-decay energy spectrum of 219Rn and its decay products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' From the α-decay energy spectrum shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='20, we see that the parent isotope, 219Rn(t 1 2 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='96s) is peaked at 6790 KeV, 6600 KeV, 6530 KeV, and 16 Alpha Decay Counts 8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Counts 2 0 6400 6600 6800 7000 7200 7400 7600 7800 Alpha Decay Energy (KeV)Alpha Decay Counts 500- 400 Counts 300 200 - 100 4000 4500 5000 5500 6000 6500 7000 7500 8000 8500 9000 9500 10000 Alpha Decay Energy (KeV)6400 KeV with probability to decay with these energies are 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='4%, 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='9%, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='12%, and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='5% respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' All α-decay energies have the same ABR=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The α-decay product of 219Rn is 215Po(t 1 2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='78ms), and is peaked at 7360 KeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 211Pb(t 1 2 = 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='1months) which is the alpha-daughter of 215Po could not be seen in this α- decay energy spectrum because it only undergoes β− decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' However, its daughter nucleus, 211Bi(t 1 2 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='14months) is peaked at 6250 KeV, with ABR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='99 and probability to decay with this energy is 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='23%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='4 HEATMAP OF Rn ISOTOPES (48Ca + 242Pu) Figure 21: Heatmap of Rn isotopes using MNTR 48Ca + 242Pu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The energy-position graph of Rn isotopes produced in MNTR 48Ca + 242Pu is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 212Rn and 219Rn isotopes were largely produced in the reaction with high counts, while 218Rn isotope with few counts is barely visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' This is because of low half-life of 218Rn(t 1 2 = 35ms), that major part of it decayed before reaching the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The α-daughter of 219Rn, 215Po was also detected at some strip number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' A huge gap between 212Rn and 218Rn isotopes could be explained in terms of less half-life of Rn isotopes from A=213-217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Their half-life is even less than 35 ms which is more or less the average separation time used by ISOL method for this reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' So, we conclude that only long-lived Rn isotopes were detected which lived at least 35 ms [1], while others decayed before reaching the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 17 9,98 Counts 920,0 215 9,00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 218 Rn 219Rn Energy (MeV) 736,3 8,02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 552.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='6 7,04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 368,9 6,05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='2 5,07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 4,09 1,500 0 a 3 4 5 0 110 3 140 Detector Strip NumberTable 3: Comparison between theoretical and experimental values of Ealpha(in KeV) of Rn isotopes produced in reaction 48Ca + 242Pu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Nucleus Theo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Ealpha Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Ealpha ∆% 212Rn 6264 6250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='223 218Rn 7129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='2 7110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='269 219Rn 6819.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='1 6790 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='427 The theoretical and experimental values of Ealpha for Rn isotopes produced via MNTR 48Ca + 242Pu is summarized in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='3% change in these values is observed for 212Rn and 218Rn isotopes, while for 219Rn, the % change is < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' So, our analysis on this reaction is almost accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 7 RESULTS AND CONCLUSIONS In this entire work, the production and spectroscopic investigation of Hg and Rn isotopes was performed using full fusion reactions 148Sm(40Ar, xn)188−xHg, 166Er(40Ar, xn)206−xRn and multi-nucleon transfer reaction 48Ca + 242Pu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The final product in all these reactions were isotopes of Hg and Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The experimen- tal data obtained from the MASHA setup were analysed and 1D α-decay energy spectrum graphs were plotted for those strips of detector which had detected any isotopic product of nuclear reaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Further, this 1D histograms were used to plot a 2D energy-position graph, separately for Hg and Rn isotopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The masses of super-heavy nuclei which were detected at different strips of Si based Position Sensitive Detector (PSD), have also been identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Using 1D histograms and nu- clide chart, the values of Ealpha, ABR, Counts, and the probability to decay with a specific amount of energy were calculated for all isotopic products of nuclear reactions studied in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 8 ACKNOWLEDGEMENTS The author expresses his gratitude to Mr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Viacheslav Vedeneev, Flerov Labo- ratory of Nuclear Reactions, Joint Institute for Nuclear Research for providing experimental data obtained from the MASHA setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' The author is also grateful to his parents for motivating him throughout his research work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 9 STATEMENTS AND DECLARATIONS The author has no competing interests to declare that are relevant to the content of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 18 References [1] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' Y.' metadata={'source': 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International Symposium on Exotic Nuclei, World Scientific, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 437–443.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} +page_content=' 19' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vdAzT4oBgHgl3EQfB_qZ/content/2301.00952v1.pdf'} diff --git a/wNE3T4oBgHgl3EQflArR/content/tmp_files/2301.04603v1.pdf.txt b/wNE3T4oBgHgl3EQflArR/content/tmp_files/2301.04603v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0bc8211ea9b2d1a80bfed15e54d832fdd1b1e475 --- /dev/null +++ b/wNE3T4oBgHgl3EQflArR/content/tmp_files/2301.04603v1.pdf.txt @@ -0,0 +1,850 @@ +Feasibility and Regularity Analysis of Safe Stabilizing Controllers +under Uncertainty +Pol Mestres +Jorge Cort´es +Abstract— This paper studies the problem of safe stabiliza- +tion of control-affine systems under uncertainty. Our starting +point is the availability of worst-case or probabilistic error +descriptions for the dynamics and a control barrier function +(CBF). These descriptions give rise to second-order cone con- +straints (SOCCs) whose simultaneous satisfaction guarantees +safe stabilization. We study the feasibility of such SOCCs and +the regularity properties of various controllers satisfying them. +I. INTRODUCTION +The last years have seen a dramatic increase in the +deployment of robotic systems in diverse areas like home +automation and autonomous driving. In these applications, it +is critical that robots satisfy simultaneously safety and per- +formance specifications in the presence of model uncertainty. +Controllers that achieve these goals are usually defined using +tools from stability analysis and Lyapunov theory. However, +this faces several challenges, such as the development of +understanding about the level of model uncertainty that can +be tolerated while still being able to meet safety and sta- +bility requirements and the characterization of the regularity +properties, along with the identification of suitable conditions +to ensure them, in order to be implementable in real-world +scenarios. These are the questions that motivate this paper. +Literature Review: Control Lyapunov functions (CLFs) [1] +are a well-established tool for designing stabilizing con- +trollers for nonlinear systems. More recently, control barrier +functions (CBFs) [2], [3] have been introduced as a tool to +render a certain predefined set safe. If the system is control +affine, the CLF and CBF conditions can be incorporated in a +quadratic program (QP) [4], [5] that can be efficiently solved +online. However, this design assumes complete knowledge of +the dynamics and safe set. Several recent papers have pro- +posed alternative formulations of the CLF-CBF QP for sys- +tems with uncertainty or learned dynamics. In [6], the CLF- +CBF QP is transformed into another QP to account for worst- +case uncertainty bounds. However, imperfect knowledge of +the system dynamics or safety constraints transforms the +affine-in-the-input inequalities arising from CBFs and CLFs +into second-order cone constraints (SOCCs). The papers [7], +[8] leverage Gaussian Processes (GPs) to learn the system +dynamics from data and show that the mean and variance +of the estimated GP can be used to formulate two SOCCs +whose pointwise satisfaction implies safe stabilization of the +true system with a prescribed probability. However, during +the control design stage, the SOCC associated to stability is +P. Mestres and J. Cort´es are with the Department of Mechanical and +Aerospace Engineering, UC San Diego, {pomestre,cortes}@ucsd.edu +relaxed and hence the resulting controller does not have sta- +bility guarantees. In the case where worst-case error bounds +for the dynamics and the CBF are known, [9], [10] show how +the satisfaction of two SOCCs can yield a safe stabilizing +controller valid for all models consistent with these error +bounds. [11] uses the framework of distributionally robust +optimization to formulate a second-order convex program +that achieves safe stabilization for systems where only a finite +set of model parametric uncertainty samples are available. +However, these works lack guarantees on the simultaneous +feasibility of these SOCCs and the regularity of controllers +satisfying them. Hence, the resulting controllers might be +undefined in practice, resulting in unsafe, unstable, or dis- +continuous behaviors. Our work also relies on techniques to +design CLFs without an accurate model of the system. For +instance, for polynomial systems with uncertainty, [12] uses +sum-of-squares techniques to find robust Lyapunov functions +based on known error bounds. Alternatively, [13] describes +a method that only requires knowledge of the degree of +actuation. Finally, the papers [14], [15] utilize online data to +improve the estimates of the dynamics and synthesize (also +via SOCCs) less conservative controllers. +Statement of Contributions: We study1 the problem of safe +stabilization of control-affine systems under uncertainty. We +consider two scenarios for the estimates of the dynamics +and safe set: either worst-case error bounds or probabilistic +descriptions in the form of Gaussian Processes (GPs) are +available. In both cases, the problem of designing a safe +stabilizing controller can be reduced to satisfying two SOCCs +1We use the following notation. We denote by Z>0, R, and R≥0 the set of +positive integers, real, and nonnegative real numbers, resp. We denote by 0n +the n-dimensional zero vector. We write int(S) and ∂S for the interior and +the boundary of the set S, resp. Given x ∈ Rn, ∥x∥ denotes the Euclidean +norm of x. Given f : Rn → Rn, g : Rn → Rn×m and a smooth function +W : Rn → R, LfW : Rn → R (resp. LgW : Rn → Rm) denotes the +Lie derivative of W with respect to f (resp. g), that is LfW = ∇W T f +(resp. ∇W T g). We use GP(µ(x), K(x, x′)) to denote a Gaussian Process +distribution with mean function µ(x) and covariance function K(x, x′). We +denote by Cl(A) the set of l-times continuously differentiable functions on +an open set A ⊆ Rn. A function β : R → R is of class K if β(0) = 0 +and β is strictly increasing. If moreover limt→∞ β(t) = ∞, then β is of +class K∞. A function V : Rn → R is positive definite if V (0) = 0 and +V (x) > 0 for all x ̸= 0. V is proper in a set Γ if the set {x ∈ Γ : V (x) ≤ +c} is compact for any c ≥ 0. A set C ⊆ Rn is forward invariant under +the dynamical system ˙x = f(x) if any trajectory with initial condition in +C at time t = 0 remains in C for all positive times. A set C is safe for +˙x = f(x, u) if there exists a locally Lipschitz control u : Rn → Rm +such that C is forward invariant for ˙x = f(x, u(x)). Given m × n matrix +A and two integers i, j such that 1 ≤ i < j ≤ n, Ai:j denotes the +m × (j − i + 1) matrix obtained by selecting the columns from i to j of +A, and σmax(A) denotes the maximum singular value of A. We denote +by Br(p) = {y ∈ Rn +: +∥y − p∥ < r}. Given A ∈ Rq×n, b ∈ Rq, +c ∈ Rn, d ∈ R, the inequality ∥Ax + b∥ ≤ cT x + d is a second-order +cone constraint (SOCC) in the variable x ∈ Rn. +arXiv:2301.04603v1 [math.OC] 11 Jan 2023 + +at every point in the safe set. Our first contribution consists +of giving sufficient conditions for the feasibility of each pair +of SOCCs. Second, if each pair of SOCCs is feasible, we +show that there exists a smooth safe stabilizing controller +and that the minimum-norm controller satisfying each pair +of SOCCs is locally Lipschitz. Third, we provide a universal +formula for satisfying a single SOCC and hence achieving +either safety or stability. We illustrate our results in the safe +stabilization of a planar system. +II. PRELIMINARIES +This section presents preliminaries on control Lyapunov +and barrier functions, and safe stabilization using worst-case +and probabilistic estimates of the dynamics. +A. Control Lyapunov and Barrier Functions +Consider the control-affine system +˙x = f(x) + g(x)u, +(1) +where f : Rn → Rn and g : Rn → Rm×n are locally +Lipschitz functions, with x ∈ Rn the state and u ∈ Rm the +input. We assume without loss of generality that f(0n) = 0n. +Definition 2.1: (Control Lyapunov Function [16]): Given +a set Γ ⊆ Rn, with 0n ∈ Γ, a continuously differentiable +function V : Rn → R is a CLF on Γ for the system (1) if it +is proper in Γ, positive definite, and there exists a continuous +positive definite function W : Rn → R such that, for each +x ∈ Γ\{0}, there exists a control u ∈ Rm satisfying +LfV (x) + LgV (x)u ≤ −W(x). +(2) +A Lipschitz controller k : Rn → Rm such that u = k(x) +satisfies (2) for all x ∈ D\{0} makes the origin of the closed- +loop system asymptotically stable. Hence, CLFs provide a +way to guarantee asymptotic stability. +Definition 2.2: (Robust Control Barrier Function [6, Def- +inition 6]): Let C ⊂ Rn and h : Rn → R continuously +differentiable such that +C = {x ∈ Rn : h(x) ≥ 0}, +(3a) +∂C = {x ∈ Rn : h(x) = 0}. +(3b) +Given ηh > 0, h is an ηh-robust CBF if there exists a K∞ +function α such that for all x ∈ C, there exists u ∈ Rm with +Lfh(x) + Lgh(x)u + α(h(x)) ≥ ηh. +(4) +When ηh = 0, this definition reduces to the notion of +CBF [3, Definition 2], and the inequality reduces to +Lfh(x) + Lgh(x)u + α(h(x)) ≥ 0. +(5) +Note that all robust CBFs are CBFs. A Lipschitz controller +k : Rn → Rm such that u = k(x) satisfies (5) for all x ∈ C +makes C forward invariant. Hence, CBFs provide a way to +guarantee safety. +Remark 2.3: (Slack in CLF and CBF conditions): Without +loss of generality, if V is a CLF on an open set Γ, we can +assume there exists a positive definite function S such that, +for all x ∈ Γ, there is u ∈ Rm with +LfV (x) + LgV (x)u + W(x) ≤ −S(x). +(6) +This is because we can always define a positive definite +function +˜W(x) +:= +1 +2W(x) in Definition 2.1 and take +S(x) := 1 +2W(x). Similarly, if h is an ηh-robust CBF, then +there exists a class K∞ function ζ such that for all x ∈ C, +there is u ∈ Rm with +Lfh(x) + Lgh(x)u + α(h(x)) ≥ ηh + ζ(h(x)). +• +(7) +B. Robust and Probabilistic Safe Stabilization +We are interested in the design of controllers that ensure +simultaneously stability and safety in the presence of uncer- +tainty. Assume that the maps f, g in (1) and the CBF h +and its gradient ∇h are unknown. Instead, estimates ˆf, ˆg, ˆh +and � +∇h are available. We also assume that a CLF V for the +system (1) is available. We consider two types of models for +the errors between the estimates and the true quantities. First, +for x ∈ Rn, consider worst-case error bounds as follows: +∥f(x) − ˆf(x)∥ ≤ ef(x), ∥g(x) − ˆg(x)∥ ≤ eg(x), +∥h(x) − ˆh(x)∥ ≤ eh(x), ∥∇h(x) − � +∇h(x)∥ ≤ e∇h(x). +Since the exact dynamics and CBF are unknown, one can not +certify the inequalities (2) and (5) directly. Instead, using the +error bounds above, define +aV (x) = ∥∇V (x)∥eg(x), +bV (x) = −∇V (x)T ˆg(x), +cV (x) = −∥∇V (x)∥ef(x) − ∇V (x)T ˆf(x) − W(x), +ah(x) = e∇h(x)eg(x) + e∇h(x)∥ˆg(x)∥ + ∥ � +∇h(x)∥eg(x), +bh(x) = � +∇h(x)T ˆg(x), +ch(x) = −e∇h(x)ef(x) − e∇h(x)∥ ˆf(x)∥ − ∥ � +∇h(x)∥ef(x) ++ � +∇h(x)T ˆf(x) + α(ˆh(x) − eh(x)). +According to [9, Proposition V.I], if the two (state-dependent) +SOCCs (in u): +aV (x)∥u∥ ≤ bV (x)u + cV (x), +(8a) +ah(x)∥u∥ ≤ bh(x)u + ch(x), +(8b) +are satisfied for all x ∈ C, then (2) and (5) hold for all +x ∈ C. This result provides a way of designing controllers +that simultaneously satisfy (2) and (5). +Second, suppose that GP estimates are available for the +following quantities [7]: +∆V (x, u) = LfV (x) + LgV (x)u − L ˆ +fV (x) − LˆgV (x)u, +∆h(x, u) = Lfh(x) + Lgh(x)u + α(h(x)) − � +∇h(x) +T ˆf(x) +− � +∇h(x) +T +ˆg(x)u − α(ˆh(x)). +Let µV (x, u) and s2 +V (x, u) denote the mean and variance, +resp., of the GP prediction of ∆V , which we assume affine +and quadratic in u, resp. Therefore, there exist γV (x) : Rn → +Rm+1 and GV (x) ∈ R(m+1)×(m+1) such that +µV (x, u) = γV (x)T +� +1 +u +� +, +sV (x, u) = ∥GV (x) +� +1 +u +� +∥2. +For the GP prediction of ∆h, let γh(x), and Gh(x) be +defined analogously. Since the exact dynamics and CBF are + +unknown, one cannot certify the inequalities (2) and (5). +However, for δ ∈ (0, 1), and using the GP predictions, define +QV (x) = β(δ)GV,2:(m+1)(x) ∈ R(m+1)×m, +rV (x) = β(δ)GV,1(x) ∈ R(m+1)×1, +bV (x) = −LˆgV (x) − γT +V,2:(m+1)(x) ∈ R1×m, +cV (x) = −L ˆ +fV (x) − W(x) − γV,1(x) ∈ R, +and similarly Qh, rh, bh and ch (the exact form of β(δ) is +given in [8, Theorem 2]). Then, according to [7, Section IV], +if the two SOCCs +∥QV (x)u + rV (x)∥ ≤ bV (x)u + cV (x), +(9a) +∥Qh(x)u + rh(x)∥ ≤ bh(x)u + ch(x), +(9b) +are satisfied for all x ∈ C, then (2) and (5) each hold for all +x ∈ C with probability at least 1 − δ. +Remark 2.4: (General form of SOCCs): By taking in (9a), +QV (x) = aV (x) +�Im +0T +m +� +, rV (x) = 0m+1, +we obtain (8a). Hence, in the rest of the paper, we derive the +results for SOCCs of the most general form (9). +• +III. PROBLEM STATEMENT +We consider a control-affine system of the form (1) and a +safe set C of the form (3), with f, g, h and ∇h unknown. We +assume that either worst-case or probabilistic descriptions of +the dynamics and barrier function are available, as described +in Section II-B. We make the following assumptions. +Assumption 1: The function h is a CBF of C and ∂h +∂x(x) ̸= +0 for all x ∈ ∂C. +By [3, Theorem 2], Assumption 1 implies that C is safe. +However, since the true h is unknown, a safe controller is +not readily computable. +Assumption 2: A CLF V on an open set containing the +origin is known. +Given this setup, our goals are to (i) derive conditions that +ensure the feasibility of the pair of robust stability (8a) and +safety-(8b) (resp., probabilistic stability (9a) and safety (9b)) +inequalities and, building on this, (ii) design controllers that +jointly satisfy the inequalities pointwise in C and charac- +terize their regularity properties. The latter is motivated by +both theoretical (guarantee the existence and uniqueness of +solutions to the closed-loop system) and practical (ease of +implementation of feedback control on digital platforms and +avoidance of chattering behavior) considerations. +IV. COMPATIBILITY OF PAIRS OF SECOND-ORDER CONE +CONSTRAINTS +In this section, we derive sufficient conditions that guar- +antee the feasibility of the pairs of inequalities in (8) and +in (9), resp. The following definition extends the notion of +compatibility [17, Definition 3] to any set of inequalities. +Definition 4.1: (Compatibility of a set of inequalities): +Given functions qi : Rn × Rm → R for i ∈ {1, . . . , p}, +the inequalities qi(x, u) ≤ 0, i ∈ {1, . . . , p} are (strictly) +compatible at a point x ∈ Rn if there exists a corresponding +u ∈ Rm satisfying all inequalities (strictly). The same +inequalities are (strictly) compatible on a set G if they are +(strictly) compatible at every x ∈ G. +As the estimation errors (resp. the variances) approach +zero, the inequalities in (8) (resp. (9)) approach (2) and (5). +If (6) and (7) are compatible, the next result provides explicit +bounds for the estimation errors such that (8a)-(8b) and (9a)- +(9b) are strictly compatible. +Proposition 4.2: (Compatibility in worst-case and prob- +abilistic SOCCs): Let h be an ηh-robust CBF. Assume (6) +and (7) are compatible on C, and let B : Rn → R be an upper +bound on the norm of a control satisfying both inequalities. +(i) If, for all x ∈ C\{0}, +∥∇V (x)∥(ef(x) + eg(x)B(x)) < 1 +2S(x), +(10a) +(e∇h(x) + ∥ � +∇h(x)∥)(ef(x) + eg(x)B(x)) + Kαeh(x) ++ e∇h(x)(∥ ˆf(x)∥ + ∥ˆg(x)∥B(x)) < 1 +2(ηh + ζ(h(x))), +(10b) +where Kα is the Lipschitz constant of α, then (8a) +and (8b) are strictly compatible on C\{0}; +(ii) If for all x ∈ C\{0}, +σmax(GV (x)) < +S(x) +2β(δ) +� +1 + B2(x) +, +(11a) +σmax(Gh(x)) < +ηh + ζ(h(x)) +2β(δ) +� +1 + B2(x) +, +(11b) +then (9a) and (9b) are strictly compatible on C\{0} +with probability greater than or equal to 1 − 2δ. +Proof: (i) The proof follows by applying the definition +of the upper bounds ef, eg, eh, e∇h given in Section II-B. +(ii) Note that (9a), (9b) can be equivalently written as: +β(δ)∥GV (x) +� +1 +u +� +∥2 ≤ −∇V (x)T ˆf(x) − γV,1(x) − W(x) +− (∇V (x)T ˆg(x) + γT +V,2:(m+1)(x))u, +β(δ)∥Gh(x) +� +1 +u +� +∥2 ≤ � +∇h(x) +T ˆf(x) + γh,1(x) + α(h(x))+ +( � +∇h(x) +T +ˆg(x) + γT +h,2:(m+1)(x))u. +Now, +note +that +−L ˆ +fV (x) − LˆgV (x)u − γV,1(x) − +γT +V,2:(m+1)(x)u = −LfV (x) − LgV (x)u + ∆V (x, u) − +γV,1(x) − γT +V,2:(m+1)(x)u, and similarly for the safety con- +straint. Define then the events EV += {|γV (x)T +� +1 +u +� +− +∆V (x, u)| ≤ βsV (x, u), ∀x ∈ C, u ∈ Rm} and Eh = +{|γh(x)T +� +1 +u +� +− ∆h(x, u)| ≤ β(δ)sh(x, u), ∀ x ∈ C, u ∈ +Rm}. By [18, Theorem 6], P(EV ) ≥ 1−δ and P(Eh) ≥ 1−δ. +Therefore, P(EV ∩ Eh) = P(EV ) + P(Eh) − P(EV ∪ Eh) ≥ +1 − 2δ. Hence, if we can find u ∈ Rm satisfying +Lfh(x) + Lgh(x)u + α(h(x)) ≥ 2β(δ)sh(x, u), +(12a) +−LfV (x) − LgV (x)u − W(x) ≥ 2β(δ)sV (x, u), +(12b) +then (9a), (9b) are compatible at x with probability at +least 1 − 2δ. Let u∗(x) be a control satisfying (6)-(7) + +with ∥u∗(x)∥ ≤ B(x). Let us show that u∗(x) satis- +fies (12). By using the characterization of the matrix norm +induced by the Euclidean norm in [19, Example 5.6.6], +we get ∥Gh(x) +� +1 +u∗(x) +� +∥2 ≤ σmax(Gh(x)) +� +1 + B2(x) and +∥GV (x) +� +1 +u∗(x) +� +∥2 +≤ +σmax(GV (x)) +� +1 + B2(x). Using +now (11), we deduce that u∗(x) satisfies (12). +Remark 4.3: (Tightness of conditions for SOCC compati- +bility): The assumption that h is an ηh-robust CBF makes it +possible for the conditions (10b) and (11b) to be satisfied +at ∂C. Note also that (10) are trivially satisfied if the +estimation errors are zero and (11) are trivially satisfied +if the matrices GV (x), Gh(x) are zero, which holds if +the variances s2 +V , s2 +h are zero. Larger values of S(x) and +ζ(h(x)), and smaller values of B(x), lead to conditions that +are easier to satisfy. Closer to the origin, S(x) becomes +smaller, thus making (10a) and (11a) harder to satisfy. If +ζ(h(x)) is unknown, a known lower bound for it (e.g., 0) +can be used at the expense of more conservativeness. +• +Remark 4.4: (Computation of upper bound of safe stabi- +lizing controller): One can obtain B in Proposition 4.2 by +relying on the expression for a safe stabilizing controller +provided in [17], together with upper and lower bounds on +the norms of f, g, h and ∇h, to obtain B(x). +• +V. DESIGN AND REGULARITY ANALYSIS OF +CONTROLLERS SATISFYING SOCCS +In this section, we study the existence and regularity +properties of controllers satisfying sets of SOCCs. Our first +result establishes that, if a set of state-dependent SOCCs +are strictly compatible, then there exists a smooth controller +satisfying all of them simultaneously. +Proposition 5.1: (Existence of a smooth controller satis- +fying a finite number of SOCCs): For i ∈ {1, . . . , p}, let +Qi : Rn → R(m+1)×m, ri : Rn → Rm+1, bi : Rn → +Rm, ci : Rn → R be continuous functions on an open +set G ⊂ Rn. If the p SOCC inequalities ∥Qi(x)u+ri(x)∥ ≤ +bi(x)u + ci, i ∈ {1, . . . , p}, are strictly compatible on G, +then there exists a C∞(G) function k : G → Rm such +that ∥Qi(x)k(x) + ri(x)∥ ≤ bi(x)k(x) + ci(x) for all +i ∈ {1, . . . , p} and all x ∈ G. +This result is an extension of [20, Proposition 3.1] to +a finite set of SOCCs. Since SOCCs define convex sets, +the proof follows an identical argument and we omit it for +space reasons. The combination of Propositions 4.2 and 5.1 +guarantees the smooth safe stabilization of (1) under either +worst-case or probabilistic uncertainty. +Corollary 5.2: (Smooth safe stabilization under uncer- +tainty): Let ˜C be a neighborhood of C, h be an ηh-robust +CBF, and assume (6) and (7) are compatible on C and +continuous in C: +• If (10) holds for all x ∈ ˜C\{0}, then there exists a +controller k : ˜C → Rm, smooth on ˜C\{0}, such that all +trajectories of ˙x = f(x)+g(x)k(x) starting at C remain +in C and asymptotically converge to the origin; +• If (11) holds for all x ∈ ˜C\{0}, then there exists a +controller k : ˜C → Rm, smooth on ˜C\{0}, such that +all trajectories of ˙x = f(x) + g(x)k(x) starting at C +remain in C and asymptotically converge to the origin +with probability at least 1 − 2δ. +Corollary 5.2 establishes the existence of a smooth safe +stabilizing controller under uncertainty, but does not provide +an explicit closed-form design that can be used for imple- +mentation. In what follows, we provide controller designs +that are explicit but have weaker regularity properties. Let +u∗(x) = arg min +u∈Rm +1 +2∥u∥2, +(13) +s.t. +∥Qi(x)u + ri(x)∥ ≤ bi(x)u + ci(x), i ∈ {1, . . . , p}. +Note that this program can be written as a second-order +convex program (SOCP), as shown in [21, Section 2.2]. If +the constraints in (13) are either (8) or (9), we refer to (13) +as CLF-CBF-SOCP. The following result establishes that u∗ +is locally Lipschitz if the constraints are strictly compatible. +Proposition 5.3: (Lipschitzness of SOCP solution): Let +{Qi, ri, bi, ci}p +i=1 be twice continuously differentiable and +assume the constraints in (13) are strictly compatible on an +open set G ⊆ Rn. Then u∗ is locally Lipschitz on G. +Proof: +First consider the points x +∈ +G where +∥Qi(x)u∗(x) + ri(x)∥ ̸= 0. At these points, the constraints +of (13) are twice continuously differentiable in x and u +in a neighborhood of the optimizer. Moreover, since the +constraints in (13) are strictly compatible, for any ϵ > +0 there exists ˆux +ϵ satisfying them strictly and such that +∥u∗(x) − ˆux +ϵ ∥ < ϵ. Since none of the constraints are active +at ˆux +ϵ , the Mangasarian-Fromovitz Constraint Qualification +(MFCQ) holds at ˆux +ϵ . By [22, Lemma 6.1] this implies that +MFCQ also holds at u∗(x). Furthermore, since the objective +function in (13) is strongly convex and the constraints are +convex, the second-order condition (SOC2) [22, Definition +6.1] holds and by [22, Theorem 6.4], u∗ is locally Lipschitz +at x. Next, consider any point x ∈ G where Ix = {i ∈ +{1, . . . , p} +: +∥Qi(x)u∗(x) + ri(x)∥ = 0} is nonempty. +Since the constraint ∥Qi(x)u + ri(x)∥ ≤ bi(x)u + ci(x) +is not differentiable at those points, we square the SOCCs +in (13) associated to Ix to obtain the equivalent formulation +with twice-continuously differentiable constraints: +u∗(x) = arg min +u∈Rm +1 +2∥u∥2, +(14) +s.t. +gi,1(x, u) ≤ 0, gi,2(x, u) ≤ 0, i ∈ Ix, +∥Qi(x)u + ri(x)∥ ≤ bi(x)u + ci(x), i ∈ {1, . . . , p}\Ix, +where +gi,1(x, u) = uT (Qi(x)T Qi(x) − bi(x)bi(x)T )u + ri(x)2 ++ 2(Qi(x)T ri(x) − ci(x)bi(x))T u − ci(x)2, +gi,2(x, u) = −bi(x)T u − ci(x). +Strict compatibility of the constraints in (13) implies the +strict compatibility of the constraints in (14) and, by the +same argument as before, MFCQ holds at the optimizer. To +show that SOC2 also holds for (14), note that the constraints +gi,1(x, u) ≤ 0 for i ∈ Ix cannot be active at the optimizer +(otherwise, that would imply that bi(x)u∗(x) + ci(x) = 0, +implying that MFCQ is violated at the optimizer, reaching a + +contradiction). Thus, by the strict complementarity condition, +the Lagrange multipliers associated with the constraints +gi,1(x, u), i ∈ Ix are zero and the Hessian of the Lagrangian +L of (14) at the optimizer takes the form +∇2 +uL(u∗, {λi}i∈J0)x = Im + +� +i∈J0(x) +λi(x)∇2 +u˜gi(x, u∗(x)), +where J0(x) ⊂ {1, . . . , p}\Ix is the set of active constraints +at x, λi is the Lagrange multiplier associated with the +constraint ∥Qi(x)u + ri(x)∥ ≤ bi(x)u + ci for i ̸∈ Ix +and ˜gi(x, u) = ∥Qi(x)u + ri(x)∥ − bi(x)u − ci(x). Since +∥Qi(x)u∗(x) + ri(x)∥ ̸= 0 for the active constraints, their +Hessian is well-defined and is positive semidefinite due to +their convexity, making ∇2 +uL(u∗, {λi}i∈J0)x positive def- +inite. Hence, SOC2 holds for (14) at the optimizer and, +by [22, Theorem 6.4], u∗ is locally Lipschitz at x. +As a consequence of Proposition 5.3, we conclude that if +∇V , the estimates ˆf, ˆg, ˆh, � +∇h, and worst-case error bounds +(resp. means and variances) that appear in (8) (resp. (9)) +are twice continuously differentiable and the conditions (10) +(resp. (11)) hold, then the corresponding CLF-CBF-SOCP +controller is locally Lipschitz, safe, and stabilizing. +Next we provide a formula, inspired by Sontag’s universal +formula [23], for a smooth controller satisfying a single +SOCC defined by smooth functions. +Proposition 5.4: (Universal formula for a controller sat- +isfying one SOCC): Let l ∈ Z>0 and assume Q : Rn → +R(m+1)×m, r : Rn → Rm+1, b : Rn → Rm, and c : Rn → +R are l-continuously differentiable on an open set G ⊆ Rn. +Suppose that the SOCC ∥Q(x)u + r(x)∥ ≤ b(x)u + c(x) +is strictly feasible on G and Q(x)T Q(x) is invertible for +all x ∈ G. Let ˜b(x) = b(x)(QT (x)Q(x))−1QT (x), ˜c(x) = +c(x) − ˜b(x)r(x), ¯b(x) = (∥˜b(x)∥ − 1)∥˜b(x)∥, and +vs(x) = +�0 +if ∥˜b(x)∥ ≤ 1, +−˜c(x)+√ +˜c(x)2+¯b(x)2 +¯b(x) +˜b(x) +if ∥˜b(x)∥ > 1. +(15) +Further assume vs(x)−r(x) ∈ Im(Q(x)) for all x ∈ G. Then +us(x) := (QT (x)Q(x))−1QT (x)(vs(x) − r(x)), +is l-continuously differentiable for all x ∈ G. Moreover, +∥Q(x)us(x) + r(x)∥ ≤ b(x)us(x) + c(x) for all x ∈ G. +Proof: +Let v = Q(x)u + r(x). Since QT (x)Q(x) is +invertible and ∥Q(x)u + r(x)∥ ≤ b(x)u + c(x) is strictly +feasible on G, the resulting SOCC ∥v∥ ≤ ˜b(x)v + ˜c(x) is +also strictly feasible on G. Moreover, vs satisfies it. Indeed, +if ∥˜b(x)∥ ≤ 1, since the SOCC is feasible there exists v∗ +such that ∥v∗∥ ≤ ˜b(x)v∗ +˜c(x) and it follows that ˜c(x) ≥ 0. +The case ∥˜b(x)∥ > 1 follows from a direct calculation. If +∥˜b(x)∥ ̸= 1, vs is Cl at x because ˜b and ˜c are Cl at x. If +∥˜b(x)∥ = 1, then ˜c(x) ̸= 0 (otherwise, if ˜c(x) = 0, since +the SOCC ∥v∥ ≤ ˜b(x)v + ˜c(x) is strictly compatible, there +would exist ˆv such that ∥ˆv∥ < ˜b(x)v ≤ ∥ˆv∥, which is a +contradiction). Now, from the proof of [23, Theorem 1], the +function +φ(c, α) := +� +0 +if α ≤ 0, +−c+ +√ +c2+α2 +α +else, +is analytic at points of the form (c, 0), with c ̸= 0, so vs is +Cl for all x ∈ G. Moreover, since vs(x) − r(x) ∈ Im(Q(x)) +for all x ∈ G, it also follows that ∥Q(x)us(x) + r(x)∥ ≤ +b(x)us(x) + c(x) for all x ∈ G and us is Cl for all x ∈ G. +From the proof of Proposition 5.4, we observe that in +the case where the SOCC takes the form (8), a simpler +expression is available for a controller satisfying it. As a +result of Proposition 5.4, if uncertainty estimates of the +dynamics and barrier function are available as described in +Section II-B, the proposed formula can be used to guarantee +either safety or stability. +VI. SIMULATIONS +In this section we illustrate our results in an example. For +simplicity, we focus on the case of worst-case error estimates. +Consider the planar linear system +d +dt +� +x +y +� += +� +1 +0 +0 +1 +� � +x +y +� +� +�� +� +f(x,y) ++ +� +1 +0 +0 +1 +� +� +�� +� +g(x,y) +� +u1 +u2 +� +. +(16) +We consider the CBF h(x, y) = x2 + (y − 4)2 − 4. +From data to estimates and error bounds: We obtain +here worst-case error models, cf. Section II-B, from data. +For simplicity, we assume the estimates of the CBF are +exact, ˆh = h, � +∇h = ∇h, and that prior knowledge of +(not necessarily tight) Lipschitz constants (Kf = 3.0 and +Kg = 0.5) of f and g is available. We have access to an +oracle that, given a query point y ∈ C, returns noiseless +measurements (f(y), g(y)) of the functions in (1) (the noisy +case can be considered without major modifications). Given a +set of N measurements D = {xi, f(xi), g(xi)}N +i=1 obtained +by querying the oracle, we estimate f at x ∈ Rn as +ˆf(x) = f(pcl(x)), where pcl(x) is the closest datapoint +to x, with corresponding worst-case error bound ef(x) := +Kf∥x − pcl(x)∥. We do similarly for ˆg and eg. +Performance dependency on error estimates: Here we +illustrate how smaller estimation errors lead to improved +performance. We use different datasets with different number +of data points N to generate ˆf, ˆg, ef and eg. We solve the +resulting CLF-CBF-SOCP every 0.01s with initial condition +at (2.0, 6.0) and plot the trajectories until it becomes unfeasi- +ble. We compare the results for different N in Figure 1(left). +Larger datasets with data from a neighborhood of the origin +allow trajectories to converge closer to the origin before the +problem becomes unfeasible. In the plot, we also observe +that the conditions (10) serve as a good indicator of when +the SOCP actually becomes unfeasible, hence illustrating +how Proposition 4.2 can be used to infer when the available +estimates are insufficient to guarantee that the controller is +well defined. +Online safe stabilization: We illustrate also the case where +data is collected online. We start from an initial set of 25 +measurements of f and g near the initial condition obtained +by querying the oracle. Given an initial condition, we find +the CLF-CBF-SOCP controller every 0.01s. If during the +execution (13) becomes infeasible at some point ¯x, we query +the oracle to obtain measurements of f and g at ¯x (making it + +Fig. 1: Safe stabilization of a planar system with worst-case uncertainty error bounds. The green ball is the set of unsafe states and black dots denote +initial conditions. (Left) Dashed lines enclose the region where data is located for different N. Solid lines show the evolution under the CLF-CBF-SOCP +controller in (13) computed with different N. Black triangles indicate points where the sufficient conditions for feasibility (10) do not hold. (Right) The +dashed lines display the evolution of a model-based min-norm controller (CLF-CBF QP) [4], and the solid lines display the evolution of a controller +obtained by solving the CLF-CBF-SOCP (13). Black stars denote points where measurements have been taken. +feasible) and a small neighborhood around it (for improved +performance). Figure 1(right) illustrates executions of this +procedure for three different initial conditions. As trajectories +approach the origin, more measurements need to be taken +because the conditions for feasibility of the CLF-CBF-SOCP, +cf. (10) in Proposition 4.2, become harder to satisfy. +VII. CONCLUSIONS +We have studied conditions to ensure the safe stabilization +of a nonlinear affine control system under uncertainty. Given +a known control Lyapunov function and either worst-case or +probabilistic estimates of the dynamics and CBF, SOCCs +encode the impact of uncertainty on the ability to guarantee +stability and safety. We have provided explicit bounds on the +error estimates that ensure the relevant SOCCs are compat- +ible and built on them to ensure the existence of a smooth +safe stabilizing controller, the Lipschitz regularity of the min- +norm CLF-CBF-SOCP-based controller, and the regularity +of a universal controller for the satisfaction of a single +SOCC. Future work will characterize the conditions for +compatibility in terms of data, design online safe stabilization +mechanisms that balance computational effort, sampling rate, +and performance using resource-aware control, explore the +design of universal formulas for more than one SOCC, and +implement our results on physical testbeds. +REFERENCES +[1] Z. Artstein, “Stabilization with relaxed controls,” Nonlinear Analysis, +vol. 7, no. 11, pp. 1163–1173, 1983. +[2] P. Wieland and F. Allg¨ower, “Constructive safety using control barrier +functions,” IFAC Proceedings Volumes, vol. 40, no. 12, pp. 462–467, +2007. +[3] A. D. Ames, S. Coogan, M. Egerstedt, G. Notomista, K. Sreenath, +and P. Tabuada, “Control barrier functions: theory and applications,” +in European Control Conference, Naples, Italy, Jun. 2019, pp. 3420– +3431. +[4] X. Xu, P. Tabuada, J. W. Grizzle, and A. D. Ames, “Robustness of con- +trol barrier functions for safety critical control,” IFAC-PapersOnLine, +vol. 48, no. 27, pp. 54–61, 2015. +[5] A. D. Ames, X. Xu, J. W. Grizzle, and P. Tabuada, “Control barrier +function based quadratic programs for safety critical systems,” IEEE +Transactions on Automatic Control, vol. 62, no. 8, pp. 3861–3876, +2017. +[6] M. Jankovic, “Robust control barrier functions for constrained sta- +bilization of nonlinear systems,” Automatica, vol. 96, pp. 359–367, +2018. +[7] F. Casta˜neda, J. J. Choi, B. Zhang, C. J. Tomlin, and K. Sreenath, +“Pointwise feasibility of Gaussian process-based safety-critical control +under model uncertainty,” in IEEE Conf. on Decision and Control, +Austin, Texas, USA, 2021, pp. 6762–6769. +[8] ——, “Gaussian process-based min-norm stabilizing controller for +control-affine systems with uncertain input effects and dynamics,” +in American Control Conference, New Orleans, LA, May 2021, pp. +3683–3690. +[9] K. Long, V. Dhiman, M. Leok, J. Cort´es, and N. Atanasov, “Safe +control synthesis with uncertain dynamics and constraints,” IEEE +Robotics and Automation Letters, vol. 7, no. 3, pp. 7295–7302, 2022. +[10] K. Long, C. Qian, J. Cort´es, and N. Atanasov, “Learning barrier +functions with memory for robust safe navigation,” IEEE Robotics +and Automation Letters, vol. 6, no. 3, pp. 4931–4938, 2021. +[11] K. Long, Y. Yi, J. Cort´es, and N. Atanasov, “Safe and stable control +synthesis for uncertain system models via distributionally robust +optimization,” https://arxiv.org/pdf/2210.01341.pdf, 2022. +[12] A. A. Ahmadi and A. Majumdar, “Some applications of polynomial +optimization in operations research and real-time decision making,” +Optimization Letters, vol. 10, no. 4, p. 709–729, 2016. +[13] A. J. Taylor, V. D. Dorobantu, H. M. Le, Y. Yue, and A. D. Ames, +“Episodic learning with control Lyapunov functions for uncertain +robotic systems,” in IEEE/RSJ Int. Conf. on Intelligent Robots & +Systems, Macau, 2019, pp. 6878–6884. +[14] V. Dhiman, M. J. Khojasteh, M. Franceschetti, and N. Atanasov, +“Control barriers in Bayesian learning of system dynamics,” IEEE +Transactions on Automatic Control, 2021, early access. +[15] F. Casta˜neda, J. J. Choi, W. Jung, B. Zhang, C. J. Tomlin, and +K. Sreenath, “Probabilistic safe online learning with control barrier +functions,” https://arxiv.org/pdf/2208.10733.pdf, 2022. +[16] E. D. Sontag, Mathematical Control Theory: Deterministic Finite +Dimensional Systems, 2nd ed., ser. TAM. +Springer, 1998, vol. 6. +[17] P. Mestres and J. Cort´es, “Optimization-based safe stabilizing feedback +with guaranteed region of attraction,” IEEE Control Systems Letters, +vol. 7, pp. 367–372, 2023. +[18] N. Srinivas, A. Krause, S. Kakade, and M. Seeger, “Gaussian process +optimization in the bandit setting: no regret and experimental design,” +in International Conference on Machine Learning, 2010, pp. 1015– +1022. +[19] R. A. Horn and C. R. Johnson, Matrix Analysis. Cambridge University +Press, 2012. +[20] P. Ong and J. Cort´es, “Universal formula for smooth safe stabilization,” +in IEEE Conf. on Decision and Control, Nice, France, Dec. 2019, pp. +2373–2378. +[21] F. Alizadeh and D. Goldfarb, “Second-order cone programming,” +Mathematical Programming, vol. 95, no. 1, pp. 3–51, 2003. +[22] G. Still, “Lectures on Parametric Optimization: An Introduction,” +Optimization Online, 2018. +[23] E. D. Sontag, “A universal construction of Artstein’s theorem on +nonlinear stabilization,” Systems & Control Letters, vol. 13, no. 2, +pp. 117–123, 1989. + +N= 100 +6 +N= 169 +N= 256 +5 +N=361 +4 +3 +2 +1 : +0 +-1 +-2 +-6 +-2 +0 +2 +4 +6 +X6 +4 +2 +0 +-2 +-4 +-2 +0 +2 +4 +X \ No newline at end of file diff --git a/wNE3T4oBgHgl3EQflArR/content/tmp_files/load_file.txt b/wNE3T4oBgHgl3EQflArR/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dd95c0f7380b2bbf6162c3c46b713c5e649d53a7 --- /dev/null +++ b/wNE3T4oBgHgl3EQflArR/content/tmp_files/load_file.txt @@ -0,0 +1,464 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf,len=463 +page_content='Feasibility and Regularity Analysis of Safe Stabilizing Controllers under Uncertainty Pol Mestres Jorge Cort´es Abstract— This paper studies the problem of safe stabiliza- tion of control-affine systems under uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Our starting point is the availability of worst-case or probabilistic error descriptions for the dynamics and a control barrier function (CBF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' These descriptions give rise to second-order cone con- straints (SOCCs) whose simultaneous satisfaction guarantees safe stabilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' We study the feasibility of such SOCCs and the regularity properties of various controllers satisfying them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' INTRODUCTION The last years have seen a dramatic increase in the deployment of robotic systems in diverse areas like home automation and autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' In these applications, it is critical that robots satisfy simultaneously safety and per- formance specifications in the presence of model uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Controllers that achieve these goals are usually defined using tools from stability analysis and Lyapunov theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' However, this faces several challenges, such as the development of understanding about the level of model uncertainty that can be tolerated while still being able to meet safety and sta- bility requirements and the characterization of the regularity properties, along with the identification of suitable conditions to ensure them, in order to be implementable in real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' These are the questions that motivate this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Literature Review: Control Lyapunov functions (CLFs) [1] are a well-established tool for designing stabilizing con- trollers for nonlinear systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' More recently, control barrier functions (CBFs) [2], [3] have been introduced as a tool to render a certain predefined set safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' If the system is control affine, the CLF and CBF conditions can be incorporated in a quadratic program (QP) [4], [5] that can be efficiently solved online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' However, this design assumes complete knowledge of the dynamics and safe set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Several recent papers have pro- posed alternative formulations of the CLF-CBF QP for sys- tems with uncertainty or learned dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' In [6], the CLF- CBF QP is transformed into another QP to account for worst- case uncertainty bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' However, imperfect knowledge of the system dynamics or safety constraints transforms the affine-in-the-input inequalities arising from CBFs and CLFs into second-order cone constraints (SOCCs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' The papers [7], [8] leverage Gaussian Processes (GPs) to learn the system dynamics from data and show that the mean and variance of the estimated GP can be used to formulate two SOCCs whose pointwise satisfaction implies safe stabilization of the true system with a prescribed probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' However, during the control design stage, the SOCC associated to stability is P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Mestres and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Cort´es are with the Department of Mechanical and Aerospace Engineering, UC San Diego, {pomestre,cortes}@ucsd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='edu relaxed and hence the resulting controller does not have sta- bility guarantees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' In the case where worst-case error bounds for the dynamics and the CBF are known, [9], [10] show how the satisfaction of two SOCCs can yield a safe stabilizing controller valid for all models consistent with these error bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' [11] uses the framework of distributionally robust optimization to formulate a second-order convex program that achieves safe stabilization for systems where only a finite set of model parametric uncertainty samples are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' However, these works lack guarantees on the simultaneous feasibility of these SOCCs and the regularity of controllers satisfying them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Hence, the resulting controllers might be undefined in practice, resulting in unsafe, unstable, or dis- continuous behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Our work also relies on techniques to design CLFs without an accurate model of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' For instance, for polynomial systems with uncertainty, [12] uses sum-of-squares techniques to find robust Lyapunov functions based on known error bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Alternatively, [13] describes a method that only requires knowledge of the degree of actuation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Finally, the papers [14], [15] utilize online data to improve the estimates of the dynamics and synthesize (also via SOCCs) less conservative controllers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Statement of Contributions: We study1 the problem of safe stabilization of control-affine systems under uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' We consider two scenarios for the estimates of the dynamics and safe set: either worst-case error bounds or probabilistic descriptions in the form of Gaussian Processes (GPs) are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' In both cases, the problem of designing a safe stabilizing controller can be reduced to satisfying two SOCCs 1We use the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' We denote by Z>0, R, and R≥0 the set of positive integers, real, and nonnegative real numbers, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' We denote by 0n the n-dimensional zero vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' We write int(S) and ∂S for the interior and the boundary of the set S, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Given x ∈ Rn, ∥x∥ denotes the Euclidean norm of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Given f : Rn → Rn, g : Rn → Rn×m and a smooth function W : Rn → R, LfW : Rn → R (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' LgW : Rn → Rm) denotes the Lie derivative of W with respect to f (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' g), that is LfW = ∇W T f (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' ∇W T g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' We use GP(µ(x), K(x, x′)) to denote a Gaussian Process distribution with mean function µ(x) and covariance function K(x, x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' We denote by Cl(A) the set of l-times continuously differentiable functions on an open set A ⊆ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' A function β : R → R is of class K if β(0) = 0 and β is strictly increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' If moreover limt→∞ β(t) = ∞, then β is of class K∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' A function V : Rn → R is positive definite if V (0) = 0 and V (x) > 0 for all x ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' V is proper in a set Γ if the set {x ∈ Γ : V (x) ≤ c} is compact for any c ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' A set C ⊆ Rn is forward invariant under the dynamical system ˙x = f(x) if any trajectory with initial condition in C at time t = 0 remains in C for all positive times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' A set C is safe for ˙x = f(x, u) if there exists a locally Lipschitz control u : Rn → Rm such that C is forward invariant for ˙x = f(x, u(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Given m × n matrix A and two integers i, j such that 1 ≤ i < j ≤ n, Ai:j denotes the m × (j − i + 1) matrix obtained by selecting the columns from i to j of A, and σmax(A) denotes the maximum singular value of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' We denote by Br(p) = {y ∈ Rn : ∥y − p∥ < r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Given A ∈ Rq×n, b ∈ Rq, c ∈ Rn, d ∈ R, the inequality ∥Ax + b∥ ≤ cT x + d is a second-order cone constraint (SOCC) in the variable x ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='04603v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='OC] 11 Jan 2023 at every point in the safe set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Our first contribution consists of giving sufficient conditions for the feasibility of each pair of SOCCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Second, if each pair of SOCCs is feasible, we show that there exists a smooth safe stabilizing controller and that the minimum-norm controller satisfying each pair of SOCCs is locally Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Third, we provide a universal formula for satisfying a single SOCC and hence achieving either safety or stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' We illustrate our results in the safe stabilization of a planar system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' PRELIMINARIES This section presents preliminaries on control Lyapunov and barrier functions, and safe stabilization using worst-case and probabilistic estimates of the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Control Lyapunov and Barrier Functions Consider the control-affine system ˙x = f(x) + g(x)u, (1) where f : Rn → Rn and g : Rn → Rm×n are locally Lipschitz functions, with x ∈ Rn the state and u ∈ Rm the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' We assume without loss of generality that f(0n) = 0n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='1: (Control Lyapunov Function [16]): Given a set Γ ⊆ Rn, with 0n ∈ Γ, a continuously differentiable function V : Rn → R is a CLF on Γ for the system (1) if it is proper in Γ, positive definite, and there exists a continuous positive definite function W : Rn → R such that, for each x ∈ Γ\\{0}, there exists a control u ∈ Rm satisfying LfV (x) + LgV (x)u ≤ −W(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' (2) A Lipschitz controller k : Rn → Rm such that u = k(x) satisfies (2) for all x ∈ D\\{0} makes the origin of the closed- loop system asymptotically stable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Hence, CLFs provide a way to guarantee asymptotic stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='2: (Robust Control Barrier Function [6, Def- inition 6]): Let C ⊂ Rn and h : Rn → R continuously differentiable such that C = {x ∈ Rn : h(x) ≥ 0}, (3a) ∂C = {x ∈ Rn : h(x) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' (3b) Given ηh > 0, h is an ηh-robust CBF if there exists a K∞ function α such that for all x ∈ C, there exists u ∈ Rm with Lfh(x) + Lgh(x)u + α(h(x)) ≥ ηh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' (4) When ηh = 0, this definition reduces to the notion of CBF [3, Definition 2], and the inequality reduces to Lfh(x) + Lgh(x)u + α(h(x)) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' (5) Note that all robust CBFs are CBFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' A Lipschitz controller k : Rn → Rm such that u = k(x) satisfies (5) for all x ∈ C makes C forward invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Hence, CBFs provide a way to guarantee safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='3: (Slack in CLF and CBF conditions): Without loss of generality, if V is a CLF on an open set Γ, we can assume there exists a positive definite function S such that, for all x ∈ Γ, there is u ∈ Rm with LfV (x) + LgV (x)u + W(x) ≤ −S(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' (6) This is because we can always define a positive definite function ˜W(x) := 1 2W(x) in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='1 and take S(x) := 1 2W(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Similarly, if h is an ηh-robust CBF, then there exists a class K∞ function ζ such that for all x ∈ C, there is u ∈ Rm with Lfh(x) + Lgh(x)u + α(h(x)) ≥ ηh + ζ(h(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' (7) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Robust and Probabilistic Safe Stabilization We are interested in the design of controllers that ensure simultaneously stability and safety in the presence of uncer- tainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Assume that the maps f, g in (1) and the CBF h and its gradient ∇h are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Instead, estimates ˆf, ˆg, ˆh and � ∇h are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' We also assume that a CLF V for the system (1) is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' We consider two types of models for the errors between the estimates and the true quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' First, for x ∈ Rn, consider worst-case error bounds as follows: ∥f(x) − ˆf(x)∥ ≤ ef(x), ∥g(x) − ˆg(x)∥ ≤ eg(x), ∥h(x) − ˆh(x)∥ ≤ eh(x), ∥∇h(x) − � ∇h(x)∥ ≤ e∇h(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Since the exact dynamics and CBF are unknown, one can not certify the inequalities (2) and (5) directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Instead, using the error bounds above, define aV (x) = ∥∇V (x)∥eg(x), bV (x) = −∇V (x)T ˆg(x), cV (x) = −∥∇V (x)∥ef(x) − ∇V (x)T ˆf(x) − W(x), ah(x) = e∇h(x)eg(x) + e∇h(x)∥ˆg(x)∥ + ∥ � ∇h(x)∥eg(x), bh(x) = � ∇h(x)T ˆg(x), ch(x) = −e∇h(x)ef(x) − e∇h(x)∥ ˆf(x)∥ − ∥ � ∇h(x)∥ef(x) + � ∇h(x)T ˆf(x) + α(ˆh(x) − eh(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' According to [9, Proposition V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='I], if the two (state-dependent) SOCCs (in u): aV (x)∥u∥ ≤ bV (x)u + cV (x), (8a) ah(x)∥u∥ ≤ bh(x)u + ch(x), (8b) are satisfied for all x ∈ C, then (2) and (5) hold for all x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' This result provides a way of designing controllers that simultaneously satisfy (2) and (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Second, suppose that GP estimates are available for the following quantities [7]: ∆V (x, u) = LfV (x) + LgV (x)u − L ˆ fV (x) − LˆgV (x)u, ∆h(x, u) = Lfh(x) + Lgh(x)u + α(h(x)) − � ∇h(x) T ˆf(x) − � ∇h(x) T ˆg(x)u − α(ˆh(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Let µV (x, u) and s2 V (x, u) denote the mean and variance, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=', of the GP prediction of ∆V , which we assume affine and quadratic in u, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Therefore, there exist γV (x) : Rn → Rm+1 and GV (x) ∈ R(m+1)×(m+1) such that µV (x, u) = γV (x)T � 1 u � , sV (x, u) = ∥GV (x) � 1 u � ∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' For the GP prediction of ∆h, let γh(x), and Gh(x) be defined analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Since the exact dynamics and CBF are unknown, one cannot certify the inequalities (2) and (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' However, for δ ∈ (0, 1), and using the GP predictions, define QV (x) = β(δ)GV,2:(m+1)(x) ∈ R(m+1)×m, rV (x) = β(δ)GV,1(x) ∈ R(m+1)×1, bV (x) = −LˆgV (x) − γT V,2:(m+1)(x) ∈ R1×m, cV (x) = −L ˆ fV (x) − W(x) − γV,1(x) ∈ R, and similarly Qh, rh, bh and ch (the exact form of β(δ) is given in [8, Theorem 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Then, according to [7, Section IV], if the two SOCCs ∥QV (x)u + rV (x)∥ ≤ bV (x)u + cV (x), (9a) ∥Qh(x)u + rh(x)∥ ≤ bh(x)u + ch(x), (9b) are satisfied for all x ∈ C, then (2) and (5) each hold for all x ∈ C with probability at least 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='4: (General form of SOCCs): By taking in (9a), QV (x) = aV (x) �Im 0T m � , rV (x) = 0m+1, we obtain (8a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Hence, in the rest of the paper, we derive the results for SOCCs of the most general form (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' PROBLEM STATEMENT We consider a control-affine system of the form (1) and a safe set C of the form (3), with f, g, h and ∇h unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' We assume that either worst-case or probabilistic descriptions of the dynamics and barrier function are available, as described in Section II-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' We make the following assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Assumption 1: The function h is a CBF of C and ∂h ∂x(x) ̸= 0 for all x ∈ ∂C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' By [3, Theorem 2], Assumption 1 implies that C is safe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' However, since the true h is unknown, a safe controller is not readily computable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Assumption 2: A CLF V on an open set containing the origin is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Given this setup, our goals are to (i) derive conditions that ensure the feasibility of the pair of robust stability (8a) and safety-(8b) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=', probabilistic stability (9a) and safety (9b)) inequalities and, building on this, (ii) design controllers that jointly satisfy the inequalities pointwise in C and charac- terize their regularity properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' The latter is motivated by both theoretical (guarantee the existence and uniqueness of solutions to the closed-loop system) and practical (ease of implementation of feedback control on digital platforms and avoidance of chattering behavior) considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' COMPATIBILITY OF PAIRS OF SECOND-ORDER CONE CONSTRAINTS In this section, we derive sufficient conditions that guar- antee the feasibility of the pairs of inequalities in (8) and in (9), resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' The following definition extends the notion of compatibility [17, Definition 3] to any set of inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='1: (Compatibility of a set of inequalities): Given functions qi : Rn × Rm → R for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' , p}, the inequalities qi(x, u) ≤ 0, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' , p} are (strictly) compatible at a point x ∈ Rn if there exists a corresponding u ∈ Rm satisfying all inequalities (strictly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' The same inequalities are (strictly) compatible on a set G if they are (strictly) compatible at every x ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' As the estimation errors (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' the variances) approach zero, the inequalities in (8) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' (9)) approach (2) and (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' If (6) and (7) are compatible, the next result provides explicit bounds for the estimation errors such that (8a)-(8b) and (9a)- (9b) are strictly compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='2: (Compatibility in worst-case and prob- abilistic SOCCs): Let h be an ηh-robust CBF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Assume (6) and (7) are compatible on C, and let B : Rn → R be an upper bound on the norm of a control satisfying both inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' (i) If, for all x ∈ C\\{0}, ∥∇V (x)∥(ef(x) + eg(x)B(x)) < 1 2S(x), (10a) (e∇h(x) + ∥ � ∇h(x)∥)(ef(x) + eg(x)B(x)) + Kαeh(x) + e∇h(x)(∥ ˆf(x)∥ + ∥ˆg(x)∥B(x)) < 1 2(ηh + ζ(h(x))), (10b) where Kα is the Lipschitz constant of α, then (8a) and (8b) are strictly compatible on C\\{0};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' (ii) If for all x ∈ C\\{0}, σmax(GV (x)) < S(x) 2β(δ) � 1 + B2(x) , (11a) σmax(Gh(x)) < ηh + ζ(h(x)) 2β(δ) � 1 + B2(x) , (11b) then (9a) and (9b) are strictly compatible on C\\{0} with probability greater than or equal to 1 − 2δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Proof: (i) The proof follows by applying the definition of the upper bounds ef, eg, eh, e∇h given in Section II-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' (ii) Note that (9a), (9b) can be equivalently written as: β(δ)∥GV (x) � 1 u � ∥2 ≤ −∇V (x)T ˆf(x) − γV,1(x) − W(x) − (∇V (x)T ˆg(x) + γT V,2:(m+1)(x))u, β(δ)∥Gh(x) � 1 u � ∥2 ≤ � ∇h(x) T ˆf(x) + γh,1(x) + α(h(x))+ ( � ∇h(x) T ˆg(x) + γT h,2:(m+1)(x))u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Now, note that −L ˆ fV (x) − LˆgV (x)u − γV,1(x) − γT V,2:(m+1)(x)u = −LfV (x) − LgV (x)u + ∆V (x, u) − γV,1(x) − γT V,2:(m+1)(x)u, and similarly for the safety con- straint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Define then the events EV = {|γV (x)T � 1 u � − ∆V (x, u)| ≤ βsV (x, u), ∀x ∈ C, u ∈ Rm} and Eh = {|γh(x)T � 1 u � − ∆h(x, u)| ≤ β(δ)sh(x, u), ∀ x ∈ C, u ∈ Rm}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' By [18, Theorem 6], P(EV ) ≥ 1−δ and P(Eh) ≥ 1−δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Therefore, P(EV ∩ Eh) = P(EV ) + P(Eh) − P(EV ∪ Eh) ≥ 1 − 2δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Hence, if we can find u ∈ Rm satisfying Lfh(x) + Lgh(x)u + α(h(x)) ≥ 2β(δ)sh(x, u), (12a) −LfV (x) − LgV (x)u − W(x) ≥ 2β(δ)sV (x, u), (12b) then (9a), (9b) are compatible at x with probability at least 1 − 2δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Let u∗(x) be a control satisfying (6)-(7) with ∥u∗(x)∥ ≤ B(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Let us show that u∗(x) satis- fies (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' By using the characterization of the matrix norm induced by the Euclidean norm in [19, Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='6], we get ∥Gh(x) � 1 u∗(x) � ∥2 ≤ σmax(Gh(x)) � 1 + B2(x) and ∥GV (x) � 1 u∗(x) � ∥2 ≤ σmax(GV (x)) � 1 + B2(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Using now (11), we deduce that u∗(x) satisfies (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='3: (Tightness of conditions for SOCC compati- bility): The assumption that h is an ηh-robust CBF makes it possible for the conditions (10b) and (11b) to be satisfied at ∂C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Note also that (10) are trivially satisfied if the estimation errors are zero and (11) are trivially satisfied if the matrices GV (x), Gh(x) are zero, which holds if the variances s2 V , s2 h are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Larger values of S(x) and ζ(h(x)), and smaller values of B(x), lead to conditions that are easier to satisfy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Closer to the origin, S(x) becomes smaller, thus making (10a) and (11a) harder to satisfy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' If ζ(h(x)) is unknown, a known lower bound for it (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=', 0) can be used at the expense of more conservativeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='4: (Computation of upper bound of safe stabi- lizing controller): One can obtain B in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='2 by relying on the expression for a safe stabilizing controller provided in [17], together with upper and lower bounds on the norms of f, g, h and ∇h, to obtain B(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' DESIGN AND REGULARITY ANALYSIS OF CONTROLLERS SATISFYING SOCCS In this section, we study the existence and regularity properties of controllers satisfying sets of SOCCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Our first result establishes that, if a set of state-dependent SOCCs are strictly compatible, then there exists a smooth controller satisfying all of them simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='1: (Existence of a smooth controller satis- fying a finite number of SOCCs): For i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' , p}, let Qi : Rn → R(m+1)×m, ri : Rn → Rm+1, bi : Rn → Rm, ci : Rn → R be continuous functions on an open set G ⊂ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' If the p SOCC inequalities ∥Qi(x)u+ri(x)∥ ≤ bi(x)u + ci, i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' , p}, are strictly compatible on G, then there exists a C∞(G) function k : G → Rm such that ∥Qi(x)k(x) + ri(x)∥ ≤ bi(x)k(x) + ci(x) for all i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' , p} and all x ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' This result is an extension of [20, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='1] to a finite set of SOCCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Since SOCCs define convex sets, the proof follows an identical argument and we omit it for space reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' The combination of Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='1 guarantees the smooth safe stabilization of (1) under either worst-case or probabilistic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='2: (Smooth safe stabilization under uncer- tainty): Let ˜C be a neighborhood of C, h be an ηh-robust CBF, and assume (6) and (7) are compatible on C and continuous in C: If (10) holds for all x ∈ ˜C\\{0}, then there exists a controller k : ˜C → Rm, smooth on ˜C\\{0}, such that all trajectories of ˙x = f(x)+g(x)k(x) starting at C remain in C and asymptotically converge to the origin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' If (11) holds for all x ∈ ˜C\\{0}, then there exists a controller k : ˜C → Rm, smooth on ˜C\\{0}, such that all trajectories of ˙x = f(x) + g(x)k(x) starting at C remain in C and asymptotically converge to the origin with probability at least 1 − 2δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='2 establishes the existence of a smooth safe stabilizing controller under uncertainty, but does not provide an explicit closed-form design that can be used for imple- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' In what follows, we provide controller designs that are explicit but have weaker regularity properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Let u∗(x) = arg min u∈Rm 1 2∥u∥2, (13) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' ∥Qi(x)u + ri(x)∥ ≤ bi(x)u + ci(x), i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' , p}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Note that this program can be written as a second-order convex program (SOCP), as shown in [21, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' If the constraints in (13) are either (8) or (9), we refer to (13) as CLF-CBF-SOCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' The following result establishes that u∗ is locally Lipschitz if the constraints are strictly compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='3: (Lipschitzness of SOCP solution): Let {Qi, ri, bi, ci}p i=1 be twice continuously differentiable and assume the constraints in (13) are strictly compatible on an open set G ⊆ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Then u∗ is locally Lipschitz on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Proof: First consider the points x ∈ G where ∥Qi(x)u∗(x) + ri(x)∥ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' At these points, the constraints of (13) are twice continuously differentiable in x and u in a neighborhood of the optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Moreover, since the constraints in (13) are strictly compatible, for any ϵ > 0 there exists ˆux ϵ satisfying them strictly and such that ∥u∗(x) − ˆux ϵ ∥ < ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Since none of the constraints are active at ˆux ϵ , the Mangasarian-Fromovitz Constraint Qualification (MFCQ) holds at ˆux ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' By [22, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='1] this implies that MFCQ also holds at u∗(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Furthermore, since the objective function in (13) is strongly convex and the constraints are convex, the second-order condition (SOC2) [22, Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='1] holds and by [22, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='4], u∗ is locally Lipschitz at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Next, consider any point x ∈ G where Ix = {i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' , p} : ∥Qi(x)u∗(x) + ri(x)∥ = 0} is nonempty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Since the constraint ∥Qi(x)u + ri(x)∥ ≤ bi(x)u + ci(x) is not differentiable at those points, we square the SOCCs in (13) associated to Ix to obtain the equivalent formulation with twice-continuously differentiable constraints: u∗(x) = arg min u∈Rm 1 2∥u∥2, (14) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' gi,1(x, u) ≤ 0, gi,2(x, u) ≤ 0, i ∈ Ix, ∥Qi(x)u + ri(x)∥ ≤ bi(x)u + ci(x), i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' , p}\\Ix, where gi,1(x, u) = uT (Qi(x)T Qi(x) − bi(x)bi(x)T )u + ri(x)2 + 2(Qi(x)T ri(x) − ci(x)bi(x))T u − ci(x)2, gi,2(x, u) = −bi(x)T u − ci(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Strict compatibility of the constraints in (13) implies the strict compatibility of the constraints in (14) and, by the same argument as before, MFCQ holds at the optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' To show that SOC2 also holds for (14), note that the constraints gi,1(x, u) ≤ 0 for i ∈ Ix cannot be active at the optimizer (otherwise, that would imply that bi(x)u∗(x) + ci(x) = 0, implying that MFCQ is violated at the optimizer, reaching a contradiction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Thus, by the strict complementarity condition, the Lagrange multipliers associated with the constraints gi,1(x, u), i ∈ Ix are zero and the Hessian of the Lagrangian L of (14) at the optimizer takes the form ∇2 uL(u∗, {λi}i∈J0)x = Im + � i∈J0(x) λi(x)∇2 u˜gi(x, u∗(x)), where J0(x) ⊂ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' , p}\\Ix is the set of active constraints at x, λi is the Lagrange multiplier associated with the constraint ∥Qi(x)u + ri(x)∥ ≤ bi(x)u + ci for i ̸∈ Ix and ˜gi(x, u) = ∥Qi(x)u + ri(x)∥ − bi(x)u − ci(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Since ∥Qi(x)u∗(x) + ri(x)∥ ̸= 0 for the active constraints, their Hessian is well-defined and is positive semidefinite due to their convexity, making ∇2 uL(u∗, {λi}i∈J0)x positive def- inite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Hence, SOC2 holds for (14) at the optimizer and, by [22, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='4], u∗ is locally Lipschitz at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' As a consequence of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='3, we conclude that if ∇V , the estimates ˆf, ˆg, ˆh, � ∇h, and worst-case error bounds (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' means and variances) that appear in (8) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' (9)) are twice continuously differentiable and the conditions (10) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' (11)) hold, then the corresponding CLF-CBF-SOCP controller is locally Lipschitz, safe, and stabilizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Next we provide a formula, inspired by Sontag’s universal formula [23], for a smooth controller satisfying a single SOCC defined by smooth functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='4: (Universal formula for a controller sat- isfying one SOCC): Let l ∈ Z>0 and assume Q : Rn → R(m+1)×m, r : Rn → Rm+1, b : Rn → Rm, and c : Rn → R are l-continuously differentiable on an open set G ⊆ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Suppose that the SOCC ∥Q(x)u + r(x)∥ ≤ b(x)u + c(x) is strictly feasible on G and Q(x)T Q(x) is invertible for all x ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Let ˜b(x) = b(x)(QT (x)Q(x))−1QT (x), ˜c(x) = c(x) − ˜b(x)r(x), ¯b(x) = (∥˜b(x)∥ − 1)∥˜b(x)∥, and vs(x) = �0 if ∥˜b(x)∥ ≤ 1, −˜c(x)+√ ˜c(x)2+¯b(x)2 ¯b(x) ˜b(x) if ∥˜b(x)∥ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' (15) Further assume vs(x)−r(x) ∈ Im(Q(x)) for all x ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Then us(x) := (QT (x)Q(x))−1QT (x)(vs(x) − r(x)), is l-continuously differentiable for all x ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Moreover, ∥Q(x)us(x) + r(x)∥ ≤ b(x)us(x) + c(x) for all x ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Proof: Let v = Q(x)u + r(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Since QT (x)Q(x) is invertible and ∥Q(x)u + r(x)∥ ≤ b(x)u + c(x) is strictly feasible on G, the resulting SOCC ∥v∥ ≤ ˜b(x)v + ˜c(x) is also strictly feasible on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Moreover, vs satisfies it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Indeed, if ∥˜b(x)∥ ≤ 1, since the SOCC is feasible there exists v∗ such that ∥v∗∥ ≤ ˜b(x)v∗ +˜c(x) and it follows that ˜c(x) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' The case ∥˜b(x)∥ > 1 follows from a direct calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' If ∥˜b(x)∥ ̸= 1, vs is Cl at x because ˜b and ˜c are Cl at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' If ∥˜b(x)∥ = 1, then ˜c(x) ̸= 0 (otherwise, if ˜c(x) = 0, since the SOCC ∥v∥ ≤ ˜b(x)v + ˜c(x) is strictly compatible, there would exist ˆv such that ∥ˆv∥ < ˜b(x)v ≤ ∥ˆv∥, which is a contradiction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Now, from the proof of [23, Theorem 1], the function φ(c, α) := � 0 if α ≤ 0, −c+ √ c2+α2 α else, is analytic at points of the form (c, 0), with c ̸= 0, so vs is Cl for all x ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Moreover, since vs(x) − r(x) ∈ Im(Q(x)) for all x ∈ G, it also follows that ∥Q(x)us(x) + r(x)∥ ≤ b(x)us(x) + c(x) for all x ∈ G and us is Cl for all x ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' From the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='4, we observe that in the case where the SOCC takes the form (8), a simpler expression is available for a controller satisfying it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' As a result of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='4, if uncertainty estimates of the dynamics and barrier function are available as described in Section II-B, the proposed formula can be used to guarantee either safety or stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' SIMULATIONS In this section we illustrate our results in an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' For simplicity, we focus on the case of worst-case error estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Consider the planar linear system d dt � x y � = � 1 0 0 1 � � x y � � �� � f(x,y) + � 1 0 0 1 � � �� � g(x,y) � u1 u2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' (16) We consider the CBF h(x, y) = x2 + (y − 4)2 − 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' From data to estimates and error bounds: We obtain here worst-case error models, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Section II-B, from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' For simplicity, we assume the estimates of the CBF are exact, ˆh = h, � ∇h = ∇h, and that prior knowledge of (not necessarily tight) Lipschitz constants (Kf = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='0 and Kg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='5) of f and g is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' We have access to an oracle that, given a query point y ∈ C, returns noiseless measurements (f(y), g(y)) of the functions in (1) (the noisy case can be considered without major modifications).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Given a set of N measurements D = {xi, f(xi), g(xi)}N i=1 obtained by querying the oracle, we estimate f at x ∈ Rn as ˆf(x) = f(pcl(x)), where pcl(x) is the closest datapoint to x, with corresponding worst-case error bound ef(x) := Kf∥x − pcl(x)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' We do similarly for ˆg and eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Performance dependency on error estimates: Here we illustrate how smaller estimation errors lead to improved performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' We use different datasets with different number of data points N to generate ˆf, ˆg, ef and eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' We solve the resulting CLF-CBF-SOCP every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='01s with initial condition at (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='0, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='0) and plot the trajectories until it becomes unfeasi- ble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' We compare the results for different N in Figure 1(left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Larger datasets with data from a neighborhood of the origin allow trajectories to converge closer to the origin before the problem becomes unfeasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' In the plot, we also observe that the conditions (10) serve as a good indicator of when the SOCP actually becomes unfeasible, hence illustrating how Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='2 can be used to infer when the available estimates are insufficient to guarantee that the controller is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Online safe stabilization: We illustrate also the case where data is collected online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' We start from an initial set of 25 measurements of f and g near the initial condition obtained by querying the oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Given an initial condition, we find the CLF-CBF-SOCP controller every 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='01s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' If during the execution (13) becomes infeasible at some point ¯x, we query the oracle to obtain measurements of f and g at ¯x (making it Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' 1: Safe stabilization of a planar system with worst-case uncertainty error bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' The green ball is the set of unsafe states and black dots denote initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' (Left) Dashed lines enclose the region where data is located for different N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Solid lines show the evolution under the CLF-CBF-SOCP controller in (13) computed with different N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Black triangles indicate points where the sufficient conditions for feasibility (10) do not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' (Right) The dashed lines display the evolution of a model-based min-norm controller (CLF-CBF QP) [4], and the solid lines display the evolution of a controller obtained by solving the CLF-CBF-SOCP (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Black stars denote points where measurements have been taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' feasible) and a small neighborhood around it (for improved performance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Figure 1(right) illustrates executions of this procedure for three different initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' As trajectories approach the origin, more measurements need to be taken because the conditions for feasibility of the CLF-CBF-SOCP, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' (10) in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content='2, become harder to satisfy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' CONCLUSIONS We have studied conditions to ensure the safe stabilization of a nonlinear affine control system under uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Given a known control Lyapunov function and either worst-case or probabilistic estimates of the dynamics and CBF, SOCCs encode the impact of uncertainty on the ability to guarantee stability and safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' We have provided explicit bounds on the error estimates that ensure the relevant SOCCs are compat- ible and built on them to ensure the existence of a smooth safe stabilizing controller, the Lipschitz regularity of the min- norm CLF-CBF-SOCP-based controller, and the regularity of a universal controller for the satisfaction of a single SOCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/wNE3T4oBgHgl3EQflArR/content/2301.04603v1.pdf'} +page_content=' Future work will characterize the conditions for compatibility in terms of data, design online safe stabilization mechanisms that balance computational effort, sampling rate, and performance using resource-aware control, explore the design of universal formulas for more than one SOCC, and implement our results on physical testbeds.' 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--git a/yNFKT4oBgHgl3EQfLi3P/vector_store/index.faiss b/yNFKT4oBgHgl3EQfLi3P/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..c26a1b4c4caf59423c9ed5fd4010e118b1936731 --- /dev/null +++ b/yNFKT4oBgHgl3EQfLi3P/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0e9fb8a18b7243a4ad1b854ce077df1a6ac0a0523c77fec17875a7ff15a75c11 +size 3080237 diff --git a/ydFIT4oBgHgl3EQf1CsY/content/tmp_files/2301.11371v1.pdf.txt b/ydFIT4oBgHgl3EQf1CsY/content/tmp_files/2301.11371v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ac621319ab5f435c16defbf267b822224ac19c3d --- /dev/null +++ b/ydFIT4oBgHgl3EQf1CsY/content/tmp_files/2301.11371v1.pdf.txt @@ -0,0 +1,1411 @@ +arXiv:2301.11371v1 [nucl-ex] 26 Jan 2023 +138Ba(d, α) study of states in 136Cs: Implications for new physics searches with xenon +detectors +B. M. Rebeiro,1, 2 S. Triambak,1, ∗ P. E. Garrett,3, 1 G. C. Ball,4 B. A. Brown,5 J. Men´endez,6 +B. Romeo,7 P. Adsley,8 B. G. Lenardo,9 R. Lindsay,1 V. Bildstein,3 C. Burbadge,3, † +R. Coleman,3 A. Diaz Varela,3 R. Dubey,1 T. Faestermann,10 R. Hertenberger,11 M. Kamil,1 +K. G. Leach,12 C. Natzke,12 J. C. Nzobadila Ondze,1 A. Radich,3 E. Rand,3 and H. -F. Wirth11 +1Department of Physics and Astronomy, University of the Western Cape, P/B X17, Bellville 7535, South Africa +2Department of Physics, McGill University, Montr´eal, Qu´ebec H3A 2T8, Canada +3Department of Physics, University of Guelph, Guelph, Ontario N1G 2W1, Canada +4TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia V6T 2A3, Canada +5Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, +Michigan State University, East Lansing, Michigan 48824-1321, USA +6Department of Quantum Physics and Astrophysics and Institute of Cosmos Sciences, +University of Barcelona, 08028 Barcelona, Spain +7Donostia International Physics Center, 20018 San Sebasti´an, Spain +8Cyclotron Institute and Department of Physics & Astronomy, +Texas A&M University, College Station, Texas 77843, USA +9SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA +10Physik Department, Technische Universit¨at M¨unchen, D-85748 Garching, Germany +11Fakult¨at f¨ur Physik, Ludwig-Maximilians-Universit¨at M¨unchen, D-85748 Garching, Germany +12Department of Physics, Colorado School of Mines, Golden, Colorado 80401, USA +(Dated: January 30, 2023) +We used the 138Ba(d, α) reaction to carry out a high-resolution study of low-lying states in the +odd-odd nucleus 136Cs. +Our results are compared with shell-model calculations performed with +Hamiltonians that were previously used to obtain the nuclear matrix element for 136Xe neutrinoless +double beta decay. We show that one of these Hamiltonians dramatically fails to describe the low- +energy 136Cs spectrum, while the other two show reasonably good agreement. Additionally, we find +that the energy of the first excited state in 136Cs is more than three times higher than a previous +assumption that was based on shell-model calculations. +This may have significant implications +for the anticipated detection of solar-neutrino or fermionic dark matter events via charged-current +scattering in future large-scale xenon experiments. +Xenon-based detectors [1–7] have provided some of the +most sensitive searches of physics beyond the standard +model (BSM). These experimental campaigns mainly +placed emphasis on observing dark matter interactions +or lepton-number-violating neutrinoless double beta de- +cays (0ν2β). Several next-generation experiments with +xenon time projection chambers (TPCs) [8–11] aim to +build on this work and probe for BSM physics with sig- +nificantly improved sensitivity. Of particular note are the +nEXO [8] and NEXT [10] experiments, which will employ +enriched 136Xe to search for 136Xe 0ν2β. +Solar neutrino interactions [12] become a significant +consideration in such large-scale experiments, with the +dominant contributions being elastic neutrino-electron +scattering [13, 14] and charged-current (CC) interactions +on xenon nuclei [12, 15]. Axial-vector CC scattering on +136Xe will produce 1+ excited states in 136Cs [12, 15]. +Similar CC-type excitations may also occur through in- +teractions between MeV-scale fermionic dark matter can- +didates [16, 17] and 136Xe nuclei. The subsequent deex- +citations within 136Cs, via γ-ray cascades/internal con- +∗ striambak@uwc.ac.za +† Deceased. +version allow a means to discriminate such CC cap- +tures [12, 15], from multi-site interaction vertices within +the TPC volume and/or a time-correlation analysis of +events from long-lived states in 136Cs. +In this context, following previous work by Ejiri and +Elliott [12], Haselschwardt et al. [15] investigated the pos- +sibility of tagging solar neutrino captures in liquid xenon +(LXe) TPCs using both an event-topology and a delayed- +coincidence analysis. +Calculated CC capture rates on +136Xe from oscillated solar-ν fluxes showed that the dom- +inant contributions are through the first two 1+ states in +136Cs, at 591 and 845 keV respectively [18], with the 1+ +1 +state being the most significant [15]. +Despite the knowledge of several 1+ excited states in +136Cs from a previous 136Xe(3He, t) experiment [18, 19], +the authors of Ref. [15] were challenged by the meager +experimental information [20] available for the low-lying +level scheme of 136Cs. Although a rich spectrum is pre- +dicted below the 1+ +1 state at 591 keV [15], only three +states have thus far been experimentally verified, with +spin-parities Jπ = 4+, 8− and 9−, respectively [20–22]. +Because of this dearth of experimental information (c.f. +left panel, Fig. 2), Haselschwardt et al. used shell-model +calculations to predict γ-ray deexcitation paths from the +predominantly populated 1+ +1 level in 136Cs. The time- + +2 +correlation analysis method showed promise [15] for the +detection of neutrinos emitted in the solar CNO cycle as +well as determining the 7Be solar neutrino line shift [23]. +This was mainly because of the predicted long-lived first +excited 3+ state in 136Cs, which opens the possibility of +future Xe TPC experiments offering insight into the in- +nermost core of the Sun. However, it is apparent that +an elucidation of the low-lying level structure of 136Cs is +essential to make further progress in this regard. +In addition, there is widespread interest to accurately +determine the nuclear matrix elements (NMEs) for vari- +ous 0ν2β candidates [24–27], including 136Xe. The eval- +uated NMEs for individual cases are known to differ sig- +nificantly, depending on the many-body approach used +to perform the calculation [24]. This theoretical limita- +tion translates into an inevitable uncertainty band on the +BSM parameter responsible for the decay, which is hoped +to be extracted from future experiments. Consequently, +there has been a concerted effort to minimize the model +dependence [28–33] in calculating 0ν2β NMEs. 136Cs is +the intermediate odd-odd nucleus in 136Xe → 136Ba 2β +decay. The calculated 0ν2β NME for this particular case +ranges from 1.11 to 4.77 [29], for light Majorana neu- +trino exchange. Within the shell-model, the 136Xe 0ν2β +NME is in the range 1.63–2.45 [27], depending on the +Hamiltonian used for the calculation. +Although these +Hamiltonians describe reasonably well the initial and fi- +nal states for 136Xe 2β decay, they have not yet been +tested against the intermediate nucleus 136Cs, because of +scarce experimental information. +In the particle-hole picture [34, 35], the NME is ex- +pressed as a sum over intermediate states in 136Cs. If one +ignores a negligible Fermi component, the 1+ particle- +hole states in the intermediate nucleus contribute to the +standard-model-allowed two-neutrino double beta decay +(2ν2β) [24, 25]. +However these 1+ states only repre- +sent a small part of the total 0ν2β strength [34, 35], +as many other intermediate multipoles [24, 25] also con- +tribute to 0ν2β [34–37], because of its q ∼ 100 MeV +momentum-transfer. +Considering the relatively higher +density of states expected in 136Cs, any experimental in- +formation on its low-lying excitation region will be useful +to place 136Xe 0ν2β NME calculations on a more secure +footing. This information will also be beneficial for fu- +ture calculations of the NME that include higher-order +corrections [38]. These sub-leading terms are expected +to be sensitive to competing contributions from low and +high-energy intermediate states, similar to what has been +observed for 136Xe 2ν2β [39]. +It is evident from the above that an improved knowl- +edge of the nuclear structure of 136Cs is invaluable for +both current and future xenon TPC-based experiments, +particularly in the context of solar neutrino/fermionic +dark matter detection and 136Xe 0ν2β. +Motivated by +this, we report (for the first time) a detailed high- +resolution investigation of low-lying states in 136Cs, with +emphasis placed on multipoles with J ≥ 1. We used the +0 +200 +400 +600 +800 +1000 1200 1400 1600 1800 2000 2200 2400 2600 +Excitation energy (keV) +0 +50 +100 +150 +200 +250 +300 +350 +Number of counts per keV +GS +589 +Carbon +2367 +995 +2501 +658 +737 +1905 +1758 +74 +1111 +314 +1361 +1493 +2111 +FIG. 1. +Sample 138Ba(d, α) spectrum obtained at θlab = 10◦. +138Ba(d, α)136Cs two-nucleon transfer reaction, which is +well suited for such a study. +The experiment was performed at the Maier-Leibnitz +Laboratorium (MLL) in Garching, Germany. A 600 nA, +22 MeV deuteron beam was incident on a 99.8% enriched +40 µg/cm2-thick 138BaO target, evaporated on a car- +bon foil. The reaction ejectiles were momentum analyzed +with the high-resolution Q3D magnetic spectrograph [40] +and focused onto its focal-plane detector, whose posi- +tion resolution was ≈ 0.1 mm [41]. The α particles were +selected by comparing the partial energy losses of the +reaction products in two gas proportional counters and +the total energy deposited in a plastic scintillator detec- +tor at the focal plane. For energy calibration, we used +the 94Mo(d, α)92Nb and 92Zr(d, α)90Y reactions on en- +riched 94MoO3 and 92Zr targets that had thicknesses of +100 µg/cm2 and 50 µg/cm2, respectively. The calibra- +tions explicitly took into account differences in reaction +kinematics and energy losses within the target foils, as de- +scribed in Refs. [42, 43]. A sample calibrated 138Ba(d, α) +spectrum is shown in Fig. 1. The measured full widths +at half maxima (FWHM) of the α peaks were ∼ 10 keV, +significantly better than the 40 keV resolution reported +in the previous 136Xe(3He, t) work [19]. +The 138Ba(d, α) spectra were collected at different an- +gles in the range θlab = 5◦ − 45◦, at 5◦ intervals. Ad- +ditionally, 138Ba(d, d) elastic scattering data were also +acquired in the range θlab = 15◦ − 115◦, at 5◦ intervals. +We used these datasets to determine the target thickness +and obtain differential scattering cross sections, as de- +scribed in Refs. [29, 44]. The measured angular distribu- +tions were then compared to distorted wave Born approx- +imation (DWBA) predictions, provided by the DWUCK5 +computer code. +The selectivity of the (d, α) reaction is such that the +transferred np pair is in a relative l = 0 state, with spin +S = 1 and isospin T = 0 [45]. If both nucleons are picked +up from the same single-particle (j2) configuration, the +orbital angular momentum L transferred in the reaction +is even. In such a scenario, the total angular momentum +J of the final state is odd. However, if the neutron and +proton are picked up from different configurations, with + +3 +L = ln + lp, then J = L and J = L ± 1 states, with +parity (−1)ln+lp are produced [46]. +For the DWBA analysis, we chose appropriate opti- +cal model parameters (OMPs) for the incoming d+138Ba +channel by comparing our measured elastic scattering an- +gular distribution with DWBA results from using differ- +ent global OMPs. This comparison showed that the rec- +ommended OMPs by An and Cai [47] yielded best agree- +ment with our data. For the outgoing α+136Cs channel +we chose the OMPs of Burnett et al. [48], as they were +optimized for the 136Ba(α, α) reaction at 20 MeV [49]. +The 138Ba(d, α) calculations were performed assuming +the ‘cluster’ deuteron-transfer approximation [50, 51], +with form-factors for a deuteron in a Woods-Saxon po- +tential well, at the correct separation energy for each +state in 136Cs. On account of the large angular momen- +tum mismatch between the incoming deuteron and the +outgoing α, we also took into consideration finite-range +corrections [52, 53] and nonlocality effects, using the pre- +scription from Ref. [54]. In the next step, our measured +138Ba(d, α) angular distributions were overlaid with nor- +malized best-fit DWBA results. The latter were obtained +assuming various L-transfer values for given J, and al- +lowed incoherent summations of two different values L +and L′. Our identified states were then compared with +shell-model predictions and previous measurements. +For the shell-model calculations we used a configu- +ration space comprising the 0g7/2, 1d5/2, 1d3/2, 2s1/2 +and 0h11/2 orbitals for neutrons and protons, and three +different Hamiltonians: +SN100PN [55], GCN5082 [56] +and QX [57]. +The SN100PN interaction was used by +Haselschwardt et al. [15] to determine the level scheme +of 136Cs. It was also used to benchmark 136Xe 0ν2β NME +calculations [29, 30]. Independently, the GCN5082 and +QX Hamiltonians were used to calculate the 136Xe 0ν2β +NME [58–61]. +Figure 2 compares calculated energy levels of 136Cs to +those identified from this experiment. Our results, for +states up to the 1+ +1 level are summarized in Fig. 3 and +Table I. We also used two-nucleon transfer amplitudes +(TNAs) [62] obtained with the GCN5082 and SN100PN +Hamiltonians for critical comparative cross-checks. This +was feasible because most of the low-lying states had +TNA dominated by simple two-nucleon configurations. +For example, both calculations showed that the domi- +nant orbitals involved in the transfer to the Jπ = 5+ +ground state [63, 64] are g7/2 and d3/2 for proton (π) +and neutron (ν) pick-up, respectively. This state can be +produced by both L = 4 and L = 6 transfer. The relative +L contributions can be evaluated via the jj to LS trans- +formation that involves the normalized 9j coefficient [45], +� +3(2jn + 1)(2jp + 1)(2L + 1) + + + + + + + + + +ln 1/2 jn +lp 1/2 jp +L +1 +J + + + + + + + + + +. +(1) +TABLE I. Observed 136Cs levels up to the 1+ +1 state. +Refs. [18, 20] +This work +Ex (keV) +Jπ +Ex (keV) +L +L′ +Assigned Jπ +0.0 +5+ +0.0 +4 +6 +5+ +74(2) +4 +... +3+ +104.8(3) +4+ +104(2) +4 +... +4+ +140(3) +2 +4 +3+ +314(2) +4 +... +(4+) +423(3) +4 +... +(4+) +431(2) +(3+) +432(3) +2 +... +(2+) +460(3) +4 +... +(3+) +517.9(1) +8− +517(3) +7 +9 +8− +583.9(5) +9− +... +... +... +... +591(2) +1+ +589(3) +0 +2 +1+ +This yields a predominantly L = 6 transition for the +ground state, which is consistent with our observations. +The same two-nucleon configuration dominates transfer +to the 3+ +1 and 4+ +1 states. For the former, the intensity of +the L = 2 transition is nearly 17 times weaker than L = 4 +transfer. This agrees with the measured angular distribu- +tion of the first excited state, observed at 74 keV. Next, +we compared the measured cross section for this level rel- +ative to the ground state (after accounting for the differ- +ence in their predicted DWBA yields), with the relative +scaling of their calculated transfer intensities. The rea- +sonable agreement between these two values validated the +3+ +1 assignment for this state. In comparison, we identify +the 140-keV state as 3+ +2 , whose dominant TNA corre- +sponds to the (πd5/2) (νd3/2) orbitals. Both L = 2 and +L = 4 transfer contribute for this state, which agrees +well with the measured distribution. Spin-parity assign- +ments for the remaining states identified in Table I are +made through similar analysis of the shapes of the angu- +lar distributions, relative cross sections, and L-transfer +intensities predicted by theory. +We do not observe the explicit signatures of the low- +lying 2+ states, which are predicted to be weakly pop- +ulated. We also do not observe the known 9− state at +583.9(5) keV. This can be explained by the DWBA cal- +culations, which show that L = 9 transfer for this state +is significantly weaker than the dominant L = 7 transfer +to the 8− state. +A tentative 3+ state was reported at 431 keV [18], but +excluded from Ref. [20]’s compilation. We investigated +this state’s possible existence by refitting the 423 keV +peak with fixed lineshape parameters, based on previous +knowledge of the detector response [65]. This analysis +indicated a possible level at Ex = 432(3) keV, whose +angular distribution is shown in Fig. 3. Although it is + +4 +Previous work [20] +GCN5082 +This work +SN100PN +QX +0 +100 +200 +300 +400 +500 +600 +Excitation energy (keV) +5 ++ +3 ++ +4 ++ +3 ++ +(4 ++) +(2 ++) +(3 ++) +8 +- +5 ++ +3 ++ +4 ++ +2 ++ +3 ++ +3 ++ +2 ++ +2 +- +8 +-1 ++ +3 +- +4 ++ +4 ++ +3 ++ +5 ++ +3 ++ +4 ++ +2 ++ +3 ++ +2 ++ +3 ++ +4 ++ +8 +- +4 ++ +5 ++ +4 ++ +1 ++ +1 ++ +8 +- +3 ++ +1 ++ +2 ++ +3 ++ +7 +-2 ++ +8 +- +8 +- +3 ++ +5 +-4 +-5 +- +6 +-4 ++ +4 +-3 ++ +2 ++ +3 +- +3 ++ +1 ++ +2 ++ +4 ++ +5 ++ +4 ++ +4 +- +5 +-6 +- +6 +- +(2 ++, 4 ++) +9 +- +7 +- +7 +- +5 +-6 +-3 +- +4 +- +6 +- +7 +- +FIG. 2. Comparison between theory and experiment for the low-lying energy spectrum of 136Cs. The shell-model results were +obtained with the SN100PN, GCN5082 and QX effective interactions. +statistics-limited, the measured distribution is consistent +with L = 2 transfer. The intensity of this possible tran- +sition is comparable to those predicted for the 2+ +1 and 2+ +2 +levels. We also observe that the θlab = 5◦ cross section +for the 423 keV state is enhanced compared to the other +L = 4 transitions. This can be attributed to an addi- +tional L = 2 component which is ∼ 20% of the L = 4 +contribution, as shown in Fig. 3. Thus, one cannot rule +out an unresolved state at ∼ 423 keV, with an L = 2 +contribution that corresponds to one of the 2+ levels. +The dominant (πg7/2), (πd5/2), (νd3/2), (νs1/2) and +(νh11/2) configurations that describe these levels are also +consistent with other observations. The spin-parity val- +ues for first two (three) states in 137Cs (N = 82, Z = 55) +and 137Ba (N = 81, Z = 56), are 7/2+, 5/2+, and 3/2+, +1/2+, 11/2−, respectively [20]. +This information, to- +gether with independently measured 138Ba(d, 3He) spec- +troscopic factors [66], provides an important validation +of these configurations. +Figure 2 makes it obvious that while the SN100PN +and GCN5082 results are overall very similar and could +be matched to our identified levels from this experiment, +there is a stark disagreement with the QX results. Not +only does the QX interaction fail to predict the cor- +rect ground state for 136Cs (the 5+ +1 state is instead at +512 keV), it also shows several low-lying negative parity +states that are not predicted by the other two Hamil- +tonians or verified by experiment. This comparison un- +derscores the importance of testing model predictions in +intermediate odd-odd nuclei for 0ν2β candidates. Un- +der such requirement the QX interaction may be consid- +ered less reliable; which calls into question 136Xe 0ν2β +NME values determined with this Hamiltonian. +This +is particularly relevant because the QX NME is ∼ 40% +smaller [60] than the ones obtained with GCN5082 [67] +and Vlow-k [68] effective interactions. The QX interaction +was also recently used to determine γγ decay NMEs [32], +study the s-process nucleosynthesis of barium nuclei [69], +calculate the subleading NME for 136Xe 2ν2β [39], and +perform more up-to-date calculations of the 136Xe 0ν2β +NME [61]. +In the context of detecting CNO and 7Be solar neu- +trinos and/or fermionic dark matter events in Xe detec- +tors, Haselschwardt et al. [15] showed that a delayed- +coincidence analysis allows an efficient detection of CC- +type excitations in LXe TPCs, on account of the slow +3+ +1 → 5+ +1 E2 transition in 136Cs. +The SN100PN cal- +culation shows the 3+ +1 state at 23 keV. Based on this +value, the state’s lifetime was estimated to be around +624 µs [15] - long enough for a feasible time-correlation +analysis of scintillation light in LXe TPC detectors with +drift lengths ∼ 1.3 m. On the other hand the GCN5082 +interaction predicts the first excited state at 15 keV, with +a comparatively longer lifetime of 8.5 ms. Our experi- +ment shows that the first excited state in 136Cs is in fact +significantly higher in energy, at 74(2) keV. As the tran- +sition probability scales as Eγ +5, this implies that the life- +time of the 3+ +1 state is approximately 340 times shorter +than assumed in Ref. [15]. +This may have significant +implications for the anticipated detection of solar neu- +trino/fermionic dark matter events via CC excitations in +future large-scale Xe TPC experiments. +This work was partially supported by the National Re- +search Foundation (NRF), South Africa, under Grant +No. +85100, the Natural Sciences and Engineering Re- +search Council of Canada (NSERC), the U.S. National +Science Foundation under Grant No. PHY-2110365 and +the U.S. Department of Energy (DOE) under Grant No. + +5 +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +Scattering Angle, θc.m. (deg) +10 +20 +30 +40 +50 +60 +dσ/dΩ (µb/sr) +J = 5; L = 4 + L = 6 +J = 5; L = 4 +E = 0 keV (πg7/2, νd3/2) +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +Scattering Angle, θc.m. (deg) +0.0 +5.0 +10.0 +15.0 +20.0 +dσ/dΩ (µb/sr) +J=3; L=4 +E = 74 keV (πg7/2, νd3/2) +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +Scattering Angle, θc.m. (deg) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +4.5 +dσ/dΩ (µb/sr) +J = 4; L = 4 +E = 104 keV (πg7/2, νd3/2) +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +Scattering Angle, θc.m. (deg) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +dσ/dΩ (µb/sr) +J = 3; L = 2 + L = 4 +E = 140 keV (πd5/2, νd3/2) +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +Scattering Angle, θc.m. (deg) +0.0 +1.0 +2.0 +3.0 +4.0 +5.0 +6.0 +dσ/dΩ (µb/sr) +J = 4; L = 4 +E = 314 keV (πd5/2, νd3/2) +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +Scattering Angle, θc.m. (deg) +2.0 +4.0 +6.0 +8.0 +10.0 +12.0 +14.0 +16.0 +dσ/dΩ (µb/sr) +J = 4; L = 4 (πg7/2, νs1/2) +J = 2; L = 2 + J = 4; L = 4 +E = 423 keV +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +Scattering Angle, θc.m. (deg) +0.0 +1.0 +2.0 +3.0 +4.0 +5.0 +dσ/dΩ (µb/sr) +J = 2; L = 2 +E = 432 keV +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +Scattering Angle, θc.m. (deg) +0.0 +2.0 +4.0 +6.0 +8.0 +10.0 +12.0 +14.0 +16.0 +dσ/dΩ (µb/sr) +J = 3: L = 4 +E = 460 keV (πg7/2, νs1/2) +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +Scattering Angle, θc.m. (deg) +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +2.0 +dσ/dΩ (µb/sr) +J = 8: L = 7 + L = 9 +E = 517 keV (πg7/2, νh11/2) +0 +5 +10 +15 +20 +25 +30 +35 +40 +45 +50 +Scattering Angle, θc.m. (deg) +0.0 +2.0 +4.0 +6.0 +8.0 +10.0 +dσ/dΩ (µb/sr) +J = 1: L = 0 + L = 2 +E = 589 keV (πd5/2, νd3/2) +FIG. 3. Measured 138Ba(d, α) angular distributions compared +with best-fit DWUCK5 DWBA predictions (solid red curves). +The blue dashed curves are from using fixed relative L contri- +butions from Eq. (1). The dominant orbitals involved in the +pair-transfer are specified in each plot. +DE-FG02-93ER40789. +[1] G. Anton, +I. Badhrees, +P. S. Barbeau, +D. Beck, +V. Belov, T. Bhatta, M. Breidenbach, T. Brunner, +G. F. Cao, W. R. Cen, et al. (EXO-200 Collaboration), +Phys. Rev. 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Karakas, +Jacqueline +den +Hartogh, +Bing-Shui +Gao, +and +Xiao-Dong +Tang, +The Astrophysical Journal Letters 919, L19 (2021). + diff --git a/ydFIT4oBgHgl3EQf1CsY/content/tmp_files/load_file.txt b/ydFIT4oBgHgl3EQf1CsY/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3e3b567289fd7969861dbfb9d5bfc3f15d0a8798 --- /dev/null +++ b/ydFIT4oBgHgl3EQf1CsY/content/tmp_files/load_file.txt @@ -0,0 +1,1061 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf,len=1060 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='11371v1 [nucl-ex] 26 Jan 2023 138Ba(d, α) study of states in 136Cs: Implications for new physics searches with xenon detectors B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' M.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' USA 6Department of Quantum Physics and Astrophysics and Institute of Cosmos Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' University of Barcelona,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' 08028 Barcelona,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Spain 7Donostia International Physics Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' 20018 San Sebasti´an,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Spain 8Cyclotron Institute and Department of Physics & Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Texas A&M University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' College Station,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Texas 77843,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' USA 9SLAC National Accelerator Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Menlo Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' California 94025,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Germany 12Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Colorado School of Mines,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Golden,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Colorado 80401,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' USA (Dated: January 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' 2023) We used the 138Ba(d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' α) reaction to carry out a high-resolution study of low-lying states in the odd-odd nucleus 136Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Our results are compared with shell-model calculations performed with Hamiltonians that were previously used to obtain the nuclear matrix element for 136Xe neutrinoless double beta decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' We show that one of these Hamiltonians dramatically fails to describe the low- energy 136Cs spectrum, while the other two show reasonably good agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Additionally, we find that the energy of the first excited state in 136Cs is more than three times higher than a previous assumption that was based on shell-model calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' This may have significant implications for the anticipated detection of solar-neutrino or fermionic dark matter events via charged-current scattering in future large-scale xenon experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Xenon-based detectors [1–7] have provided some of the most sensitive searches of physics beyond the standard model (BSM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' These experimental campaigns mainly placed emphasis on observing dark matter interactions or lepton-number-violating neutrinoless double beta de- cays (0ν2β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Several next-generation experiments with xenon time projection chambers (TPCs) [8–11] aim to build on this work and probe for BSM physics with sig- nificantly improved sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Of particular note are the nEXO [8] and NEXT [10] experiments, which will employ enriched 136Xe to search for 136Xe 0ν2β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Solar neutrino interactions [12] become a significant consideration in such large-scale experiments, with the dominant contributions being elastic neutrino-electron scattering [13, 14] and charged-current (CC) interactions on xenon nuclei [12, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Axial-vector CC scattering on 136Xe will produce 1+ excited states in 136Cs [12, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Similar CC-type excitations may also occur through in- teractions between MeV-scale fermionic dark matter can- didates [16, 17] and 136Xe nuclei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' The subsequent deex- citations within 136Cs, via γ-ray cascades/internal con- ∗ striambak@uwc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='za † Deceased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' version allow a means to discriminate such CC cap- tures [12, 15], from multi-site interaction vertices within the TPC volume and/or a time-correlation analysis of events from long-lived states in 136Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' In this context, following previous work by Ejiri and Elliott [12], Haselschwardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' [15] investigated the pos- sibility of tagging solar neutrino captures in liquid xenon (LXe) TPCs using both an event-topology and a delayed- coincidence analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Calculated CC capture rates on 136Xe from oscillated solar-ν fluxes showed that the dom- inant contributions are through the first two 1+ states in 136Cs, at 591 and 845 keV respectively [18], with the 1+ 1 state being the most significant [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Despite the knowledge of several 1+ excited states in 136Cs from a previous 136Xe(3He, t) experiment [18, 19], the authors of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' [15] were challenged by the meager experimental information [20] available for the low-lying level scheme of 136Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Although a rich spectrum is pre- dicted below the 1+ 1 state at 591 keV [15], only three states have thus far been experimentally verified, with spin-parities Jπ = 4+, 8− and 9−, respectively [20–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Because of this dearth of experimental information (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' left panel, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' 2), Haselschwardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' used shell-model calculations to predict γ-ray deexcitation paths from the predominantly populated 1+ 1 level in 136Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' The time- 2 correlation analysis method showed promise [15] for the detection of neutrinos emitted in the solar CNO cycle as well as determining the 7Be solar neutrino line shift [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' This was mainly because of the predicted long-lived first excited 3+ state in 136Cs, which opens the possibility of future Xe TPC experiments offering insight into the in- nermost core of the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' However, it is apparent that an elucidation of the low-lying level structure of 136Cs is essential to make further progress in this regard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' In addition, there is widespread interest to accurately determine the nuclear matrix elements (NMEs) for vari- ous 0ν2β candidates [24–27], including 136Xe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' The eval- uated NMEs for individual cases are known to differ sig- nificantly, depending on the many-body approach used to perform the calculation [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' This theoretical limita- tion translates into an inevitable uncertainty band on the BSM parameter responsible for the decay, which is hoped to be extracted from future experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Consequently, there has been a concerted effort to minimize the model dependence [28–33] in calculating 0ν2β NMEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' 136Cs is the intermediate odd-odd nucleus in 136Xe → 136Ba 2β decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' The calculated 0ν2β NME for this particular case ranges from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='11 to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='77 [29], for light Majorana neu- trino exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Within the shell-model, the 136Xe 0ν2β NME is in the range 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='63–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='45 [27], depending on the Hamiltonian used for the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Although these Hamiltonians describe reasonably well the initial and fi- nal states for 136Xe 2β decay, they have not yet been tested against the intermediate nucleus 136Cs, because of scarce experimental information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' In the particle-hole picture [34, 35], the NME is ex- pressed as a sum over intermediate states in 136Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' If one ignores a negligible Fermi component, the 1+ particle- hole states in the intermediate nucleus contribute to the standard-model-allowed two-neutrino double beta decay (2ν2β) [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' However these 1+ states only repre- sent a small part of the total 0ν2β strength [34, 35], as many other intermediate multipoles [24, 25] also con- tribute to 0ν2β [34–37], because of its q ∼ 100 MeV momentum-transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Considering the relatively higher density of states expected in 136Cs, any experimental in- formation on its low-lying excitation region will be useful to place 136Xe 0ν2β NME calculations on a more secure footing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' This information will also be beneficial for fu- ture calculations of the NME that include higher-order corrections [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' These sub-leading terms are expected to be sensitive to competing contributions from low and high-energy intermediate states, similar to what has been observed for 136Xe 2ν2β [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' It is evident from the above that an improved knowl- edge of the nuclear structure of 136Cs is invaluable for both current and future xenon TPC-based experiments, particularly in the context of solar neutrino/fermionic dark matter detection and 136Xe 0ν2β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Motivated by this, we report (for the first time) a detailed high- resolution investigation of low-lying states in 136Cs, with emphasis placed on multipoles with J ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' We used the 0 200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 Excitation energy (keV) 0 50 100 150 200 250 300 350 Number of counts per keV GS 589 Carbon 2367 995 2501 658 737 1905 1758 74 1111 314 1361 1493 2111 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Sample 138Ba(d, α) spectrum obtained at θlab = 10◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' 138Ba(d, α)136Cs two-nucleon transfer reaction, which is well suited for such a study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' The experiment was performed at the Maier-Leibnitz Laboratorium (MLL) in Garching, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' A 600 nA, 22 MeV deuteron beam was incident on a 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='8% enriched 40 µg/cm2-thick 138BaO target, evaporated on a car- bon foil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' The reaction ejectiles were momentum analyzed with the high-resolution Q3D magnetic spectrograph [40] and focused onto its focal-plane detector, whose posi- tion resolution was ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='1 mm [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' The α particles were selected by comparing the partial energy losses of the reaction products in two gas proportional counters and the total energy deposited in a plastic scintillator detec- tor at the focal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' For energy calibration, we used the 94Mo(d, α)92Nb and 92Zr(d, α)90Y reactions on en- riched 94MoO3 and 92Zr targets that had thicknesses of 100 µg/cm2 and 50 µg/cm2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' The calibra- tions explicitly took into account differences in reaction kinematics and energy losses within the target foils, as de- scribed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' [42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' A sample calibrated 138Ba(d, α) spectrum is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' The measured full widths at half maxima (FWHM) of the α peaks were ∼ 10 keV, significantly better than the 40 keV resolution reported in the previous 136Xe(3He, t) work [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' The 138Ba(d, α) spectra were collected at different an- gles in the range θlab = 5◦ − 45◦, at 5◦ intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Ad- ditionally, 138Ba(d, d) elastic scattering data were also acquired in the range θlab = 15◦ − 115◦, at 5◦ intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' We used these datasets to determine the target thickness and obtain differential scattering cross sections, as de- scribed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' [29, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' The measured angular distribu- tions were then compared to distorted wave Born approx- imation (DWBA) predictions, provided by the DWUCK5 computer code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' The selectivity of the (d, α) reaction is such that the transferred np pair is in a relative l = 0 state, with spin S = 1 and isospin T = 0 [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' If both nucleons are picked up from the same single-particle (j2) configuration, the orbital angular momentum L transferred in the reaction is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' In such a scenario, the total angular momentum J of the final state is odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' However, if the neutron and proton are picked up from different configurations, with 3 L = ln + lp, then J = L and J = L ± 1 states, with parity (−1)ln+lp are produced [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' For the DWBA analysis, we chose appropriate opti- cal model parameters (OMPs) for the incoming d+138Ba channel by comparing our measured elastic scattering an- gular distribution with DWBA results from using differ- ent global OMPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' This comparison showed that the rec- ommended OMPs by An and Cai [47] yielded best agree- ment with our data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' For the outgoing α+136Cs channel we chose the OMPs of Burnett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' [48], as they were optimized for the 136Ba(α, α) reaction at 20 MeV [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' The 138Ba(d, α) calculations were performed assuming the ‘cluster’ deuteron-transfer approximation [50, 51], with form-factors for a deuteron in a Woods-Saxon po- tential well, at the correct separation energy for each state in 136Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' On account of the large angular momen- tum mismatch between the incoming deuteron and the outgoing α, we also took into consideration finite-range corrections [52, 53] and nonlocality effects, using the pre- scription from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' In the next step, our measured 138Ba(d, α) angular distributions were overlaid with nor- malized best-fit DWBA results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' The latter were obtained assuming various L-transfer values for given J, and al- lowed incoherent summations of two different values L and L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Our identified states were then compared with shell-model predictions and previous measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' For the shell-model calculations we used a configu- ration space comprising the 0g7/2, 1d5/2, 1d3/2, 2s1/2 and 0h11/2 orbitals for neutrons and protons, and three different Hamiltonians: SN100PN [55], GCN5082 [56] and QX [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' The SN100PN interaction was used by Haselschwardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' [15] to determine the level scheme of 136Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' It was also used to benchmark 136Xe 0ν2β NME calculations [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Independently, the GCN5082 and QX Hamiltonians were used to calculate the 136Xe 0ν2β NME [58–61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Figure 2 compares calculated energy levels of 136Cs to those identified from this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Our results, for states up to the 1+ 1 level are summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' 3 and Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' We also used two-nucleon transfer amplitudes (TNAs) [62] obtained with the GCN5082 and SN100PN Hamiltonians for critical comparative cross-checks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' This was feasible because most of the low-lying states had TNA dominated by simple two-nucleon configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' For example, both calculations showed that the domi- nant orbitals involved in the transfer to the Jπ = 5+ ground state [63, 64] are g7/2 and d3/2 for proton (π) and neutron (ν) pick-up, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' This state can be produced by both L = 4 and L = 6 transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' The relative L contributions can be evaluated via the jj to LS trans- formation that involves the normalized 9j coefficient [45], � 3(2jn + 1)(2jp + 1)(2L + 1) \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 ln 1/2 jn lp 1/2 jp L 1 J \uf8fc \uf8f4 \uf8f4 \uf8f4 \uf8fd \uf8f4 \uf8f4 \uf8f4 \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' (1) TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Observed 136Cs levels up to the 1+ 1 state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' [18, 20] This work Ex (keV) Jπ Ex (keV) L L′ Assigned Jπ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 5+ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 4 6 5+ 74(2) 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' 3+ 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='8(3) 4+ 104(2) 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' 4+ 140(3) 2 4 3+ 314(2) 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' (4+) 423(3) 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' (4+) 431(2) (3+) 432(3) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' (2+) 460(3) 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' (3+) 517.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='9(1) 8− 517(3) 7 9 8− 583.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='9(5) 9− .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' 591(2) 1+ 589(3) 0 2 1+ This yields a predominantly L = 6 transition for the ground state, which is consistent with our observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' The same two-nucleon configuration dominates transfer to the 3+ 1 and 4+ 1 states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' For the former, the intensity of the L = 2 transition is nearly 17 times weaker than L = 4 transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' This agrees with the measured angular distribu- tion of the first excited state, observed at 74 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Next, we compared the measured cross section for this level rel- ative to the ground state (after accounting for the differ- ence in their predicted DWBA yields), with the relative scaling of their calculated transfer intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' The rea- sonable agreement between these two values validated the 3+ 1 assignment for this state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' In comparison, we identify the 140-keV state as 3+ 2 , whose dominant TNA corre- sponds to the (πd5/2) (νd3/2) orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Both L = 2 and L = 4 transfer contribute for this state, which agrees well with the measured distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Spin-parity assign- ments for the remaining states identified in Table I are made through similar analysis of the shapes of the angu- lar distributions, relative cross sections, and L-transfer intensities predicted by theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' We do not observe the explicit signatures of the low- lying 2+ states, which are predicted to be weakly pop- ulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' We also do not observe the known 9− state at 583.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='9(5) keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' This can be explained by the DWBA cal- culations, which show that L = 9 transfer for this state is significantly weaker than the dominant L = 7 transfer to the 8− state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' A tentative 3+ state was reported at 431 keV [18], but excluded from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' [20]’s compilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' We investigated this state’s possible existence by refitting the 423 keV peak with fixed lineshape parameters, based on previous knowledge of the detector response [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' This analysis indicated a possible level at Ex = 432(3) keV, whose angular distribution is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Although it is 4 Previous work [20] GCN5082 This work SN100PN QX 0 100 200 300 400 500 600 Excitation energy (keV) 5 + 3 + 4 + 3 + (4 +) (2 +) (3 +) 8 5 + 3 + 4 + 2 + 3 + 3 + 2 + 2 8 1 + 3 4 + 4 + 3 + 5 + 3 + 4 + 2 + 3 + 2 + 3 + 4 + 8 4 + 5 + 4 + 1 + 1 + 8 3 + 1 + 2 + 3 + 7 2 + 8 8 3 + 5 4 5 6 4 + 4 3 + 2 + 3 3 + 1 + 2 + 4 + 5 + 4 + 4 5 6 6 (2 +, 4 +) 9 7 7 5 6 3 4 6 7 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Comparison between theory and experiment for the low-lying energy spectrum of 136Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' The shell-model results were obtained with the SN100PN, GCN5082 and QX effective interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' statistics-limited, the measured distribution is consistent with L = 2 transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' The intensity of this possible tran- sition is comparable to those predicted for the 2+ 1 and 2+ 2 levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' We also observe that the θlab = 5◦ cross section for the 423 keV state is enhanced compared to the other L = 4 transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' This can be attributed to an addi- tional L = 2 component which is ∼ 20% of the L = 4 contribution, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Thus, one cannot rule out an unresolved state at ∼ 423 keV, with an L = 2 contribution that corresponds to one of the 2+ levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' The dominant (πg7/2), (πd5/2), (νd3/2), (νs1/2) and (νh11/2) configurations that describe these levels are also consistent with other observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' The spin-parity val- ues for first two (three) states in 137Cs (N = 82, Z = 55) and 137Ba (N = 81, Z = 56), are 7/2+, 5/2+, and 3/2+, 1/2+, 11/2−, respectively [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' This information, to- gether with independently measured 138Ba(d, 3He) spec- troscopic factors [66], provides an important validation of these configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Figure 2 makes it obvious that while the SN100PN and GCN5082 results are overall very similar and could be matched to our identified levels from this experiment, there is a stark disagreement with the QX results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Not only does the QX interaction fail to predict the cor- rect ground state for 136Cs (the 5+ 1 state is instead at 512 keV), it also shows several low-lying negative parity states that are not predicted by the other two Hamil- tonians or verified by experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' This comparison un- derscores the importance of testing model predictions in intermediate odd-odd nuclei for 0ν2β candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Un- der such requirement the QX interaction may be consid- ered less reliable;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' which calls into question 136Xe 0ν2β NME values determined with this Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' This is particularly relevant because the QX NME is ∼ 40% smaller [60] than the ones obtained with GCN5082 [67] and Vlow-k [68] effective interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' The QX interaction was also recently used to determine γγ decay NMEs [32], study the s-process nucleosynthesis of barium nuclei [69], calculate the subleading NME for 136Xe 2ν2β [39], and perform more up-to-date calculations of the 136Xe 0ν2β NME [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' In the context of detecting CNO and 7Be solar neu- trinos and/or fermionic dark matter events in Xe detec- tors, Haselschwardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' [15] showed that a delayed- coincidence analysis allows an efficient detection of CC- type excitations in LXe TPCs, on account of the slow 3+ 1 → 5+ 1 E2 transition in 136Cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' The SN100PN cal- culation shows the 3+ 1 state at 23 keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Based on this value, the state’s lifetime was estimated to be around 624 µs [15] - long enough for a feasible time-correlation analysis of scintillation light in LXe TPC detectors with drift lengths ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='3 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' On the other hand the GCN5082 interaction predicts the first excited state at 15 keV, with a comparatively longer lifetime of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='5 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Our experi- ment shows that the first excited state in 136Cs is in fact significantly higher in energy, at 74(2) keV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' As the tran- sition probability scales as Eγ 5, this implies that the life- time of the 3+ 1 state is approximately 340 times shorter than assumed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' This may have significant implications for the anticipated detection of solar neu- trino/fermionic dark matter events via CC excitations in future large-scale Xe TPC experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' This work was partially supported by the National Re- search Foundation (NRF), South Africa, under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' 85100, the Natural Sciences and Engineering Re- search Council of Canada (NSERC), the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' PHY-2110365 and the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Department of Energy (DOE) under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' 5 0 5 10 15 20 25 30 35 40 45 50 Scattering Angle, θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' (deg) 10 20 30 40 50 60 dσ/dΩ (µb/sr) J = 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' L = 4 + L = 6 J = 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' L = 4 E = 0 keV (πg7/2, νd3/2) 0 5 10 15 20 25 30 35 40 45 50 Scattering Angle, θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 dσ/dΩ (µb/sr) J=3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' L=4 E = 74 keV (πg7/2, νd3/2) 0 5 10 15 20 25 30 35 40 45 50 Scattering Angle, θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='5 dσ/dΩ (µb/sr) J = 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' L = 4 E = 104 keV (πg7/2, νd3/2) 0 5 10 15 20 25 30 35 40 45 50 Scattering Angle, θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 dσ/dΩ (µb/sr) J = 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' L = 2 + L = 4 E = 140 keV (πd5/2, νd3/2) 0 5 10 15 20 25 30 35 40 45 50 Scattering Angle, θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 dσ/dΩ (µb/sr) J = 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' L = 4 E = 314 keV (πd5/2, νd3/2) 0 5 10 15 20 25 30 35 40 45 50 Scattering Angle, θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' (deg) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 dσ/dΩ (µb/sr) J = 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' L = 4 (πg7/2, νs1/2) J = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' L = 2 + J = 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' L = 4 E = 423 keV 0 5 10 15 20 25 30 35 40 45 50 Scattering Angle, θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 dσ/dΩ (µb/sr) J = 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' L = 2 E = 432 keV 0 5 10 15 20 25 30 35 40 45 50 Scattering Angle, θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 dσ/dΩ (µb/sr) J = 3: L = 4 E = 460 keV (πg7/2, νs1/2) 0 5 10 15 20 25 30 35 40 45 50 Scattering Angle, θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 dσ/dΩ (µb/sr) J = 8: L = 7 + L = 9 E = 517 keV (πg7/2, νh11/2) 0 5 10 15 20 25 30 35 40 45 50 Scattering Angle, θc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content='0 dσ/dΩ (µb/sr) J = 1: L = 0 + L = 2 E = 589 keV (πd5/2, νd3/2) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Measured 138Ba(d, α) angular distributions compared with best-fit DWUCK5 DWBA predictions (solid red curves).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' The blue dashed curves are from using fixed relative L contri- butions from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' The dominant orbitals involved in the pair-transfer are specified in each plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' DE-FG02-93ER40789.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Anton, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Badhrees, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Barbeau, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Beck, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Belov, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Bhatta, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Breidenbach, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Brunner, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Cao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Cen, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' (EXO-200 Collaboration), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' [69] Kuo-Ang Li, Chong Qi, Maria Lugaro, Andr´es Yag¨ue L´opez, Amanda I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} +page_content=' Karakas, Jacqueline den Hartogh, Bing-Shui Gao, and Xiao-Dong Tang, The Astrophysical Journal Letters 919, L19 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ydFIT4oBgHgl3EQf1CsY/content/2301.11371v1.pdf'} diff --git a/ytAyT4oBgHgl3EQfO_Y5/content/tmp_files/2301.00015v1.pdf.txt b/ytAyT4oBgHgl3EQfO_Y5/content/tmp_files/2301.00015v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..aec2f5e0e5be310df218155609791c16cdf0edc4 --- /dev/null +++ b/ytAyT4oBgHgl3EQfO_Y5/content/tmp_files/2301.00015v1.pdf.txt @@ -0,0 +1,1230 @@ +Self-organization Preserved Graph Structure Learning +with Principle of Relevant Information +Qingyun Sun12, Jianxin Li12, Beining Yang12, Xingcheng Fu12, Hao Peng1, Philip S. Yu3 +1 Beijing Advanced Innovation Center for Big Data and Brain Computing, Beihang University, Beijing 100191, China +2 School of Computer Science and Engineering, Beihang University, Beijing 100191, China +3 Department of Computer Science, University of Illinois at Chicago, Chicago, USA +{sunqy, lijx, yangbeining, fuxc, penghao}@buaa.edu.cn, psyu@uic.edu +Abstract +Most Graph Neural Networks follow the message-passing +paradigm, assuming the observed structure depicts the +ground-truth node relationships. However, this fundamental +assumption cannot always be satisfied, as real-world graphs +are always incomplete, noisy, or redundant. How to re- +veal the inherent graph structure in a unified way remains +under-explored. We proposed PRI-GSL, a Graph Structure +Learning framework guided by the Principle of Relevant +Information, providing a simple and unified framework for +identifying the self-organization and revealing the hidden +structure. PRI-GSL learns a structure that contains the most +relevant yet least redundant information quantified by von +Neumann entropy and Quantum Jensen-Shannon divergence. +PRI-GSL incorporates the evolution of quantum continuous +walk with graph wavelets to encode node structural roles, +showing in which way the nodes interplay and self-organize +with the graph structure. Extensive experiments demonstrate +the superior effectiveness and robustness of PRI-GSL. +Introduction +Graph Neural Networks (GNNs) (Wu et al. 2020) have +gained popularity in recent years due to their remarkable +success in representing graph data in diverse tasks and ap- +plications. Most of the existing GNNs follow the message- +passing paradigm (Gilmer et al. 2017), i.e., exchanging +information between neighbors along the graph structure. +They take the raw graph structure as the path of information +flow, assuming the observed structure perfectly depicts the +ground-truth relations between nodes. However, these raw +graphs are naturally collected from network-structure data +(e.g., social networks), which are often noisy, incomplete, +and independent of the downstream tasks. There is a gap +between the raw structure and the optimal structure for spe- +cific tasks. The poor quality of graph structure leads to the +poor quality of representations produced by GNNs, making +GNNs prone to noise and adversarial attacks (Z¨ugner, Ak- +barnejad, and G¨unnemann 2018; Sun et al. 2018, 2021). +Graph structure learning (Zhu et al. 2022) aims to learn a +new structure of high quality simultaneously with the graph +representations, which has received growing attention for its +utility for improving representation quality and robustness. +Copyright © 2023, Association for the Advancement of Artificial +Intelligence (www.aaai.org). All rights reserved. +Ordered Network +Disordered Network +Cooperative or Competitive +Spontaneous Interactions +political +party +opinion +leader +Figure 1: Self-organization in political network. +Most existing methods optimize the structure with heuristic +assumptions (e.g., community (Wang et al. 2021)) or certain +structure constraints (e.g., sparsity, low-rank, and smooth- +ness (Jin et al. 2020; Sun et al. 2022b)). However, these as- +sumptions and constraints cannot always be applicable to all +graphs and tasks. How to reveal the inherent graph structure +in a unified way remains an under-explored question. +Most of the graph data in the real-world shows “self- +organization” property from molecules (Eigen and Schuster +1977) to social networks (Bonabeau et al. 1997), where the +nodes organize their interactions spontaneously through the +structure to create a global order amongst themselves. As +the example in Fig. 1, the emergence of opinion leaders and +the cooperation/competitive behaviors between people form +political parties, making the political network more ordered. +Since the structure navigates the information flow between +nodes and decides the graph’s fundamental mechanism, it +can be optimized to identify the organization and reduce the +disorder of the noisy graph. +In this paper, we introduce the self-organized Principle of +Relevant Information (PRI) to quantify the structure from +an information-theoretic point of view. We propose a novel +graph structure learning framework named PRI-GSL, which +inherits the merits of PRI to identify the self-organization +and reveal the inherent structure of graph. Rather than +imposing statistical constraints on the graph data, PRI- +GSL takes the structure learning as a trade-off between +structure redundancy reduction and information preserva- +tion, and then use the von Neumann entropy and the Quan- +tum Jensen-Shannon divergence to quantify them. To bet- +ter capture the contribution of nodes in the self-organization +evolution process, we use the quantum continuous walk evo- +lution with multi-scale graph wavelets to characterize node +arXiv:2301.00015v1 [cs.LG] 30 Dec 2022 + +structural roles and incorporate them into structure learning. +In this way, PRI-GSL enumerates the potential edges and +preserves the most relevant yet least redundant ones, show- +ing in which way the nodes interplay and self-organize with +the graph structure. +• We propose PRI-GSL, an information-theoretic graph +structure learning framework with the Principle of Rele- +vant Information, providing a simple yet unified way to +quantify the learned structure and unravel the graph self- +organization. +• We use the quantum continuous walk with graph wavelets +to encode node structural roles in a continuous and time- +varying way, which is incorporated in structure learning to +fully characterize the nodes in self-organization. +• Extensive experiment results demonstrate the superior ef- +fectiveness and robustness of PRI-GSL. +Related Work +Graph structure learning has gained more attention in re- +cent years (Zhu et al. 2022) to improve the quality of graph +representations by learning a better graph structure. Most ex- +isting works (Jin et al. 2020; Wang et al. 2021) optimize the +structure with assumptions or certain constraints in a heuris- +tic way. Substantial efforts have been made to give a theo- +retical quantification for the learned structure. SDRF (Top- +ping et al. 2022) refines the structure based on the Ricci cur- +vature of edges in a greedy pre-process strategy. SIB (Yu +et al. 2020a) utilizes the information bottleneck principle +to find the most predictive subgraph. VIB-GSL (Sun et al. +2022a) proposes a variational information bottleneck princi- +ple to learn a new structure for graph classification, which is +not applicable to the node-level tasks. Graph-PRI (Yu et al. +2022) advances the Principle of Relevant Information for +graph sparsification in an unsupervised way without consid- +ering node features and the specific downstream task. +Information theory provides a powerful methodology to +describe general properties of arbitrarily complex systems. +In information theory, there are two representative self- +organizing principles: Information Bottleneck (IB) (Tishby, +Pereira, and Bialek 2000) and Principle of Relevant Informa- +tion (PRI) (Principe 2010). Both IB and PRI describe differ- +ent forms of redundancy reduction and information preser- +vation. The famed IB is formulated on the mutual informa- +tion between independent and identically distributed (i.i.d.) +data, which is difficult to model the complex node interac- +tions imposed by the graph structure. PRI shares the spirit of +the IB method but its formulation addresses the entropy and +relative entropy of a single dataset (Principe 2010), which +can be applied to graph data with well-defined information- +theoretic tools. +Preliminary +Notions +Given a graph G = {V, E} where V is the set of +N nodes and E is the edge set. A ∈ RN×N is the adjacency +matrix and D is the degree matrix. The Laplacian matrix of +the graph G can be defined as L = D − A = UΛUT, +where U is the eigenvector matrix, Λ = Diag(λ1, · · · , λN) +and λ1 < λ2 ≤ · · · ≤ λN are the eigenvalues of L. +Graph Structure Learning +Given a graph G, graph struc- +ture learning (Zhu et al. 2022) aims to learn a new structure +˜G simultaneously with the graph representations with the +objective function: +L = Ltask( ˜G, Y ) + αLreg( ˜G, G), +(1) +where Ltask is the task-specific objective with respect to the +learned graph ˜G and the ground truth Y , Lreg imposes con- +straints on the learned graph and α is a hyper-parameter. +Principle of Relevant Information +PRI (Principe 2010) +is a self-organized information-theoretic principle that aims +to perform mode decomposition of a random variable to +obtain a reduced statistical representation. PRI formulates +the redundancy reduction and information preservation as a +trade-off between the entropy of reduced representation and +its relative entropy given the original data. +Definition 1 (Principle of Relevant Information). Given a +random variable X, the Principle of Relevant Information +aims to obtain a reduced representation T with: +LPRI = arg min +T +H(T) + βD(P(T)||P(X)), +(2) +where H(T) is the entropy of T and D(P(T)||P(X)) is the +divergence of distributions P(T) and P(X). +The first term H(T) measures the redundancy of repre- +sentation T and the second term D(P(T)||P(X)) measures +the allowable distortion of the original data. The hyper- +parameter β controls the level of distortion in T. PRI was +commonly used in scalar random variables (Wei et al. 2021; +Hoyos-Osorio et al. 2021) and defined by R´enyi’s formula- +tion of entropy and divergence (R´enyi et al. 1961). +Graph Structure Learning with Principle of +Relevant Information +In this work, we propose a graph structure learning frame- +work named PRI-GSL, which merits the Principle of Rel- +evant Information as a guideline for controlling the struc- +ture quality. The overall architecture of PRI-GSL is shown +in Figure 2. In this section, we first formulate the PRI loss +for structure learning, then introduce the Role-aware graph +learner and learning process of PRI-GSL. +PRI for Graph Structure Learning +In the PRI-GSL framework, PRI performs as a self- +supervised regularizer for the quality of the learned graph +structure. Motivates by the objective of PRI, the Graph +Structure Learning Principle of Relevant Information is: +Definition 2 (PRI for Graph Structure Learning). Given a +graph G, the Principle of Relevant Information for graph +structure learning aims to learn a refined graph ˜G with: +LPRI = H( ˜G) + βD( ˜G||G), +(3) +The first term H( ˜G) is the redundancy term, which mea- +sures the disorder of the learned graph ˜G. The larger the +H( ˜G), the more disordered ˜G is. The second term D( ˜G||G) + +t +Refined Graph +Original Graph ++ +Role-aware Structure Learner +QCW Evolution +Wavelets +… +Probability +t +… +Role Characteriztion +Figure 2: Overall Architecture of PRI-GSL. +is the distortion term, which measures the discrepancy be- +tween two graphs. The smaller the D( ˜G||G), the more sim- +ilar are the distributions of ˜G and G. β denotes the trade-off +between the redundancy reduction of ˜G and its discrimina- +tive description power of G. As β becomes larger, the em- +phasis is laid more on the distortion term, and more infor- +mation from G is preserved in ˜G. +Formulate PRI by von Neumann Entropy and +Quantum Jensen-Shannon Divergence +The choice of entropy and divergence in PRI is application- +specific. In this paper, we formulate the PRI loss LPRI by the +von Neumann entropy (VNE) (Nielsen and Chuang 2002) +and quantum Jensen-Shannon (QJS) divergence (Lamberti +et al. 2008) for graph data with complex interactions as +in (Yu et al. 2022). +For the first redundancy term H( ˜G), we propose to mea- +sure the structure redundancy by von Neumann entropy +(VNE) (Nielsen and Chuang 2002), which has been used +in a variety of graph learning studies (Dasoulas et al. 2020; +Yu et al. 2022). von Neumann entropy quantifies the spec- +tral complexity (or disorder) of graph structure by taking the +graph as a quantum system through a mapping between dis- +crete Laplacians and quantum states. A density matrix ρ is +a Hermitian and positive semi-definite matrix that is used +to encode the probability distributions and describe the state +of a quantum mechanical system. For the graph ˜G, the von +Neumann entropy HvN( ˜G) is defined as: +HvN( ˜G) = HvN(˜ρ) = −tr (˜ρ log ˜ρ) = − +N +� +i=1 +(λi log λi) , +(4) +where ˜ρ is the graph density matrix of ˜G, tr(·) denotes trace +and {λi} are the eigenvalues of ˜ρ. Typically, both the Lapla- +cian matrix and the normalized Laplacian matrix can be used +for the mapping from graphs to states (Minello, Rossi, and +Torsello 2019). We define the density matrix ˜ρ = +˜L +tr(˜L) = +˜L +2|E| based on the Laplacian matrix ˜L of ˜G, which models +the continuous information diffusion process (De Domenico +and Biamonte 2016). +For the second distortion term D( ˜G||G), we use the +quantum Jensen-Shannon (QJS) divergence (Lamberti et al. +2008) between the graph density matrices (˜ρ and ρ) of +G and ˜G. The quantum Jensen-Shannon divergence has +been widely used as a generalization of the classical +Jensen-Shannon divergence to quantum states of graph +data (De Domenico et al. 2015; Bai et al. 2015), which is +symmetric, negative definite and bounded (0 ≤ DQJS ≤ 1). +DQJS( ˜G||G) = HvN +� ˜ρ + ρ +2 +� +− 1 +2HvN (˜ρ) − 1 +2HvN (ρ) . +(5) +Combining the redundancy term in Eq. (4) and the distor- +tion term in Eq. (5), we can obtain the following objective: +LPRI = HvN( ˜G) + βDQJS( ˜G||G) += βHvN +� ˜ρ + ρ +2 +� ++ 2 − β +2 +HvN (˜ρ) − β +2 HvN (ρ) +≡ βHvN +� ˜ρ + ρ +2 +� ++ 2 − β +2 +HvN (˜ρ) . +(6) +We neglect HvN (ρ) in the last line because it’s a constant +value during optimization. The above formalism provides a +unified way of quantification and comparison for the learned +structure. Then we use a graph structure learner to obtain ˜G. +Role-aware Graph Structure Learner +In this section, we introduce the Role-aware Graph Structure +Learner in PRI-GSL, which aims to learn a better graph ˜G +that preserves the graph self-organization. Considering the +evolution process in self-organization, we characterize the +nodes’ roles in a continuous and time-varying way and then +incorporate both the merits of features as well as structural +roles to refine the graph. +Structural Role Encoding +The structural role of the node +represents its contribution to the overall information flow of +the graph, which can provide key insights into the identifica- +tion of graph organization. We propose to model graph state +by quantum continuous walk and use the time-evolution op- +erator with graph wavelets to generate role encodings. +(1) Model graph state by QCW. Recall that we apply the +von Neumann Entropy and the QJS divergence for PRI for- +mulation, which takes the whole graph as a quantum system. +To investigate the nodes’ roles in this quantum system, we +use the quantum continuous walk (QCW) (Childs 2010; Bai + +et al. 2015) to build maps of how information flows through +the graph in the perspective of graph state evolution. QCW is +the quantum mechanical counterpart of the continuous-time +random walk in a graph, which describes the propagation +of a quantum particle evolving continuously in time on the +nodes. The QCW on a graph G is defined as a dynamical +process over the nodes that takes place on a N-dimensional +Hilbert space H = span({|v⟩ , v ∈ V }). The evolution of +the walker is governed by the Schr¨odinger equation +d +dt |ψt⟩ = −iH |ψt⟩ . +(7) +|ψt⟩ represents the state of the walk at time t, which is a +time-dependent amplitude vector on nodes. H is the Hamil- +tonian operator, which accounts for the total energy of the +graph and governs the time evolution of the quantum con- +tinuous walk. Given an initial state |ψ0⟩ ∈ H, the state of +walker |ψt⟩ evolves in time according to +|ψt⟩ = U(t) |ψ0⟩ , +U(t) := e−iHt, +(8) +where U(t) is the unitary time-evolution operator. +(2) Diffusion by graph wavelets. To give a full char- +acterization of the structural properties, we use the spectral +graph wavelets Ψ (Hammond, Vandergheynst, and Gribon- +val 2011) as the Hamiltonian H in QCW. Then the structural +information residing in the graph is encoded in U(t). U(t) +is a polynomial in Ψ for all t, thus any matrix that com- +mutes with Ψ also commutes with U(t) (Coutinho and God- +sil 2021). We adopt the heat kernel gs(Λ) = e−Λs to obtain +spectral graph wavelets, where the scaling parameter s con- +trols the spread radii of the diffusion process and larger s +allows farther diffusion. The spectral wavelet basis is +Ψs = UΛsUT = (Ψs (1) |Ψs (2) | · · · |Ψs (N)) , +(9) +where Λs = gs(Λ). In this way, the spectral graph wavelet +Ψs(a) centered at node va associated with filter gs will be +given by an N-dimensional vector: +Ψs(a) = UΛsUTδa, +(10) +where δa is the one-hot vector of node va. Ψs(a) is a N- +dimensional vector where the b-th wavelet coefficient of +Ψs(a) represents the information that va received from vb. +Nodes playing similar roles have similar wavelet coefficient. +(3) Characterize by the time-evolution operator. Since +the time-evolution operator U(t) reflects the graph state evo- +lution, we treat the wavelets Ψ as probability distributions +over graph and use φ(Ψ, t) = E[U(t)] = E[e−iΨt] as the +characteristic function to uncover nodes’ roles in informa- +tion diffusion. The empirical characteristic function of va is: +φs(va, t) = 1 +N +N +� +n=1 +e−iΨs(a)t. +(11) +φs(va, t) can capture all the moments (including higher- +order moments) of the given distribution Ψs(a). We sample +at T different time points on the time-evolution operator and +then concatenate the values: +hs(va) = [Re (φs (va, t)) , Im (φs (va, t))]t=t1,t2,··· ,T . +(12) +In general, the nodes play different roles across different +scales. Hence we utilize a multi-scale wavelet diffusion +strategy to capture both the local and global structural roles +of nodes. We integrate information across different radii of +neighborhoods by jointly considering a set of different val- +ues of s. We can obtain multi-scale structure role encodings +ha ∈ R2T M by concatenating the structural encodings at +several different scales S : {s1, s2, · · · , sM}: +ha = (hs1 (va) |hs2 (va) | · · · |hsM (va)) . +(13) +Iterative Structure Learning +After obtaining the struc- +tural role encodings, we use a metric function that accounts +for both feature information and the role-based similarities +to measure the possibility of edge existence. PRI-GSL is ag- +nostic to various metric functions and we choose the multi- +head cosine similarity function here: +aR(e−1) +ij += 1 +m +m +� +h=1 +cos (Wh · +� +z(e−1) +i +|h(e−1) +i +� +, +Wh · +� +z(e−1) +j +|h(e−1) +j +�� +, +(14) +where m is the number of heads, Wh is the weight matrix +of the h-th head, z(e−1) +i +and hR(e−1) +i +denote the representa- +tion vector and the structural role encoding vector of node +vi in the (e − 1)-th epoch, and | denotes the concatenation +operation. With the above structure learning strategy, we can +obtain a role-aware adjacency matrix in the e-th epoch: +A(e) +R = +� +aR(e−1) +ij +� +, i, j ∈ {1, 2, · · · , N}. +(15) +Learning Process of PRI-GSL +Dynamic Structure Fusion +The input graph structure de- +termines the learning performance to a certain extent. To +avoid the non-convergence or unstable training brought by +the poor quality of learned structure at the beginning of train- +ing, we hence incorporate the original graph structure A as +supplementary to formulate an optimized graph structure ˜A: +˜A(e) = γD− 1 +2 AD− 1 +2 +(1 − γ)·RowNorm +� +A(e) +R +� +, (16) +where RowNorm(·) denotes the row-wise normalization +function, γ is a constant that control the contribution of orig- +inal structure. Here we use a dynamic decay mechanism for +γ to enable the role-aware structure A(e) +R to play a more and +more important role during training. Then the refined struc- +ture is inputted into a GNN encoder for node representation +vectors Z ∈ RN×d and classification: +Z(e) = GNN-Encoder( ˜A(e), X). +(17) +Objective of PRI-GSL +The overall loss L of PRI-GSL is +composed of two terms, the classification loss Lcls and the +structure PRI loss LPRI in Eq. (6), given by: +L = Lcls + αLPRI += HCE +� +˜G, Y +� ++ α +� +HvN +� +˜G +� ++ βDQJS +� +˜G||G +�� +, +(18) +where HCE(·) is the cross-entropy loss for classification and +α is a hyper-parameter to balance the two loss terms. The +overall process of PRI-GSL is shown in Algorithm 1. + +Algorithm 1: The overall process of PRI-GSL for +node classification +Input: Graph G with node labels Y ; Number of +training epochs Epochs; Wavelet scale set S; +Timepoints T; Hyper-parameters α, β and γ. +Output: Refined graph ˜G; Predicted labels ˆY . +1 Parameter initialization; +2 for e = 1, 2, · · · , Epochs do +// Structural Role Encoding +3 +for s ∈ S do +4 +Ψ(e−1) +s +← Eq. (9), φs(va, t) ←Eq. (11); +5 +h(e−1) +s +(va) ← Eq. (12); +6 +end +7 +h(e−1) +a +← Eq. (13); +// Graph Structure Learning +8 +aR(e) +ij +← Eq. (14), A(e) +R = +� +aR(e) +ij +� +; +9 +˜A(e) ←Eq. (16), ˜G(e) ← (X, ˜A(e)); +// Learn Node Representations +10 +Z(e) = GNN-Encoder( ˜A(e), X); +// Optimize +11 +L(e) +PRI = HvN( ˜G(e)) + βDQJS( ˜G(e)||G); +12 +L(e) = L(e) +cls( ˜G(e), Y ) + αL(e) +PRI; +13 +Update model parameters to minimize L(e). +14 end +Approximation +Recall that the computation of von Neu- +mann entropy and spectral wavelets requires the full eigen- +value decomposition of the Laplacian matrix, which takes +O(N 3) time. The von Neumann entropy can be approxi- +mated with linear complexity O(|V | + |E|) (Chen et al. +2019). In the experiments, we still use the basic von Neu- +mann entropy. As for the graph wavelets, we use the Cheby- +shev polynomial approximation (Shuman, Vandergheynst, +and Frossard 2011) to compute Ψ, reducing the computa- +tional complexity to O(K|E|), where K is the order of +Chebyshev polynomials. +Properties of ˜G Learned by PRI-GSL +PRI-GSL provides a unified way to control the quality of +learned graph structure in terms of sparsity, centrality, and +nuisance invariance property. +Sparsity and Centrality +The von Neumann entropy has +close connections with the structure sparsity and centrality. +As indicated in (Passerini and Severini 2008), given a graph +G, let G′ = G + {x, y} with V (G′) = V (G) and E(G′) = +E(G) ∪ {x, y}, then HvN(ρG′) ≥ dG′−2 +dG′ HvN(ρG). The von +Neumann entropy tends to grow with the increasing num- +ber of edges. The graph centrality (i.e., the extent to which a +graph is organized around some central nodes) can be mea- +sured as a quantum relative entropy between the relative +degree distribution and the uniform distribution (Simmons, +Coon, and Datta 2018), which is given by D(ρG||IN) := +tr(ρG(log ρG − log I/N)) = log N − HvN(G). The von +Neumann entropy tends to grow with the increasing regular- +ity of the graph. The above conclusions suggest that mini- +mizing HvN( ˜G) leads to a sparse and centralized structure. +Nuisance Invariance +˜G only preserves the most relevant +yet least redundant information in the observed graph G and +is invariant to nuisances in data. Suppose Gn ∈ G the task- +irrelevant nuisance in G, the relevance of ˜G and Gn can +be formulated as the divergence between their conditional +distributions predictions of the desired labels Y (Yu et al. +2020b): E +� +DKL +� +p +� +Y | ˜G +�� +||p (Y |Gn) +� +. Since Gn is ir- +relevant with Y , we have p(Y |Gn) = p(Y ). Minimizing the +cross-entropy loss (i.e., the mutual information I(Y ; ˜G) be- +tween ˜G and Y ) is equivalent to minimizing the relevance +between ˜G and Gn: +E +� +DKL +� +p +� +Y | ˜G +�� +||p (Y |Gn) +� += E +� +DKL +� +p +� +Y | ˜G +�� +||p (Y ) +� += +�� � +�p +� +Y | ˜G +� +log +p +� +Y | ˜G +� +p (Y ) +� +� p +� +˜G +� += +�� +p +� +Y | ˜G +� +log +p +� +Y, ˜G +� +p (Y ) p +� +˜G +� += I(Y ; ˜G). +(19) +Experiments +We evaluate PRI-GSL on node classification and graph de- +noising tasks to verify its capability of improving the ef- +fectiveness and robustness of graph representation learning. +Then we provide the analyses of the PRI loss, the structural +role encodings, and the learned structure. +Experimental Settings +Datasets +We select datasets with different homophily ra- +tios h (Pei et al. 2020) to analyze methods’ generalization +on graphs with different properties. The evaluation datasets +are Squirrel, Chameleon (Rozemberczki, Allen, and Sarkar +2021), Actor (Pei et al. 2020), CiteSeer, PubMed, Cora (Sen +et al. 2008) and Photo (Shchur et al. 2018). +Baselines +We consider three types of baselines: (1) +Graph neural networks: GCN (Kipf and Welling 2016), +GAT (Veliˇckovi´c et al. 2017) and GraphSAGE (Hamilton, +Ying, and Leskovec 2017); (2) Graph sparsification meth- +ods: DropEdge (Rong et al. 2019), NeuralSparse (Zheng +et al. 2020), and Graph-PRI (Yu et al. 2022); (3) Graph +structure learning methods: IDGL (Chen, Wu, and Zaki +2020), Pro-GNN (Jin et al. 2020), SDRF (Topping et al. +2022), and SLAPS (Fatemi, El Asri, and Kazemi 2021). +Parameter +Settings +We +re-implement +the +Neu- +ralSparse +(Zheng +et +al. +2020) +and +SDRF +(Topping +et al. 2022). The parameters of baseline methods are set to +the suggested value in their papers or carefully tuned for + +Table 1: Accuracy ± standard deviation (%) of node classification. (Bold: best result; Underlined: runner up. ) +Method +Squirrel +h=0.22 +Actor +h=0.24 +Chameleon +h=0.25 +CiteSeer +h=0.72 +PubMed +h=0.79 +Cora +h=0.83 +Photo +h=0.83 +GCN +22.22±1.24 +22.85±1.64 +32.16±2.76 +66.31±1.12 +74.11±3.65 +79.10±0.77 +85.61±2.20 +GAT +22.64±1.25 +23.53±1.33 +32.05±2.56 +63.85±2.45 +73.02±2.52 +77.86±1.48 +87.97±2.73 +GraphSAGE +28.79±1.74 +24.22±1.44 +37.05±2.35 +64.80±1.83 +72.61±2.95 +75.23±1.31 +86.23±2.53 +DropEdge +22.35±1.12 +23.84±1.20 +32.62±2.75 +66.68±1.38 +75.97±0.82 +79.30±0.84 +86.05±1.78 +NeuralSparse +29.02±1.10 +24.50±1.42 +47.30±2.22 +67.82±1.18 +74.87±2.77 +81.47±1.43 +89.40±1.85 +Graph-PRI +28.44±2.10 +23.81±2.31 +42.39±1.99 +69.24±1.25 +76.25±1.44 +79.07±1.12 +88.30±2.11 +IDGL +29.13±2.94 +27.44±5.80 +49.80±4.80 +67.94±0.28 +75.32±1.45 +83.23±0.62 +88.89±2.55 +Pro-GNN +27.18±1.28 +24.82±2.81 +48.54±4.87 +66.68±2.02 +75.44±3.54 +82.14±0.58 +87.28±1.85 +SDRF +>1 day +>1 day +41.05±1.17 +69.97±0.28 +>1 day +81.94±0.59 +>1 day +SLAPS +25.29±1.06 +23.10±3.39 +40.24±1.80 +68.58±1.46 +75.64±0.77 +79.27±1.54 +88.48±2.47 +PRI-GSL +33.87±2.08 +28.55±2.04 +51.83±2.44 +69.34±2.64 +76.77±3.20 +83.67±2.09 +92.40±1.15 +0.00 +0.25 +0.50 +0.75 +Deleting edges +65 +70 +75 +80 +85 +90 +Accuracy (%) +83.67 +82.95 +80.63 +76.67 +0.00 +0.25 +0.50 +0.75 +Adding edges +55 +60 +65 +70 +75 +80 +85 +90 +Accuracy (%) +83.67 +79.00 +75.62 +71.33 +GCN +Graph-PRI +IDGL +PRI-GSL +Figure 3: PRI-GSL on noisy graphs. +fairness. For the GNN encoders, we use a 2-layer GCN for +node classification. We set the representation dimension +d=32, the Chebyshev polynomial order K=10, the number +of time points T=4, the number of scales M=2, and the +number of heads m=4. The other hyper-parameters (α, β, +and γ) are tuned for each dataset. +Evaluation Results and Analysis +Node Classification +We set the number of nodes in each +class to be 20/30 for training/validation and take the remain- +ing nodes for the test. The accuracy and standard deviation +on 10 randomly split are shown in Table 1. The best results +are shown in bold and the runner-ups are underlined. Our +PRI-GSL achieves the best performance on both homophilic +and heterophilic datasets, showing the effectiveness of uti- +lizing the self-organization property when mining the la- +tent inherent structure. Generally, the graph structure learn- +ing methods show better performance than GNNs and graph +sparsification methods. Pro-GNN and SLAPS perform well +on the homophilic graphs but show unsatisfactory perfor- +mance on the heterophilic graphs (Actor, Chameleon, and +Squirrel), demonstrating the limitation of using heuristic as- +sumptions for structural constraints. Although Graph-PRI +also uses PRI for graph sparsification, it achieves fewer im- +provements since it does not take the node feature and the +downstream task into consideration. +Cora +CiteSeer +Chameleon +40 +50 +60 +70 +80 +90 +Accuracy (%) +83.7 +69.9 +50.9 +82.2 +69.2 +50.0 +81.2 +68.4 +48.4 +80.0 +67.5 +42.4 +PRI-GSL +PRI-GSL (w/o PRI) +PRI-GSL (w/o RE) +GSL +Figure 4: Ablation study results of PRI-GSL. +Graph Denoising +To evaluate PRI-GSL’s ability to move +noisy information, we generate synthetics noisy datasets by +adding/deleting edges on Cora following (Chen, Wu, and +Zaki 2020; Sun et al. 2022a). Specifically, we randomly ad- +d/delete 25%, 50%, 75% edges for 5 times and show the +mean accuracy (solid line) and standard deviation (shaded +region) in Fig. 3. The performance of GCN and Graph- +PRI decreases dramatically with the increasing noise level. +The structure learning methods, IDGL and PRI-GSL, show +more robustness compared to vanilla GCN and Graph-PRI. +Adding edge hurts more than deleting edges, indicating the +importance of removing redundant information in the struc- +ture. PRI-GSL consistently shows better performance under +different levels of external noise than the other baselines. +Even though Graph-PRI shares the same spirit with PRI- +GSL, it fails to distinguish whether the noise is task-relevant +and shows the same poor robustness as GCN. The perfor- +mance of PRI-GSL on noisy graphs also demonstrates the +nuisance invariance property. +Ablation Study +To illustrate the advantages of the guid- +ance of PRI and structural role information, we compare +PRI-GSL with three variants: (1) PRI-GSL (w/o PRI) that +removes the PRI loss, (2) PRI-GSL (w/o RE) that removes +the role encodings, and (3) GSL that removes both the PRI +loss and the role encoding. The results of variants on 5 ran- +dom split datasets are shown in Fig. 4. As we can observe, +both the PRI loss and the structural role encoding benefit + +(a) Original Graph. +(b) Graph-PRI. +(c) Pro-GNN. +(d) IDGL. +(e) PRI-GSL. +Figure 5: Visualization of the original graph of Cora and learned graphs by Graph-PRI, Pro-GNN, IDGL, and PRI-GSL. +0 +25 +50 +75 +100 +125 +150 +Epochs +0.000029 +0.000030 +0.000031 +0.000032 +0.000033 +HvN( ̃G) ++2.9187 +converge at 2.9187301 +0 +25 +50 +75 +100 +125 +150 +Epochs +2.9185 +2.9190 +2.9195 +DQJS( ̃G||G) +Figure 6: The variations of HvN( ˜G) and DQJS( ˜G||G). +the classification, where the structural role encoding brings +more improvement. This suggests that it’s important to cap- +ture the node’s contribution to the graph information flow +when identifying the graph organization. +Empirical Behavior of HvN( ˜G) and DQJS( ˜G||G) +We +analyze the learning dynamics of PRI-GSL by measuring +the variations of HvN( ˜G) and DQJS( ˜G||G) on Cora with +α=0.1 and β=1 in Fig. 6. The shadowed area is enclosed by +the min and max value of four training runs. The solid line +in the middle is the mean value of each epoch. HvN( ˜G) first +increases for about 50 epochs with the structure exploration +and then decreases to converge after about the 80-th epoch, +indicating that the learned structure is with high certainty. +DQJS( ˜G||G) bumps during the training process. This may +be because the model continues to seek a balance of struc- +ture redundancy and distortion during the training process. +Hyper-parameter Analysis +We analyze the impact of +hyper-parameters including α controlling the importance of +the PRI loss in Eq. (18) and β trading off redundancy and +distortion in Eq. (6). The results are shown in Fig. 7. PRI- +GSL achieves the best performance with α=1 on Cora and +α=0.4 on CiteSeer, which indicates that PRI-GSL benefits +from the PRI loss. As for the β in the PRI loss, when the dis- +tortion term has a weight more than 2 compared to the redun- +dancy term, PRI-GSL could reach satisfactory performance +on both datasets. This suggests that the distortion term dom- +inates the PRI loss in PRI-GSL. That is to say, the learned +0.1 +0.2 +0.4 +1.0 +2.0 +α +65 +70 +75 +80 +85 +90 +Accuracy (%) +83.28 +83.22 +83.33 +84.1 +83.28 +69.67 +69.03 +70.42 +69.03 +69.03 +Cora +CiteSeer +1 +2 +3 +4 +5 +β +65 +70 +75 +80 +85 +90 +Accuracy (%) +82.33 +83.05 +83.24 +83.34 +83.24 +69.11 +69.11 +69.56 +70.11 +69.67 +Cora +CiteSeer +Figure 7: Parameter sensitivity of α and β. +structure should preserve enough information from the orig- +inal graph to perform well on the downstream task. +Visualization +In Figure 5, we visualize the original graph structure of Cora +and the graphs learned by PRI-Graph, Pro-GNN, IDGL, and +PRI-GSL using networkx. The nodes’ colors indicate their +classes, the labeled nodes are solid and the unlabeled nodes +are hollow. The edges are not shown for clarity and the lay- +out of nodes represents their connectivities. Graph-PRI has +little effect on the overall property of graph structure. Even +though Pro-GNN and IDGL can make nodes within different +classes more separate, there are still some overlapping and +entangled areas. Benefiting from the structural role encod- +ing, PRI-GSL can obtain the structure with separate clusters +with similar shapes and clearer class boundaries, showing +how the nodes within a class are organized. +Conclusion +In this work, we propose an information-theoretic frame- +work for graph structure learning named PRI-GSL. We for- +mulate the Principle of Relevant Information for graph data +to quantify the structure redundancy and distortion, which +acts as a potentially unifying guidance of structure learn- +ing. We propose a role-aware structure learner based on +the quantum continuous walk evolution to unravel the self- +organization of the graph. The learned structure enjoys the +property of sparse, centralized, and nuisance invariance. Ex- +tensive experiments demonstrate the superior effectiveness +and robustness of PRI-GSL. + +O +O +O +00000O +C +CO +CAcknowledgments +The corresponding author is Jianxin Li. The authors are +supported by the NSFC through grant No.U20B2053, and +in part by NSF under grants III-1763325, III-1909323, III- +2106758, and SaTC-1930941. +References +Bai, L.; Rossi, L.; Torsello, A.; and Hancock, E. R. 2015. +A quantum Jensen–Shannon graph kernel for unattributed +graphs. Pattern Recognition, 48(2): 344–355. +Bonabeau, E.; Theraulaz, G.; Deneubourg, J.-L.; Aron, S.; +and Camazine, S. 1997. Self-organization in social insects. +Trends in ecology & evolution, 12(5): 188–193. +Chen, P.-Y.; Wu, L.; Liu, S.; and Rajapakse, I. 2019. 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In ACM +SIGKDD, 2847–2856. + diff --git a/ytAyT4oBgHgl3EQfO_Y5/content/tmp_files/load_file.txt b/ytAyT4oBgHgl3EQfO_Y5/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6908cc2b9b9b67ec926c248d96aeb98e335d27e8 --- /dev/null +++ b/ytAyT4oBgHgl3EQfO_Y5/content/tmp_files/load_file.txt @@ -0,0 +1,1050 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf,len=1049 +page_content='Self-organization Preserved Graph Structure Learning with Principle of Relevant Information Qingyun Sun12, Jianxin Li12, Beining Yang12, Xingcheng Fu12, Hao Peng1, Philip S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Yu3 1 Beijing Advanced Innovation Center for Big Data and Brain Computing, Beihang University, Beijing 100191, China 2 School of Computer Science and Engineering, Beihang University, Beijing 100191, China 3 Department of Computer Science, University of Illinois at Chicago, Chicago, USA {sunqy, lijx, yangbeining, fuxc, penghao}@buaa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='cn, psyu@uic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='edu Abstract Most Graph Neural Networks follow the message-passing paradigm, assuming the observed structure depicts the ground-truth node relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' However, this fundamental assumption cannot always be satisfied, as real-world graphs are always incomplete, noisy, or redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' How to re- veal the inherent graph structure in a unified way remains under-explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' We proposed PRI-GSL, a Graph Structure Learning framework guided by the Principle of Relevant Information, providing a simple and unified framework for identifying the self-organization and revealing the hidden structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' PRI-GSL learns a structure that contains the most relevant yet least redundant information quantified by von Neumann entropy and Quantum Jensen-Shannon divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' PRI-GSL incorporates the evolution of quantum continuous walk with graph wavelets to encode node structural roles, showing in which way the nodes interplay and self-organize with the graph structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Extensive experiments demonstrate the superior effectiveness and robustness of PRI-GSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Introduction Graph Neural Networks (GNNs) (Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2020) have gained popularity in recent years due to their remarkable success in representing graph data in diverse tasks and ap- plications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Most of the existing GNNs follow the message- passing paradigm (Gilmer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2017), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=', exchanging information between neighbors along the graph structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' They take the raw graph structure as the path of information flow, assuming the observed structure perfectly depicts the ground-truth relations between nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' However, these raw graphs are naturally collected from network-structure data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=', social networks), which are often noisy, incomplete, and independent of the downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' There is a gap between the raw structure and the optimal structure for spe- cific tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The poor quality of graph structure leads to the poor quality of representations produced by GNNs, making GNNs prone to noise and adversarial attacks (Z¨ugner, Ak- barnejad, and G¨unnemann 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2018, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Graph structure learning (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2022) aims to learn a new structure of high quality simultaneously with the graph representations, which has received growing attention for its utility for improving representation quality and robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Copyright © 2023, Association for the Advancement of Artificial Intelligence (www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='aaai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='org).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' All rights reserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Ordered Network Disordered Network Cooperative or Competitive Spontaneous Interactions political party opinion leader Figure 1: Self-organization in political network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Most existing methods optimize the structure with heuristic assumptions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=', community (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2021)) or certain structure constraints (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=', sparsity, low-rank, and smooth- ness (Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2022b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' However, these as- sumptions and constraints cannot always be applicable to all graphs and tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' How to reveal the inherent graph structure in a unified way remains an under-explored question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Most of the graph data in the real-world shows “self- organization” property from molecules (Eigen and Schuster 1977) to social networks (Bonabeau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 1997), where the nodes organize their interactions spontaneously through the structure to create a global order amongst themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' As the example in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 1, the emergence of opinion leaders and the cooperation/competitive behaviors between people form political parties, making the political network more ordered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Since the structure navigates the information flow between nodes and decides the graph’s fundamental mechanism, it can be optimized to identify the organization and reduce the disorder of the noisy graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' In this paper, we introduce the self-organized Principle of Relevant Information (PRI) to quantify the structure from an information-theoretic point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' We propose a novel graph structure learning framework named PRI-GSL, which inherits the merits of PRI to identify the self-organization and reveal the inherent structure of graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Rather than imposing statistical constraints on the graph data, PRI- GSL takes the structure learning as a trade-off between structure redundancy reduction and information preserva- tion, and then use the von Neumann entropy and the Quan- tum Jensen-Shannon divergence to quantify them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' To bet- ter capture the contribution of nodes in the self-organization evolution process, we use the quantum continuous walk evo- lution with multi-scale graph wavelets to characterize node arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='00015v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='LG] 30 Dec 2022 structural roles and incorporate them into structure learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' In this way, PRI-GSL enumerates the potential edges and preserves the most relevant yet least redundant ones, show- ing in which way the nodes interplay and self-organize with the graph structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' We propose PRI-GSL, an information-theoretic graph structure learning framework with the Principle of Rele- vant Information, providing a simple yet unified way to quantify the learned structure and unravel the graph self- organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' We use the quantum continuous walk with graph wavelets to encode node structural roles in a continuous and time- varying way, which is incorporated in structure learning to fully characterize the nodes in self-organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Extensive experiment results demonstrate the superior ef- fectiveness and robustness of PRI-GSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Related Work Graph structure learning has gained more attention in re- cent years (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2022) to improve the quality of graph representations by learning a better graph structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Most ex- isting works (Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2021) optimize the structure with assumptions or certain constraints in a heuris- tic way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Substantial efforts have been made to give a theo- retical quantification for the learned structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' SDRF (Top- ping et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2022) refines the structure based on the Ricci cur- vature of edges in a greedy pre-process strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' SIB (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2020a) utilizes the information bottleneck principle to find the most predictive subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' VIB-GSL (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2022a) proposes a variational information bottleneck princi- ple to learn a new structure for graph classification, which is not applicable to the node-level tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Graph-PRI (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2022) advances the Principle of Relevant Information for graph sparsification in an unsupervised way without consid- ering node features and the specific downstream task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Information theory provides a powerful methodology to describe general properties of arbitrarily complex systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' In information theory, there are two representative self- organizing principles: Information Bottleneck (IB) (Tishby, Pereira, and Bialek 2000) and Principle of Relevant Informa- tion (PRI) (Principe 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Both IB and PRI describe differ- ent forms of redundancy reduction and information preser- vation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The famed IB is formulated on the mutual informa- tion between independent and identically distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=') data, which is difficult to model the complex node interac- tions imposed by the graph structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' PRI shares the spirit of the IB method but its formulation addresses the entropy and relative entropy of a single dataset (Principe 2010), which can be applied to graph data with well-defined information- theoretic tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Preliminary Notions Given a graph G = {V, E} where V is the set of N nodes and E is the edge set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' A ∈ RN×N is the adjacency matrix and D is the degree matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The Laplacian matrix of the graph G can be defined as L = D − A = UΛUT, where U is the eigenvector matrix, Λ = Diag(λ1, · · · , λN) and λ1 < λ2 ≤ · · · ≤ λN are the eigenvalues of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Graph Structure Learning Given a graph G, graph struc- ture learning (Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2022) aims to learn a new structure ˜G simultaneously with the graph representations with the objective function: L = Ltask( ˜G, Y ) + αLreg( ˜G, G), (1) where Ltask is the task-specific objective with respect to the learned graph ˜G and the ground truth Y , Lreg imposes con- straints on the learned graph and α is a hyper-parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Principle of Relevant Information PRI (Principe 2010) is a self-organized information-theoretic principle that aims to perform mode decomposition of a random variable to obtain a reduced statistical representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' PRI formulates the redundancy reduction and information preservation as a trade-off between the entropy of reduced representation and its relative entropy given the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Definition 1 (Principle of Relevant Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Given a random variable X, the Principle of Relevant Information aims to obtain a reduced representation T with: LPRI = arg min T H(T) + βD(P(T)||P(X)), (2) where H(T) is the entropy of T and D(P(T)||P(X)) is the divergence of distributions P(T) and P(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The first term H(T) measures the redundancy of repre- sentation T and the second term D(P(T)||P(X)) measures the allowable distortion of the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The hyper- parameter β controls the level of distortion in T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' PRI was commonly used in scalar random variables (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Hoyos-Osorio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2021) and defined by R´enyi’s formula- tion of entropy and divergence (R´enyi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 1961).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Graph Structure Learning with Principle of Relevant Information In this work, we propose a graph structure learning frame- work named PRI-GSL, which merits the Principle of Rel- evant Information as a guideline for controlling the struc- ture quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The overall architecture of PRI-GSL is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' In this section, we first formulate the PRI loss for structure learning, then introduce the Role-aware graph learner and learning process of PRI-GSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' PRI for Graph Structure Learning In the PRI-GSL framework, PRI performs as a self- supervised regularizer for the quality of the learned graph structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Motivates by the objective of PRI, the Graph Structure Learning Principle of Relevant Information is: Definition 2 (PRI for Graph Structure Learning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Given a graph G, the Principle of Relevant Information for graph structure learning aims to learn a refined graph ˜G with: LPRI = H( ˜G) + βD( ˜G||G), (3) The first term H( ˜G) is the redundancy term, which mea- sures the disorder of the learned graph ˜G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The larger the H( ˜G), the more disordered ˜G is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The second term D( ˜G||G) t Refined Graph Original Graph + Role-aware Structure Learner QCW Evolution Wavelets … Probability t … Role Characteriztion Figure 2: Overall Architecture of PRI-GSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' is the distortion term, which measures the discrepancy be- tween two graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The smaller the D( ˜G||G), the more sim- ilar are the distributions of ˜G and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' β denotes the trade-off between the redundancy reduction of ˜G and its discrimina- tive description power of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' As β becomes larger, the em- phasis is laid more on the distortion term, and more infor- mation from G is preserved in ˜G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Formulate PRI by von Neumann Entropy and Quantum Jensen-Shannon Divergence The choice of entropy and divergence in PRI is application- specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' In this paper, we formulate the PRI loss LPRI by the von Neumann entropy (VNE) (Nielsen and Chuang 2002) and quantum Jensen-Shannon (QJS) divergence (Lamberti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2008) for graph data with complex interactions as in (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' For the first redundancy term H( ˜G), we propose to mea- sure the structure redundancy by von Neumann entropy (VNE) (Nielsen and Chuang 2002), which has been used in a variety of graph learning studies (Dasoulas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' von Neumann entropy quantifies the spec- tral complexity (or disorder) of graph structure by taking the graph as a quantum system through a mapping between dis- crete Laplacians and quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' A density matrix ρ is a Hermitian and positive semi-definite matrix that is used to encode the probability distributions and describe the state of a quantum mechanical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' For the graph ˜G, the von Neumann entropy HvN( ˜G) is defined as: HvN( ˜G) = HvN(˜ρ) = −tr (˜ρ log ˜ρ) = − N � i=1 (λi log λi) , (4) where ˜ρ is the graph density matrix of ˜G, tr(·) denotes trace and {λi} are the eigenvalues of ˜ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Typically, both the Lapla- cian matrix and the normalized Laplacian matrix can be used for the mapping from graphs to states (Minello, Rossi, and Torsello 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' We define the density matrix ˜ρ = ˜L tr(˜L) = ˜L 2|E| based on the Laplacian matrix ˜L of ˜G, which models the continuous information diffusion process (De Domenico and Biamonte 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' For the second distortion term D( ˜G||G), we use the quantum Jensen-Shannon (QJS) divergence (Lamberti et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2008) between the graph density matrices (˜ρ and ρ) of G and ˜G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The quantum Jensen-Shannon divergence has been widely used as a generalization of the classical Jensen-Shannon divergence to quantum states of graph data (De Domenico et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Bai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2015), which is symmetric, negative definite and bounded (0 ≤ DQJS ≤ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' DQJS( ˜G||G) = HvN � ˜ρ + ρ 2 � − 1 2HvN (˜ρ) − 1 2HvN (ρ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' (5) Combining the redundancy term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' (4) and the distor- tion term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' (5), we can obtain the following objective: LPRI = HvN( ˜G) + βDQJS( ˜G||G) = βHvN � ˜ρ + ρ 2 � + 2 − β 2 HvN (˜ρ) − β 2 HvN (ρ) ≡ βHvN � ˜ρ + ρ 2 � + 2 − β 2 HvN (˜ρ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' (6) We neglect HvN (ρ) in the last line because it’s a constant value during optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The above formalism provides a unified way of quantification and comparison for the learned structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Then we use a graph structure learner to obtain ˜G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Role-aware Graph Structure Learner In this section, we introduce the Role-aware Graph Structure Learner in PRI-GSL, which aims to learn a better graph ˜G that preserves the graph self-organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Considering the evolution process in self-organization, we characterize the nodes’ roles in a continuous and time-varying way and then incorporate both the merits of features as well as structural roles to refine the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Structural Role Encoding The structural role of the node represents its contribution to the overall information flow of the graph, which can provide key insights into the identifica- tion of graph organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' We propose to model graph state by quantum continuous walk and use the time-evolution op- erator with graph wavelets to generate role encodings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' (1) Model graph state by QCW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Recall that we apply the von Neumann Entropy and the QJS divergence for PRI for- mulation, which takes the whole graph as a quantum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' To investigate the nodes’ roles in this quantum system, we use the quantum continuous walk (QCW) (Childs 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Bai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2015) to build maps of how information flows through the graph in the perspective of graph state evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' QCW is the quantum mechanical counterpart of the continuous-time random walk in a graph, which describes the propagation of a quantum particle evolving continuously in time on the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The QCW on a graph G is defined as a dynamical process over the nodes that takes place on a N-dimensional Hilbert space H = span({|v⟩ , v ∈ V }).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The evolution of the walker is governed by the Schr¨odinger equation d dt |ψt⟩ = −iH |ψt⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' (7) |ψt⟩ represents the state of the walk at time t, which is a time-dependent amplitude vector on nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' H is the Hamil- tonian operator, which accounts for the total energy of the graph and governs the time evolution of the quantum con- tinuous walk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Given an initial state |ψ0⟩ ∈ H, the state of walker |ψt⟩ evolves in time according to |ψt⟩ = U(t) |ψ0⟩ , U(t) := e−iHt, (8) where U(t) is the unitary time-evolution operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' (2) Diffusion by graph wavelets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' To give a full char- acterization of the structural properties, we use the spectral graph wavelets Ψ (Hammond, Vandergheynst, and Gribon- val 2011) as the Hamiltonian H in QCW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Then the structural information residing in the graph is encoded in U(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' U(t) is a polynomial in Ψ for all t, thus any matrix that com- mutes with Ψ also commutes with U(t) (Coutinho and God- sil 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' We adopt the heat kernel gs(Λ) = e−Λs to obtain spectral graph wavelets, where the scaling parameter s con- trols the spread radii of the diffusion process and larger s allows farther diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The spectral wavelet basis is Ψs = UΛsUT = (Ψs (1) |Ψs (2) | · · · |Ψs (N)) , (9) where Λs = gs(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' In this way, the spectral graph wavelet Ψs(a) centered at node va associated with filter gs will be given by an N-dimensional vector: Ψs(a) = UΛsUTδa, (10) where δa is the one-hot vector of node va.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Ψs(a) is a N- dimensional vector where the b-th wavelet coefficient of Ψs(a) represents the information that va received from vb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Nodes playing similar roles have similar wavelet coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' (3) Characterize by the time-evolution operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Since the time-evolution operator U(t) reflects the graph state evo- lution, we treat the wavelets Ψ as probability distributions over graph and use φ(Ψ, t) = E[U(t)] = E[e−iΨt] as the characteristic function to uncover nodes’ roles in informa- tion diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The empirical characteristic function of va is: φs(va, t) = 1 N N � n=1 e−iΨs(a)t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' (11) φs(va, t) can capture all the moments (including higher- order moments) of the given distribution Ψs(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' We sample at T different time points on the time-evolution operator and then concatenate the values: hs(va) = [Re (φs (va, t)) , Im (φs (va, t))]t=t1,t2,··· ,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' (12) In general, the nodes play different roles across different scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Hence we utilize a multi-scale wavelet diffusion strategy to capture both the local and global structural roles of nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' We integrate information across different radii of neighborhoods by jointly considering a set of different val- ues of s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' We can obtain multi-scale structure role encodings ha ∈ R2T M by concatenating the structural encodings at several different scales S : {s1, s2, · · · , sM}: ha = (hs1 (va) |hs2 (va) | · · · |hsM (va)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' (13) Iterative Structure Learning After obtaining the struc- tural role encodings, we use a metric function that accounts for both feature information and the role-based similarities to measure the possibility of edge existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' PRI-GSL is ag- nostic to various metric functions and we choose the multi- head cosine similarity function here: aR(e−1) ij = 1 m m � h=1 cos (Wh · � z(e−1) i |h(e−1) i � , Wh · � z(e−1) j |h(e−1) j �� , (14) where m is the number of heads, Wh is the weight matrix of the h-th head, z(e−1) i and hR(e−1) i denote the representa- tion vector and the structural role encoding vector of node vi in the (e − 1)-th epoch, and | denotes the concatenation operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' With the above structure learning strategy, we can obtain a role-aware adjacency matrix in the e-th epoch: A(e) R = � aR(e−1) ij � , i, j ∈ {1, 2, · · · , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' (15) Learning Process of PRI-GSL Dynamic Structure Fusion The input graph structure de- termines the learning performance to a certain extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' To avoid the non-convergence or unstable training brought by the poor quality of learned structure at the beginning of train- ing, we hence incorporate the original graph structure A as supplementary to formulate an optimized graph structure ˜A: ˜A(e) = γD− 1 2 AD− 1 2 +(1 − γ)·RowNorm � A(e) R � , (16) where RowNorm(·) denotes the row-wise normalization function, γ is a constant that control the contribution of orig- inal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Here we use a dynamic decay mechanism for γ to enable the role-aware structure A(e) R to play a more and more important role during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Then the refined struc- ture is inputted into a GNN encoder for node representation vectors Z ∈ RN×d and classification: Z(e) = GNN-Encoder( ˜A(e), X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' (17) Objective of PRI-GSL The overall loss L of PRI-GSL is composed of two terms, the classification loss Lcls and the structure PRI loss LPRI in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' (6), given by: L = Lcls + αLPRI = HCE � ˜G, Y � + α � HvN � ˜G � + βDQJS � ˜G||G �� , (18) where HCE(·) is the cross-entropy loss for classification and α is a hyper-parameter to balance the two loss terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The overall process of PRI-GSL is shown in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Algorithm 1: The overall process of PRI-GSL for node classification Input: Graph G with node labels Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Number of training epochs Epochs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Wavelet scale set S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Timepoints T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Hyper-parameters α, β and γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Output: Refined graph ˜G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Predicted labels ˆY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 1 Parameter initialization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2 for e = 1, 2, · · · , Epochs do // Structural Role Encoding 3 for s ∈ S do 4 Ψ(e−1) s ← Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' (9), φs(va, t) ←Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' (11);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 5 h(e−1) s (va) ← Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' (12);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 6 end 7 h(e−1) a ← Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' (13);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' // Graph Structure Learning 8 aR(e) ij ← Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' (14), A(e) R = � aR(e) ij � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 9 ˜A(e) ←Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' (16), ˜G(e) ← (X, ˜A(e));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' // Learn Node Representations 10 Z(e) = GNN-Encoder( ˜A(e), X);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' // Optimize 11 L(e) PRI = HvN( ˜G(e)) + βDQJS( ˜G(e)||G);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 12 L(e) = L(e) cls( ˜G(e), Y ) + αL(e) PRI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 13 Update model parameters to minimize L(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 14 end Approximation Recall that the computation of von Neu- mann entropy and spectral wavelets requires the full eigen- value decomposition of the Laplacian matrix, which takes O(N 3) time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The von Neumann entropy can be approxi- mated with linear complexity O(|V | + |E|) (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' In the experiments, we still use the basic von Neu- mann entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' As for the graph wavelets, we use the Cheby- shev polynomial approximation (Shuman, Vandergheynst, and Frossard 2011) to compute Ψ, reducing the computa- tional complexity to O(K|E|), where K is the order of Chebyshev polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Properties of ˜G Learned by PRI-GSL PRI-GSL provides a unified way to control the quality of learned graph structure in terms of sparsity, centrality, and nuisance invariance property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Sparsity and Centrality The von Neumann entropy has close connections with the structure sparsity and centrality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' As indicated in (Passerini and Severini 2008), given a graph G, let G′ = G + {x, y} with V (G′) = V (G) and E(G′) = E(G) ∪ {x, y}, then HvN(ρG′) ≥ dG′−2 dG′ HvN(ρG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The von Neumann entropy tends to grow with the increasing num- ber of edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The graph centrality (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=', the extent to which a graph is organized around some central nodes) can be mea- sured as a quantum relative entropy between the relative degree distribution and the uniform distribution (Simmons, Coon, and Datta 2018), which is given by D(ρG||IN) := tr(ρG(log ρG − log I/N)) = log N − HvN(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The von Neumann entropy tends to grow with the increasing regular- ity of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The above conclusions suggest that mini- mizing HvN( ˜G) leads to a sparse and centralized structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Nuisance Invariance ˜G only preserves the most relevant yet least redundant information in the observed graph G and is invariant to nuisances in data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Suppose Gn ∈ G the task- irrelevant nuisance in G, the relevance of ˜G and Gn can be formulated as the divergence between their conditional distributions predictions of the desired labels Y (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2020b): E � DKL � p � Y | ˜G �� ||p (Y |Gn) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Since Gn is ir- relevant with Y , we have p(Y |Gn) = p(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Minimizing the cross-entropy loss (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=', the mutual information I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' ˜G) be- tween ˜G and Y ) is equivalent to minimizing the relevance between ˜G and Gn: E � DKL � p � Y | ˜G �� ||p (Y |Gn) � = E � DKL � p � Y | ˜G �� ||p (Y ) � = �� � �p � Y | ˜G � log p � Y | ˜G � p (Y ) � � p � ˜G � = �� p � Y | ˜G � log p � Y, ˜G � p (Y ) p � ˜G � = I(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' ˜G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' (19) Experiments We evaluate PRI-GSL on node classification and graph de- noising tasks to verify its capability of improving the ef- fectiveness and robustness of graph representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Then we provide the analyses of the PRI loss, the structural role encodings, and the learned structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Experimental Settings Datasets We select datasets with different homophily ra- tios h (Pei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2020) to analyze methods’ generalization on graphs with different properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The evaluation datasets are Squirrel, Chameleon (Rozemberczki, Allen, and Sarkar 2021), Actor (Pei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2020), CiteSeer, PubMed, Cora (Sen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2008) and Photo (Shchur et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Baselines We consider three types of baselines: (1) Graph neural networks: GCN (Kipf and Welling 2016), GAT (Veliˇckovi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2017) and GraphSAGE (Hamilton, Ying, and Leskovec 2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' (2) Graph sparsification meth- ods: DropEdge (Rong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2019), NeuralSparse (Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2020), and Graph-PRI (Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' (3) Graph structure learning methods: IDGL (Chen, Wu, and Zaki 2020), Pro-GNN (Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2020), SDRF (Topping et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2022), and SLAPS (Fatemi, El Asri, and Kazemi 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Parameter Settings We re-implement the Neu- ralSparse (Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2020) and SDRF (Topping et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The parameters of baseline methods are set to the suggested value in their papers or carefully tuned for Table 1: Accuracy ± standard deviation (%) of node classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' (Bold: best result;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Underlined: runner up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' ) Method Squirrel h=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='22 Actor h=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='24 Chameleon h=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='25 CiteSeer h=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='72 PubMed h=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='79 Cora h=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='83 Photo h=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='83 GCN 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='77 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='27±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='54 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='48±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='47 PRI-GSL 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='87±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='08 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='55±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='04 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='83±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='44 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='34±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='64 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='77±3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='20 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='67±2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='09 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='40±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='75 Deleting edges 65 70 75 80 85 90 Accuracy (%) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='67 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='95 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='63 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='67 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='75 Adding edges 55 60 65 70 75 80 85 90 Accuracy (%) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='67 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='00 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='62 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='33 GCN Graph-PRI IDGL PRI-GSL Figure 3: PRI-GSL on noisy graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' fairness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' For the GNN encoders, we use a 2-layer GCN for node classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' We set the representation dimension d=32, the Chebyshev polynomial order K=10, the number of time points T=4, the number of scales M=2, and the number of heads m=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The other hyper-parameters (α, β, and γ) are tuned for each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Evaluation Results and Analysis Node Classification We set the number of nodes in each class to be 20/30 for training/validation and take the remain- ing nodes for the test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The accuracy and standard deviation on 10 randomly split are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The best results are shown in bold and the runner-ups are underlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Our PRI-GSL achieves the best performance on both homophilic and heterophilic datasets, showing the effectiveness of uti- lizing the self-organization property when mining the la- tent inherent structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Generally, the graph structure learn- ing methods show better performance than GNNs and graph sparsification methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Pro-GNN and SLAPS perform well on the homophilic graphs but show unsatisfactory perfor- mance on the heterophilic graphs (Actor, Chameleon, and Squirrel), demonstrating the limitation of using heuristic as- sumptions for structural constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Although Graph-PRI also uses PRI for graph sparsification, it achieves fewer im- provements since it does not take the node feature and the downstream task into consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Cora CiteSeer Chameleon 40 50 60 70 80 90 Accuracy (%) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='7 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='9 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='9 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='2 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='2 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='0 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='2 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='4 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='4 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='0 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='5 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='4 PRI-GSL PRI-GSL (w/o PRI) PRI-GSL (w/o RE) GSL Figure 4: Ablation study results of PRI-GSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Graph Denoising To evaluate PRI-GSL’s ability to move noisy information, we generate synthetics noisy datasets by adding/deleting edges on Cora following (Chen, Wu, and Zaki 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Specifically, we randomly ad- d/delete 25%, 50%, 75% edges for 5 times and show the mean accuracy (solid line) and standard deviation (shaded region) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The performance of GCN and Graph- PRI decreases dramatically with the increasing noise level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The structure learning methods, IDGL and PRI-GSL, show more robustness compared to vanilla GCN and Graph-PRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Adding edge hurts more than deleting edges, indicating the importance of removing redundant information in the struc- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' PRI-GSL consistently shows better performance under different levels of external noise than the other baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Even though Graph-PRI shares the same spirit with PRI- GSL, it fails to distinguish whether the noise is task-relevant and shows the same poor robustness as GCN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The perfor- mance of PRI-GSL on noisy graphs also demonstrates the nuisance invariance property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Ablation Study To illustrate the advantages of the guid- ance of PRI and structural role information, we compare PRI-GSL with three variants: (1) PRI-GSL (w/o PRI) that removes the PRI loss, (2) PRI-GSL (w/o RE) that removes the role encodings, and (3) GSL that removes both the PRI loss and the role encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The results of variants on 5 ran- dom split datasets are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' As we can observe, both the PRI loss and the structural role encoding benefit (a) Original Graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' (b) Graph-PRI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' (c) Pro-GNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' (d) IDGL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' (e) PRI-GSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Figure 5: Visualization of the original graph of Cora and learned graphs by Graph-PRI, Pro-GNN, IDGL, and PRI-GSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 0 25 50 75 100 125 150 Epochs 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='000029 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='000030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='000031 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='000032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='000033 HvN( ̃G) +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='9187 converge at 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='9187301 0 25 50 75 100 125 150 Epochs 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='9185 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='9190 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='9195 DQJS( ̃G||G) Figure 6: The variations of HvN( ˜G) and DQJS( ˜G||G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' the classification, where the structural role encoding brings more improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' This suggests that it’s important to cap- ture the node’s contribution to the graph information flow when identifying the graph organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Empirical Behavior of HvN( ˜G) and DQJS( ˜G||G) We analyze the learning dynamics of PRI-GSL by measuring the variations of HvN( ˜G) and DQJS( ˜G||G) on Cora with α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='1 and β=1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The shadowed area is enclosed by the min and max value of four training runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The solid line in the middle is the mean value of each epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' HvN( ˜G) first increases for about 50 epochs with the structure exploration and then decreases to converge after about the 80-th epoch, indicating that the learned structure is with high certainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' DQJS( ˜G||G) bumps during the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' This may be because the model continues to seek a balance of struc- ture redundancy and distortion during the training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Hyper-parameter Analysis We analyze the impact of hyper-parameters including α controlling the importance of the PRI loss in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' (18) and β trading off redundancy and distortion in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' PRI- GSL achieves the best performance with α=1 on Cora and α=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='4 on CiteSeer, which indicates that PRI-GSL benefits from the PRI loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' As for the β in the PRI loss, when the dis- tortion term has a weight more than 2 compared to the redun- dancy term, PRI-GSL could reach satisfactory performance on both datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' This suggests that the distortion term dom- inates the PRI loss in PRI-GSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' That is to say, the learned 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='0 α 65 70 75 80 85 90 Accuracy (%) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='28 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='22 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='33 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='1 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='28 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='67 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='03 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='42 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='03 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='03 Cora CiteSeer 1 2 3 4 5 β 65 70 75 80 85 90 Accuracy (%) 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='33 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='05 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='24 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='34 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='24 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='11 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='11 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='56 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='11 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='67 Cora CiteSeer Figure 7: Parameter sensitivity of α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' structure should preserve enough information from the orig- inal graph to perform well on the downstream task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Visualization In Figure 5, we visualize the original graph structure of Cora and the graphs learned by PRI-Graph, Pro-GNN, IDGL, and PRI-GSL using networkx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The nodes’ colors indicate their classes, the labeled nodes are solid and the unlabeled nodes are hollow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The edges are not shown for clarity and the lay- out of nodes represents their connectivities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Graph-PRI has little effect on the overall property of graph structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Even though Pro-GNN and IDGL can make nodes within different classes more separate, there are still some overlapping and entangled areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Benefiting from the structural role encod- ing, PRI-GSL can obtain the structure with separate clusters with similar shapes and clearer class boundaries, showing how the nodes within a class are organized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Conclusion In this work, we propose an information-theoretic frame- work for graph structure learning named PRI-GSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' We for- mulate the Principle of Relevant Information for graph data to quantify the structure redundancy and distortion, which acts as a potentially unifying guidance of structure learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' We propose a role-aware structure learner based on the quantum continuous walk evolution to unravel the self- organization of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The learned structure enjoys the property of sparse, centralized, and nuisance invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Ex- tensive experiments demonstrate the superior effectiveness and robustness of PRI-GSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' O O O 00000O C CO CAcknowledgments The corresponding author is Jianxin Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' The authors are supported by the NSFC through grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content='U20B2053, and in part by NSF under grants III-1763325, III-1909323, III- 2106758, and SaTC-1930941.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' References Bai, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Rossi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=';' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' Ad- versarial attacks on neural networks for graph data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} +page_content=' In ACM SIGKDD, 2847–2856.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ytAyT4oBgHgl3EQfO_Y5/content/2301.00015v1.pdf'} diff --git a/zNE2T4oBgHgl3EQfMgZz/vector_store/index.faiss b/zNE2T4oBgHgl3EQfMgZz/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..f8795127948e37f46be9d50925308f1d4e82857d --- /dev/null +++ b/zNE2T4oBgHgl3EQfMgZz/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:302c2b52ab6df566cc021f78d71a4202366f6b3aaf2dc0c1b115563c27e86904 +size 3670061